MEAN CURVATURE MOTION OF NON-PARAMETRIC
HYPERSURFACES WITH CONTACT ANGLE CONDITION
BO GUAN
1. Introduction
Let Ω be a bounded domain in Rn with smooth boundary ∂Ω and denote by γ the
inner unit normal to ∂Ω. Let φ ∈ C ∞ (∂Ω) with |φ| ≤ φ0 < 1, and u0 ∈ C ∞ (Ω) satisfy
ν0 · γ = φ on ∂Ω, where ν0 denotes the unit normal to graph(u0 ). The boundary value
problem
(1.1)
(1.2)
∂u
− aij (Du)Di Dj u = 0 in Ω × [0, ∞)
∂t
ν · γ = φ, on ∂Ω × [0, ∞);
u(·, 0) = u0 ,
where ν denotes the downward unit normal to graph(u(·, t)) and
pi pj
aij (p) = δij −
, for p ∈ Rn ,
1 + |p|2
describes the evolution of graph(u(·, t)) by its mean curvature in the direction of the
unit normal with prescribed contact angle (given by cos−1 φ) at boundary.
This problem has been studied by G. Huisken [3] for φ ≡ 0, that is, the surfaces
have vertical contact angle at the boundary, and by Altschuler and Wu [1] for the case
that n = 2 and Ω is strictly convex. The main result of [3] states that for φ = 0 the
solution of (1.1)-(1.2) remains smooth and bounded, and asymptotically converges to
a constant function. For n = 2, Altschuler and Wu prove that if Ω is strictly convex
and |Dφ| < min κ(∂Ω), the curvature of ∂Ω, then the solution either converges to a
R
minimal surface (when ∂Ω φ = 0), or behaves like moving by a vertical translation as t
approaches infinity.
In this note we will consider this problem for general Ω and φ. We will derive the
following a priori estimates for the solution
∂u ≤ C, MT ≡ max |u − u0 | ≤ CT,
(1.3)
∂t Ω×[0,T ]
and
(1.4)
|Du| ≤ C1 eC2 MT in Ω × [0, T ],
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2
BO GUAN
where C, C1 , C2 are uniform constants independent of T . Consequently, Problem (1.1)(1.2) has a smooth solution u ∈ C ∞ (Ω × [0, ∞)).
It also follows from these estimates that the solution u is bounded uniformly in t (thus
so is |Du|) if and only if there is a minimal graph over Ω satisfying the same contact
R
angle condition at the boundary. In this case, necessarily ∂Ω φ = 0, and the solution
converges to the minimal graph up to a translation. This includes as a special case the
result of Huisken [3] mentioned above.
We will derive (1.3) and (1.4) in Section 2 below for the following slightly more general
equation
∂u
− aij (Du)Di Dj u = −ψ(u, Du), in Ω × [0, ∞),
∂t
where ψ is a smooth function. In Section 3 we consider the asymptotic behavior of the
p
solutions as t → ∞ for two special cases: (i) ψ(u, p) = κu 1 + |p|2 with κ > 0 and,
(ii) ψ(u, p) = −n/u with u > 0. We show in both cases the solution asymptotically
approaches the solution to the corresponding stationary equation. As a consequence
of the second case, we obtain some existence result for minimal surfaces in hyperbolic
space Hn+1 with a prescribed contact angle condition (see Theorem 3.2 below).
For convenience, we will use subscript indices to denote derivatives for functions and
adopt the summation convention.
I would like to thank Joel Spruck for bringing [1] to my attention and for useful
discussions.
(1.5)
2. A priori estimates
Throughout this section let u be a smooth solution to (1.5)-(1.2). We will derive (1.3)
and (1.4) under suitable assumptions on function ψ. First we recall some formulae. Set
aij = aij (Du),
1
w = (1 + |Du|2 ) 2 .
Then
uj
1
, aij ui uj = 1 − 2 .
2
w
w
The mean curvature H and the curvature norm |A| of graph(u(·, t)) are given by
(2.1)
aij ui =
1 ij
1
a uij , |A|2 = 2 aij akl uil ukj ,
w
w
The unit normal to graph(u(·, t)) is
Du −1
uk
−1
ν=
,
.
, i.e., ν k = , 1 ≤ k ≤ n, ν n+1 =
w w
w
w
(2.2)
H=
MEAN CURVATURE MOTION
3
By a direct calculation, we see that
1
1
(2.3)
νjk = akl ulj , ν j νjk = akl wl , k = 1, . . . , n
w
w
and
1
(2.4)
νkj νjk = 2 aij akl uil ukj = |A|2 .
w
Now consider the following linearized operator of (1.5)
L = aij Di Dj − ψpj Dj −
∂
.
