Tài liệu Đề tài " Boundary regularity for the Monge-Amp`ere and affine maximal surface equations " docx

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BOUNDARY REGULARITY
995
The affine mean curvature equation (1.4) is the Euler equation of the
functional
J[u]=A(u) −

Ω
fu,(1.7)
where
A(u)=

Ω
[detD
2
u]
1/(n+2)
(1.8)
is the affine surface area functional. The natural or variational boundary value
problem for (1.4), (1.7) is to prescribe u and ∇u on ∂Ω and is treated in [21].
Regularity at the boundary is a major open problem in this case.
Note that the operator L in (1.4) possesses much stronger invariance prop-
erties than its Monge-Amp`ere counterpart (1.1) in that L is invariant under
unimodular affine transformations in R
n+1
(of the dependent and independent
variables).
Although the statement of Theorem 1.1 is reasonably succinct, its proof
is technically very complicated. For interior estimates one may assume by
affine transformation that a section of a convex solution is of good shape; that
is, it lies between two concentric balls whose radii ratio is controlled. This
is not possible for sections centered on the boundary and most of our proof
is directed towards showing that such sections are of good shape. After that
we may apply a similar perturbation argument to the interior case [4]. To
show sections at the boundary are of good shape we employ a different type
of perturbation which proceeds through approximation and extension of the
trace of the inhomogeneous term f. The technical realization of this approach
constitutes the core of our proof. Theorem 1.1 may also be seen as a companion
result to the global regularity result of Caffarelli [6] for the natural boundary
value problem for the Monge-Amp`ere equation, that is the prescription of the
image of the gradient of the solution, but again the perturbation arguments
are substantially different.
The organization of the paper is as follows. In the next section, we in-
troduce our perturbation of the inhomogeneous term f and prove some pre-
liminary second derivative estimates for the approximating problems. We also
show that the shape of a section of a solution at the boundary can be controlled
by its mixed tangential-normal second derivatives. In Section 3, we establish
a partial control on the shape of sections, which yields C
1,α
estimates at the
boundary for any α ∈ (0, 1) (Theorem 3.1). In order to proceed further, we
need a modulus of continuity estimate for second derivatives for smooth data
and here it is convenient to employ a lemma from [8], which we formulate in
Section 4. In Section 5, we conclude our proof that sections at the boundary
are of good shape, thereby reducing the proof of Theorem 1.1 to analogous
perturbation considerations to the interior case [4], which we supply in Sec-
tion 6 (Theorem 6.1). Finally in Section 7, we consider the application of our
996 NEIL S. TRUDINGER AND XU-JIA WANG
preceding arguments to the affine maximal surface and affine mean curvature
equations, (1.4). In these cases, the global second derivative estimates follow
from a variant of the condition f ∈ C
α
(Ω) at the boundary, namely
|f(x) − f(y)|≤C|x − y|,(1.9)
for all x ∈ Ω,y ∈ ∂Ω. This is satisfied by the function w in (1.5). The
uniqueness part of Theorem 1.2 is proved directly (by an argument based on
concavity), and the existence part follows from our estimates and a degree
argument. The solvability of (1.4)–(1.6) without boundary regularity was al-
ready proved in [21] where it was used to prove interior regularity for the first
boundary value problem for (1.4).
2. Preliminary estimates
Let Ω be a uniformly convex domain in R
n
with C
3
boundary, and ϕ be
a C
3
smooth function on Ω. For small positive constant t>0, we denote
Ω
t
= {x ∈ Ω | dist(x, ∂Ω) >t} and D
t
=Ω− Ω
t
. For any point x ∈ Ω, we
will use ξ to denote a unit tangential vector of ∂Ω
δ
and γ to denote the unit
outward normal of ∂Ω
δ
at x, where δ = dist(x, ∂Ω).
Let u be a solution of (1.1), (1.2). By constructing proper sub-barriers we
have the gradient estimate
sup
x∈Ω
|Du(x)|≤C.(2.1)
We also have the second order tangential derivative estimates
C
−1
≤ u
ξξ
(x) ≤ C(2.2)
for any x ∈ ∂Ω. The upper bound in (2.2) follows directly from (2.1) and the
boundary condition (1.2). For the lower bound, one requires that ϕ be C
3
smooth, and ∂ΩbeC
3
and uniformly convex [22]. For (2.1) and (2.2) we only
need f to be a bounded positive function.
In the following we will assume that f is positive and f ∈ C
α
(Ω) for some
α ∈ (0, 1). Let f
τ
be the mollification of f on ∂Ω, namely f
τ
= η
τ
∗ f , where
η is a mollifier on ∂Ω. If t>0 is small, then for any point x ∈ D
t
, there is a
unique point ˆx ∈ ∂Ω such that dist(x, ∂Ω) = |x − ˆx| and γ =(ˆx − x)/|ˆx − x|.
Let
f
t
(x)=