∂t
We have
(2.5)
(2.6)
2 ij
a wi utj − ψu ut = 0,
w
2 ij
1
2
Lw − a wi wj = w|A| + ψu w −
,
w
w
Lut −
and
(2.7)
k
Lν = ν
k
ψu
|A| − 2
w
2
,
k = 1, . . . , n.
We now start establishing (1.3) and (1.4). The estimates in (1.3) follows readily from
the following
Lemma 2.1. Assume ψu ≥ 0. Then
max |ut | = max |ut (·, 0)|.
Ω×[0,∞)
Ω
Proof. It suffices to prove the following: For any fixed T > 0, if
ut (x0 , t0 ) = max ut ≥ 0 for some (x0 , t0 ) ∈ Ω × [0, T ],
Ω×[0,T ]
then t0 = 0. Now suppose t0 > 0. From (2.5) we see that ut satisfies the maximum
principle. Thus x0 ∈ ∂Ω. Choose the coordinates in Rn such that the positive xn -axis is
the interior normal direction to ∂Ω at x0 . Then at the point (x0 , t0 ),
ukt = utk = 0, for k = 1, . . . , n − 1
and hence
un unt
= φ(x0 )unt .
w
On the other hand, using the boundary condition (1.2) we find
wt =
unt = φ(x0 )wt .
Since |φ| < 1, it follows that utn (x0 , t0 ) = 0. But this contradicts the Hopf Lemma.
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BO GUAN
Turning to the gradient bound (1.4), we first recall that the distance function
d(x) = min |x − y|,
y∈∂Ω
x∈Ω
is smooth near ∂Ω and Dd = γ on ∂Ω. In the rest of this section, we assume d is
extended to be a smooth function on Ω satisfying
d ≥ 0, |Dd| ≤ 1, in Ω.
We will also assume φ is extended to a smooth function (still denoted by φ) on Ω with
|φ| ≤ φ0 < 1. Now following an idea of N. Korevaar [4], we set
η = eKu (N d + 1 − φν · Dd)
where K and N are positive numbers to be determined.
Lemma 2.2. For N > 0 sufficiently large independent of K and t, if for some t ≥ 0
fixed, wη(·, t) attains a local maximum value at a point x0 ∈ ∂Ω, then w(x0 , t) ≤ K.
Proof. Assume x0 = 0 ∈ ∂Ω and t ≥ 0 fixed. In the following all the derivatives are
evaluated at (0, t). We choose the coordinates in Rn so that the positive xn -axis is the
interior normal direction to ∂Ω at 0, and such that
u1 ≥ 0,
uα = 0, for 2 ≤ α ≤ n − 1.
Write
wη = eKu (N wd + w − φdk uk ) .
Note that at 0,
dα = 0, for 1 ≤ α ≤ n − 1,
dn = 1, and dkn = 0, for 1 ≤ k ≤ n.
We find
(2.8)
0 ≥ (wη)n = eKu N w + wn − φunn − un φn + Kw(1 − φ2 )un .
Now,
u1 u1n + un unn
u1 u1n
=
+ φunn .
w
w
From the boundary condition (1.2) we have dk uk = φw on ∂Ω. This yields
(2.9)
(2.10)
wn =
un1 = −u1 d11 + wφ1 + φw1 .
Since when restricted to ∂Ω, wη(·, t) achieves a local maximum value at 0, we have
0 = (wη)1 = wη1 + ηw1 .
On ∂Ω, η = eKu (1 − φ2 ). Thus
η1 = eKu (1 − φ2 )Ku1 − 2φφ1 ,
MEAN CURVATURE MOTION
5
and therefore
w1 = −Kwu1 +
(2.11)
2wφφ1
.
1 − φ2
Finally, combine (2.8)-(2.11) and note that u21 = w2 (1 − φ2 ) − 1 to obtain
0≥N−
u21 d11
u1 φ1 (1 + φ2 ) K
K
−
φφ
+
−
≥N −C − .
n
2
2
w
w(1 − φ )
w
w
It follows that w(0, t) ≤ K if N is chosen large enough.
From now on we assume N is fixed so that Lemma 2.2 holds.
Lemma 2.3. Assume ψ(u, p) satisfies
(2.12)
ψu (u, p) ≥ 0, |ψpj (u, p)| ≤ C,
ψ(u, p) − ψpj (u, p)pj ≥ −C.
Then one can choose K > 0 sufficiently large so that if
wη(x0 , t0 ) = max wη
Ω×[0,T ]
for some (x0 , t0 ) ∈ Ω × [0, T ], then w(x0 , t0 ) ≤ C.