f(x)inΩ
2t
,
f
τ
(ˆx) − Cτ
α
in D
t
,
(2.3)
where
τ = t
ε
0
,ε
0
=1/4n.
BOUNDARY REGULARITY
997
We define f
t
properly in the remaining part Ω
t
−Ω
2t
such that, with a proper
choice of the constant C = C
t
> 0, f
t
≤ f in Ω and f
t
is H¨older continuous in
Ω with H¨older exponent α

= ε
0
α,
|f
t
− f|≤Cτ
α
= Ct
α

in Ω,
f
t

C
α

(Ω)
≤Cf
C
α
(Ω)
for some C>0 independent of t. From (2.3), f
t
is smooth in D
t
,
|Df
t
|≤Cτ
α−1
, |D
2
f
t
|≤Cτ
α−2
, and |∂
γ
f
t
| =0 in D
t
.(2.4)
Let u
t
be the solution of the Dirichlet problem,
detD
2
u = f
t
in Ω,(2.5)
u = ϕ on ∂Ω.
First we establish some a priori estimates for u
t
in D
t
. Note that by the local
strict convexity [3] and the a priori estimates for the Monge-Amp`ere equation
[18], u
t
is smooth in D
t
.
For any given boundary point, we may suppose it is the origin such that
Ω ⊂{x
n
> 0}, and locally ∂Ω is given by
x
n
= ρ(x

)(2.6)
for some C
3
smooth, uniformly convex function ρ satisfying ρ(0)=0, Dρ(0)=0,
where x

=(x
1
, ··· ,x
n−1
). By subtracting a linear function we may also
suppose that
u
t
(0) = 0,Du
t
(0) = 0.(2.7)
We make the linear transformation T : x → y such that
y
i
= x
i
/
√
t, i =1, ··· ,n− 1,(2.8)
y
n
= x
n
/t,
v = u
t
/t.
Then v satisfies the equation
detD
2
v = tf
t
in T (Ω).(2.9)
Let G = T(Ω) ∩{y
n
< 1}.InG we have 0 ≤ v ≤ C since v is bounded on
∂G ∩{y
n
< 1}. Observe that the boundary of G in {y
n
< 1} is smooth and
uniformly convex. Hence
|v
γ
|≤C in ∂G ∩

y
n
<
7
8

.
From (2.2) we have
C
−1
≤ v
ξξ
≤ C on ∂G ∩

y
n
<
7
8

.
998 NEIL S. TRUDINGER AND XU-JIA WANG
The mixed derivative estimate
|v
γξ
|≤C on ∂G ∩

y
n
<
3
4

,
where v
ξγ
=

ξ
i
γ
j
v
y
i
y
j
, is found for example in [8] and [13]. For the mixed
derivative estimate we need f
t
∈ C
0,1
, with
|Df
t
|≤Cτ
α−1
t
1/2
≤ C.
From (2.2) and equation (2.9) we have also
v
γγ
≤ C on ∂G ∩

y
n
<
3
4

.
Next we derive an interior estimate for v.
Lemma 2.1. Let v be as above. Then
|D
2
v|≤C(1 + M) in G ∩

y
n
<
1
2

,(2.10)
where M = sup
{y
n
<7/8}
|Dv|
2
, C>0 is independent of M .
Proof. First we show v
ii
≤ C for i =1, ··· ,n− 1. Let
w(y)=ρ
4
η

1
2
v
2
1

v
11
,
where v
1
= v
y
1
, v
11
= v
y
1
y
1
, and ρ(y)=2− 3y
n
is a cut-off function, η(t)=
(1 −
t
M
)
−1/8
.Ifw attains its maximum at a boundary point, by the above
boundary estimates we have w ≤ C.Ifw attains its maximum at an interior
point y
0
, by the linear transformation
y
i
= y
i
,i=2,··· ,n,
y
1
= y
1
−
v
1i
(y
0
)
v
11
(y
0
)
y
i
,
which leaves w unchanged, one may suppose D
2
v(y
0
) is diagonal. Then at y
0
we have
0=(logw)
i
=4
ρ
i
ρ
+
η
i
η
+
v
11i
u
11
,(2.11)
0 ≥(log w)
ii
=4

ρ
ii
ρ
−
ρ
2
i
ρ
2

+

η
ii
η
−
η
2
i
η
2

+

v
11ii
v
11
−
v
2
11i
v
2
11

.(2.12)
Inserting (2.11) into (2.12) in the form
ρ
i
ρ
= −
1
4

η
i
η
+
v
11i
v
11

for i =2, ··· ,n
and
v
11i
v
11
= −(4
ρ
i
ρ
+
η
i
η
) for i = 1, we obtain
0 ≥v
ii
(log w)
ii
(2.13)
≥v
ii

η
ii
η
− 3
η
2
i
η
2

− 36v
11
ρ
2
1
ρ
2
+ v
ii
v
11ii
v
11
−
3
2
n

i=2
v
ii
v
2
11i
v
2
11
,
where (v
ij
) is the inverse matrix of (v
ij
).
BOUNDARY REGULARITY
999
It is easy to verify that
v
ii

η
ii
η
− 3
η
2
i
η
2

≥
C
M
v
11
−
C
M
,
where C>0 is independent of M. Differentiating the equation
log detD
2
v = log(tf
t
)
twice with respect to y
1
, and observing that |∂
1
f
t
|≤Cτ
α−1
t
1/2
≤ C and
|∂
2
1
f
t
|≤Cτ
α−2
t ≤ C after the transformation (2.8), we see the last two terms
in (2.13) satisfy
v
ii
v
11ii
v
11
−
3
2
n

i=2
v
ii
v
2
11i
v
2
11
≥−
1
v
11
(log f
t
)
11
≥−C.
We obtain
ρ
4
v
11
≤ C(1 + M).
Hence v
ii
≤ C for i =1, ··· ,n− 1inG ∩{y
n
<
1
2
}.
Next we show that v
nn
≤ C. Let w(y)=ρ
4
η

1
2
v
2
n

v
nn
with the same
ρ and η as above. If w attains its maximum at a boundary point, we have
v
nn
≤ C by the boundary estimates. Suppose w attains its maximum at an
interior point y
0
. As above we introduce a linear transformation
y
i
= y
i
,i=1, ··· ,n− 1,
y
n
= y
n
−
v
in
(y
0
)
v
nn
(y
0
)
y
i
,
which leaves w unchanged. Then
w(y)=(2− α
i
y
i
)
4
η

1
2
v
2
n

v
nn
and D
2
v(y
0
) is diagonal. By the estimates for v
ii
, i =1, ··· ,n−1, the constants
α
i
are uniformly bounded. Therefore the above argument applies.
Scaling back to the coordinates x, we therefore obtain
∂
2
ξ
u
t
(x) ≤ C in D
t/2
,(2.14a)
|∂
ξ
∂
γ
u
t
(x)|≤C/
√
t in D
t/2
,(2.14b)
∂
2
γ
u
t
(x) ≤ C/t in D
t/2
,(2.14c)
where C is independent of t, ξ is any unit tangential vector to ∂Ω
δ
and γ is
the unit normal to ∂Ω
δ
(δ = dist(x, ∂Ω)), and ∂
ξ
∂
γ
u =