Proof. By Lemma 2.2 we may assume x0 ∈ Ω and t0 > 0. At (x0 , t0 ), we have
(2.13)
0 = (wη)i = wi η + wηi , 1 ≤ i ≤ n;
0 ≤ (wη)t = wt η + wηt ,
and
2 ij
0 ≥ L(wη) = wLη + η Lw − a wi wj .
w
It follows from (2.6) that
1
1
2
(2.14)
Lη + |A| + ψu 1 − 2 ≤ 0.
η
w
Write η = eKu h and h = N d + 1 − φdk ν k . We have
1
1
2K ij
Lη = K 2 aij ui uj + KLu + Lh +
a ui hj .
η
h
h
(2.15)
We compute
ht = −φdk νtk ,
hi = N di − (φdk )i ν k − φdk νik ,
hij = N dij − (φdk )ij ν k − (φdk )i νjk − (φdk )j νik − φdk νijk .
Using (2.3) we find
aij (φdk )i νjk =
1 ij kl
1
a a (φdk )i ulj ≤ |A|2 + C,
w
4
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BO GUAN
and hence, by (2.7), (2.12),
Lh ≥ −φdk Lν k − 2aij (φdk )i νjk − C
1
ψu
k
2
≥ φdk ν |A| − 2 − |A|2 − C.
w
2
We also note that by (2.12),
Lu = ψ(u, Du) − ψpj (u, Du)uj ≥ −C
For the last term in (2.15), we use (2.1), (2.2), (2.3) and (2.13) to estimate at (x0 , t0 ),
1
uj hj
aij ui hj =
w2
φ
≥ −C − 2 dk uj νjk
w
φ
= −C − 2 dk akl wl
w
φ
hl
kl
Kul +
= −C + dk a
w
h
2
K
φ kl
≥ −C − 2 −
a dk dm νlm
w
hw
φ2 kl mj
K
a a dk dm ujl
= −C − 2 −
w
hw2
CK
1
|A|2 .
≥ −C − 2 −
w
4K
Plugging the above inequalities and (2.1) into (2.15), we conclude from (2.14)
C
1 − φ0 − 2 K 2 − C(K + 1) ≤ 0.
w
Choosing K sufficiently large then gives a bound w(x0 , t0 ) ≤ C.
The gradient estimate (1.4) now follows from Lemma 2.3, as
η(x0 , t0 )
w(x, t) ≤ w(x0 , t0 )
≤ C1 eC2 MT .
η(x, t)
Consequently, Equation (1.5) is uniformly parabolic on Ω × [0, T ) for any fixed T > 0.
By the standard theory, we obtain the long time existence of solutions.
Theorem 2.4. Assume ψ satisfies (2.12). There exists a solution u ∈ C ∞ (Ω × [0, ∞))
to Problem (1.5)-(1.2). Moreover, u satisfies estimates (1.3) and (1.4).
In particular, we have
Corollary 2.5. Problem (1.1)-(1.2) admits a solution u ∈ C ∞ (Ω × [0, ∞)) satisfying
(1.3) and (1.4).
MEAN CURVATURE MOTION
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3. Asymptotic behavior
There has been much interest in studying the asymptotic behavior of solutions to
curvature flow problems. For Problem (1.1)-(1.2), we see from the estimate (1.4) that if
|u| is bounded uniformly in t, then so is the gradient |Du|, and it can be shown as in [1]
that u asymptotically approaches a minimal surface satisfying the same contact angle
condition as t → ∞. Especially, this applies to the vertical contact angle case and thus
R
recovers the result of Huisken [3]. Note that it is a necessary condition that ∂Ω φ = 0
R
for |u| to be bounded uniformly in t. When ∂Ω φ 6= 0, very little is known about the
asymptotic behavior of the solution to (1.1)-(1.2).
Below we will see in two other special cases time-independent estimates are available.
Thus the solution asymptotically approaches a stationary graph as time goes to infinity.
(i) The capillary problem. The capillary surface equation
(3.1)
1 ij
a (Du)uij = κu in Ω,
w
has received extensive study. Here we want to discuss briefly an evolutionary approach
to the existence results for κ > 0 first proved in [2] and [5]. Consider the evolution
equation
(3.2)
ut − aij (Du)uij = −κuw.
A comparison argument shows that any solution u to Problem (3.2)-(1.2) satisfies the
time-independent a priori estimate
|u(x, t) − u0 (x)| ≤ C.
Consequently, the assumptions of Theorem 2.4 are verified, and Problem (3.2)-(1.2)
admits a solution u ∈ C ∞ (Ω × [0, ∞)) with |Du| bounded uniformly in t. We observe
that
Z Z 2
Z
ut
d
κ 2
dx =
−
u + w dx +
uφdσ .
dt
Ω w
Ω 2
∂Ω
Since u and w are bounded uniformly in t, it follows that
Z TZ 2
ut
dxdt ≤ C uniformly in T.