ξ
i
γ
j
u
x
i
x
j
.
The proof of Lemma 2.1 is essentially due to Pogorelov [18]. Here we used
a different auxiliary function, from which we obtain a linear dependence of
sup |D
2
v| on M, which will be used in the next section. The linear dependence
1000 NEIL S. TRUDINGER AND XU-JIA WANG
can also be derived from Pogorelov’s estimate by proper coordinate changes.
Taking ρ = −u in the auxiliary function w, we have the following estimate.
Corollary 2.1. Let u be a convex solution of detD
2
u = f in Ω.
Suppose inf
Ω
u = −1, and either u =0or |D
2
u|≤C
0
(1 + M) on ∂Ω. Then
|D
2
u|(x) ≤ C(1 + M ), ∀ x ∈{u<−
1
2
},(2.15)
where M = sup
{u<0}
|Du|
2
, and C is independent of M.
Next we derive some estimates on the level sets of the solution u to (1.1),
(1.2). Denote
S
0
h,u
(y)={x ∈ Ω | u(x) <u(y)+Du(y)(x − y)+h},
S
h,u
(y)={x ∈ Ω | u(x)=u(y)+Du(y)(x − y)+h}.
We will write S
h,u
= S
h,u
(y) and S
0
h,u
= S
0
h,u
(y) if no confusion arises. The set
S
0
h,u
(y) is the section of u at center y and height h [4].
Lemma 2.2. There exist positive constants C
2
>C
1
independent of h
such that
C
1
h
n/2
≤|S
0
h,u
(y)|≤C
2
h
n/2
(2.16)
for any y ∈ ∂Ω, where |K| denotes the Lebesgue measure of a set K.
Proof. It is known that for any bounded convex set K⊂R
n
, there is a
unique ellipsoid E containing K which achieves the minimum volume among
all ellipsoids containing K [3]. E is called the minimum ellipsoid of K.It
satisfies
1
n
(E − x
0
) ⊂K−x
0
⊂ E −x
0
, where x
0
is the center of E.
Suppose the origin is a boundary point of Ω, Ω ⊂{x
n
> 0}, and locally ∂Ω
is given by (2.6). By subtracting a linear function we also suppose u satisfies
(2.7). Let E be the minimum ellipsoid of S
0
h,u
(0). Let v be the solution to
detD
2
u = inf
Ω
f
t
in S
0
h,u
, v = h on ∂S
0
h,u
.If|E| >Ch
n/2
for some large C>1,
we have inf v<0. By the comparison principle, we obtain inf u ≤ inf v<0,
which is a contradiction to (2.7). Hence the second inequality of (2.16) holds.
Next we prove the first inequality. Denote
a
h
= sup{|x

||x ∈ S
h,u
(0)},(2.17)
b
h
= sup{x
n
| x ∈ S
h,u
(0)}.(2.18)
If the first inequality is not true, |S
0
h,u
| = o(h
n/2
) for a sequence h → 0.
By (2.2), we have S
0
h,u
⊃{x ∈ ∂Ω ||x| <Ch
1/2
} for some C>0. Hence
b
h
= o(h
1/2
). By (2.2) we also have u(x) ≥ C
0
|x|
2
for x ∈ ∂Ω. Hence if
a
h
≤ Ch
1/2
for some C>0, the function
v = δ
0
(|x

|
2
+

h
1/2
b
h
x
n
)
2

+ εx
n
BOUNDARY REGULARITY
1001
for some small δ
0
> 0, is a sub-solution to the equation detD
2
u = f in S
0
h,u
satisfying v ≤ u on ∂S
0
h,u
, where ε>0 can be arbitrarily small. It follows by the
comparison principle that v
n
(0) ≤ u
n
(0) = 0, which contradicts v
n
(0) = ε>0.
Hence, a
h
/h
1/2
→∞as h → 0. Let x
0
=(x
0,1
, 0, ··· , 0,x
0,n
) (after a
rotation of the coordinates x

) be the center of E, where E is the minimum
ellipsoid of S
0
h,u
. Make the linear transformation
y
1
= x
1
− (x
0,1
/x
0,n
)x
n
,y
i
= x
i
i =2, ··· ,n
such that the center of E is moved to the x
n
-axis. Let E

={

n−1
i=1
(x
i
/a
i
)
2
< 1}
be the projection of E on {x
n
=0}. Since the origin 0 ∈ S
0
h,u
and the center
of E is located on the x
n
-axis, one easily verifies that a
1
···a
n
≤ C|S
0
h,u
| =
o(h
n/2
), where a
n
= x
0,n
. Note that x
0,1
≤ a
h
and x
0,n
≤ b
h
≤ 2nx
0,n
. By the
uniform convexity of ∂Ω,
x
0,n
x
0,1
≥ C
b
h
a
h
≥ Ca
h
 h
1/2
.
Hence after the above transformation, the boundary part ∂Ω ∩ S
0
h,u
is still
uniformly convex. Also, as above, the function v = δ
0

n
i=1
(
h
1/2
a
i
y
i
)
2
+ εy
n
is a
sub-solution, and we reach a contradiction.
Next we show that the shape of the level set S
h,u
can be controlled by the
mixed derivatives u
ξγ
on ∂Ω.
Lemma 2.3. Let u be the solution of (1.1), (1.2). Suppose as above that
∂Ω is given by (2.6) and u satisfies (2.7). If
|∂
ξγ
u(x)|≤K on ∂Ω(2.19)
for some K ≥ 1, then
a
h
≤ CKh
1/2
,(2.20)
b
h
≥ Ch
1/2
/K(2.21)
for some C>0 independent of u, K and h.
Proof. We need only to prove (2.20) and (2.21) for small h>0. Suppose
the supremum a
h
is attained at x
h
=(a
h
, 0, ··· , 0,c
h
) ∈ S
h,u
(0). Let  =
S
h,u
∩{x
2
= ··· = x
n−1
=0}. Then  ⊂ Ω and it has an endpoint ˆx =
(ˆx
1
, 0, ··· , 0, ˆx
n
) ∈ ∂Ω with ˆx
1
> 0 such that u(ˆx)=h.Ifa
h
=ˆx
1
, by (2.2) we
have ˆx
1
≤ Ch
1/2
, and by the upper bound in (2.16), b
h
≥ Ch
1/2
. Hence (2.20)
and (2.21) hold.
When a
h
> ˆx
1
, let ξ =(ξ
1
, 0, ··· , 0,ξ
n
) be the unit tangential vector of
∂Ωatˆx in the x
1
x
n
-plane, and ζ =(ζ
1
, 0, ··· , 0,ζ
n
) be the unit tangential
vector of the curve  at ˆx. Then all ξ
1
,ξ
n
,ζ
1
, and ζ
n
> 0. Let θ
1
denote the
1002 NEIL S. TRUDINGER AND XU-JIA WANG
angle between ξ and ζ at ˆx, and θ
2
the angle between ξ and the x
1
-axis. By
(2.2) and (2.19),
|∂
γ
u(ˆx)|≤CK|ˆx|, |∂
ξ
u(ˆx)|≥C|ˆx|.
Hence
C
K
≤ θ
1
<π−
C
K
.(2.22)
But since all ξ
1
,ξ
n
,ζ
1
, and ζ
n
> 0, we have θ
1
+ θ
2
<
π
2
. Note that by (2.2)
and (2.16), a
h
≥ Ch
1/2
and b
h
≤ Ch
1/2
. We obtain
a
h
≤ ˆx
1
+ b
h
/tg (θ
1
+ θ
2
) ≤ CKh
1/2
,b
h
≥ a
h
tg (θ
1
+ θ
2
) ≥ Ch
1/2
/K.(2.23)
Lemma 2.3 is proved.
Lemma 2.3 shows that the shape of the sections S
0
h,u
(y) at boundary points
y can be controlled by the mixed second order derivatives of u.IfS
0
h,u
has a
good shape for small h>0, namely if the inscribed radius r is comparable to
the circumscribed radius R,
R ≤ C
0
r(2.24)
for some constant C
0
under control, the perturbation argument [4] applies and
one infers that |D
2
u(0)| is bounded. See Section 6. It follows that u ∈ C
2,α
(Ω)
by [2], [19]. Estimation of the mixed second order derivatives on the boundary
will be the key issue in the rest of the paper.
3. Mixed derivative estimates at the boundary
For t>0 small let u
t
be a solution of (2.5) and assume (2.6) (2.7) hold. As
in Section 2 we use ξ and γ to denote tangential (parallel to ∂Ω) and normal
(vertical to ∂Ω) vectors.
Lemma 3.1. Suppose
|∂
ξ
∂
γ
u
t
|≤K on ∂Ω(3.1)
for some 1 ≤ K ≤ Ct
−1/2
. Then
∂
2
i
u
t
≤ C in D
t
∩{x
n
<t/8},i=1, ··· ,n− 1,(3.2a)
|∂
i
∂
n
u
t
|≤CK in D
t
∩{x
n
<t/8},(3.2b)
∂
2
n
u
t
≤ CK
2
in D
t
∩{x
n
<t/8},(3.2c)
where C>0 is a constant independent of K and t.
Proof. By (2.14c), estimate (3.2a) is equivalent to (2.14a). The estimate
(3.2b) follows from (3.2a) and (3.2c) by the convexity of u
t
. By (2.2), (3.1), and
BOUNDARY REGULARITY
1003
equation (2.5), we obtain (3.2c) on the boundary ∂Ω. By (2.15), the interior
part of (3.2c) will follow if we have an appropriate gradient estimate for u
t
in
the set S
0
h,u
t
(0).
Let h>0 be the largest constant such that S
0
h,u
t
(0) ⊂ D
t/2
and u
t
satisfies
(2.14) in {u
t
<h}. By the Lipschitz continuity of u, we have h ≤ Ct. Let
v(y)=u
t
(x)/h, where y = x/
√
h. Then v satisfies the equation
detD
2
v = f
t
in