0
Ω w
From standard estimate results we see that ut uniformly converges to zero as t → ∞,
and u uniformly approaches a smooth solution to (3.1) with prescribed contact angle on
the boundary.
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BO GUAN
(ii) Hyperbolic minimal surfaces. Here we consider the evolution equation
(3.3)
ut − aij (Du)uij =
n
,
u
u > 0, in Ω
and the corresponding stationary equation
(3.4)
n
aij (Du)uij = − ,
u
u > 0 in Ω.
We note that (3.4) is the equation of hyperbolic minimal surfaces in the upper half space
model of Hn+1 , the n + 1 dimensional hyperbolic space.
Theorem 3.1. Assume u0 > 0 on Ω and
0 < φ0 ≤ φ ≤ φ1 < 1 on ∂Ω.
(3.5)
Problem (3.3)-(1.2) has a solution u ∈ C ∞ (Ω × [0, ∞)) satisfying the time-independent
estimates
(3.6)
0 < c0 ≤ u ≤ C, in Ω × [0, ∞).
Moreover, u converges uniformly to a smooth solution of (3.4) as t → ∞.
Proof. Note that, in the upper half space model of Hn+1 , all the upper hemispheres in
Rn+1 centered on the hyperplane xn+1 = 0 have hyperbolic mean curvature 0. It follows
from the comparison principle that if R > 0 is chosen sufficiently large so that |x| < R
for all x ∈ Ω and
−x · γ
≤ φ0 on ∂Ω,
R
1
then u(x, t) ≤ (R2 − |x|2 ) 2 ≤ R for x ∈ Ω uniformly in t. This proves the upper
bound in (3.6). For the lower bound, we note that Ω satisfies the uniform interior sphere
condition, i.e., for each point x0 ∈ ∂Ω there exists a ball Br (y0 ) ⊂ Ω of radius r such
that Br (y0 ) ∩ ∂Ω = {x0 } for some uniform r > 0 depending only on Ω. So we can choose
a positive constant δ < r such that
(y − x) · γ
≥ φ1 for x ∈ ∂Br (y) ∩ ∂Ω,
r
for any y ∈ Ω with d(y) = δ. The comparison principle then yields
1
u(x, t) ≥ (r2 − δ 2 ) 2 for x ∈ ∂Ω, uniformly in t.
By the maximum principle, we obtain the lower bound in (3.6).
Now we can apply Theorem 2.4 for the existence of solution u ∈ C ∞ (Ω × [0, ∞)) to
Problem (3.3)-(1.2). It remains to show that u converges uniformly to a smooth solution
MEAN CURVATURE MOTION
9
of (3.4) as t → ∞. To this end, we observe that
Z
Z
d
w
ui uit u − nw2 ut
dx
=
dx
dt Ω un
wun+1
ZΩ Z
ui ut
nut
dx
−
dx
=
n
un i
Ω w
Ω wu
Z
Z
u2t
ut
φdσ
−
dx.
=
n
n
Ω u w
∂Ω u
Since w is uniformly bounded and u is bounded below from 0, we find
Z TZ
u2t
dxdt ≤ C uniformly in T .
n
0
Ω u w
It follows that ut converges to zero uniformly as t → ∞ and hence u asymptotically
approaches a smooth solution of (3.4).
As a corollary of Theorem 3.1, we have
Theorem 3.2. Given φ satisfying (3.5), there exists a solution u ∈ C ∞ (Ω) to (3.4) with
Du
(3.7)
· γ = φ on ∂Ω,
w
which satisfies the a priori estimates
0 < c0 ≤ u ≤ C, |Du| ≤ C in Ω.
We remark that as a hypersurface of the hyperbolic space Hn+1 (in the upper half
space model), graph(u) is a (hyperbolic) minimal surface and, since Hn+1 is conformal
n+1
.
to the upper half space Rn+1
+ , (3.7) describes the “true” contact angle in H
References
[1] S. J. Altschuler and L. F. Wu, Translating surfaces of the non-parametric mean curvature flow with
prescribed contact angle, Calc. Var. 2 (1994), 101–111.
[2] C. Gerhardt, Global regularity of the solutions to the capillary problem, Ann. Scuola Norm. Sup.
Pisa Cl. Sci. 3 (1976), 157-176.
[3] G. Huisken, Non-parametric mean curvature evolution with boundary conditions, J. Differential
Equations, 77 (1989), 369-378.
[4] N. J. Korevaar, Maximum principle gradient estimates for the capillary problem, Comm. in Partial
Differential Equations, 13 (1988), 1-31.
[5] L. Simon and J. Spruck, Existence and regularity of a capillary surface with a prescribed contact
angle, Arch. Rat. Mech. Anal. 61 (1976), 19-34.
Department of Mathematics, University of Tennessee, Knoxville, TN 37996
E-mail address: [email protected]