Ω={x/
√
h | x ∈ Ω}.(3.3)
By (2.16),
C
1
≤|{v<1}| ≤ C
2
.(3.4)
We claim
|∂
n
v(y)|≤CK ∀ y ∈

v<
1
2

.(3.5)
If (3.5) holds, by Corollary 2.1 (with the auxiliary function w(y)=(
1
2
− v)
4
· η(
1
2
v
2
n
)v
nn
in the proof of Lemma 2.1), we obtain
∂
2
y
n
v ≤ CK
2
in {v<1/4}.
In the above estimate we have used
∂
2
y
n
log f
t
(y)=h∂
2
x
n
log f
t
(x) ≤ C in {x
n
<t}
by our definition of f
t
in (2.3). Changing back to the x-coordinates we obtain
(3.2c).
By convexity it suffices to prove (3.5) for y ∈ ∂{v<
1
2
}. Let a
h
= h
−1/2
a
h
,
where a
h
is as defined in (2.17). If a
h
≤ C, by (2.16), the set {v<1} has a
good shape. By (2.1) and (2.2), the gradient estimate in {v<
1
2
} is obvious.
If
a
h
 1(a
h
≤ CK by (2.20)), we divide ∂{v<
1
2
} into two parts. Let
∂
1
{v<
1
2
} denote the set y ∈{v =
1
2
}∩

Ω such that the outer normal line of
{v<
1
2
} at y intersects {v =1} = {y ∈

Ω | v(y)=1}, and ∂
2
{v<
1
2
} denote
the rest of ∂{v<
1
2
}, which consists of the boundary part {v<
1
2
}∩∂

Ω and
the points y ∈{v =
1
2
} at which the outer normal line of {v<
1
2
} intersects a
boundary point in {v<1}∩∂

Ω.
Observe that for any y ∈{v<1}∩∂

Ω, (3.5) holds by (3.1) since Dv(0)=0.
By convexity we obtain (3.5) on the part ∂
2
{v<
1
2
}.
To verify (3.5) on ∂
1
{v<
1
2
}, it suffices to show that
dist

{v =1},

v<
1
2

>
C
K
.(3.6)
By the convexity of v we then have |Dv| <CK on ∂
1
{v<
1
2
}. From the last
paragraph, dist({v =1}∩∂Ω, {v<
1
2
}) >C/K.
1004 NEIL S. TRUDINGER AND XU-JIA WANG
We will construct appropriate sub-barriers to prove (3.6). Our sub-barrier
will be a function defined on a cylinder U = E × (−a
n
,a
n
) ⊂ R
n
(after a
rotation of axes), where E =

n−1
i=1
x
2
i
/a
2
i
< 1 is an ellipsoid in R
n−1
.
First we derive a gradient estimate for such a sub-barrier. Suppose a
1
···a
n
= 1. Let w be the convex solution to detD
2
w =1inU with w =0on∂U.
By making the linear transformation y
i
= y
i
/a
i
for i =1, ··· ,n such that
U = {|y

| < 1}×(−1, 1), where y

=(y
1
, ··· , y
n−1
), we have the estimate
C
1
≤−inf
U
w ≤ C
2
for two constants C
2
>C
1
> 0 depending only on n.
By constructing proper sub-barriers [4], we see that w is H¨older continuous
in y. Hence for any C
0
> 0, by the convexity of w, the gradient estimate
C
1
< |D

y
w| <C
2
on {w<−C
0
}, for different C
2
>C
1
> 0 depends only on
n and C
0
. Changing back to the variable y, we obtain
C
1
a
−1
n
≤|D
y
n
w|≤C
2
a
−1
n
(3.7)
at any point y ∈{w = −C
0
} such that y

∈
1
2
E.Ifa := a
1
···a
n
= 1, then by
a dilation one sees that (3.7) holds with a
n
replaced by a
n
/a.
In order to use (3.7) to verify (3.5) on the part ∂
1
{v<
1
2
}, we first show
that
inf
|ν|=1
sup
y,z∈{v<1}
ν · (y −z) ≥ C/K,(3.8)
namely the in-radius of the convex set {v<1} is greater than C/K, where ν ·y
denotes the inner product in R
n
. To prove (3.8) we first observe that by (2.2),
B
r
1
(0) ∩ ∂

Ω ⊂{v<1}∩∂

Ω ⊂ B
r
2
(0) ∩ ∂

Ω
for some r
1
,r
2
> 0 independent of t. Let y =(0, ··· , 0, y
n
) be a point on the
positive x
n
-axis such that v(y) = 1. To prove (3.8), it suffices to show that
y
n
≥ C/K.(3.9)
Let
y =(a, 0, ···, 0, c) ∈ ∂

Ω be an arbitrary point such that v(y) = 1. Then
similarly to (2.22), the angle at
y of the triangle with vertices y, y and the
origin is larger than C/K. Hence y
n
≥ Cr
1
/K ≥ C/K. Hence (3.9) holds.
With (3.9), we can now prove (3.6). For any given point ˆy ∈{v =1}∩∂

Ω,
let P denote the tangent plane of {v =1} at ˆy. Choose a new coordinate system
z such that ˆy is the origin, P = {z
n
=0} and the inner normal of {v<1} is
the positive z
n
-axis. Let S

denote the projection {v<1} on P . By (3.4) and
(3.8) we have the volume estimate
|S

|≤CK.(3.10)
Let E ⊂ P be the minimum ellipsoid of S

with center z
0
, and E
0
⊂ P be
the translation of E such that its center is located at the origin z = 0 (the
point ˆy). Then we have S

⊂ E ⊂ 4nE
0
. The latter inclusion is true when E
is a ball and it is also invariant under linear transformations.