elliptic and parabolic equations in fractured...
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July 9, 2015
Elliptic and parabolic equations in fractured media
Li-Ming Yeh
Department of Applied Mathematics
National Chiao Tung University, Hsinchu, 30050, Taiwan, R.O.C.
The elliptic and the parabolic equations with Dirichlet boundary conditions in frac-tured media are considered. The fractured media consist of a periodic connected highpermeability sub-region and a periodic disconnected matrix block subset with low perme-ability. Let ǫ ∈ (0, 1] denote the size ratio of the matrix blocks to the whole domain andlet ω2
∈ (0, 1] denote the permeability ratio of the disconnected subset to the connectedsub-region. It is proved that the W 1,p norm of the elliptic and the parabolic solutionsin the high permeability sub-region are bounded uniformly in ω, ǫ. However, the W 1,p
norm of the solutions in the low permeability subset may not be bounded uniformly inω, ǫ. For the elliptic and the parabolic equations in periodic perforated domains, it isalso shown that the W 1,p norm of their solutions are bounded uniformly in ǫ.
Keywords: fractured media, permeability, periodic perforated domain, VMO
AMS Subject Classification: 35J05, 35J15, 35J25
1. Introduction
The W 1,p estimates for the solutions of the elliptic and the parabolic equations with
Dirichlet boundary conditions in fractured media are concerned. The problem arises
from two-phase problems, flows in fractured media, and the stress in composite ma-
terials (see [3, 9, 15]). Let Ω be a smooth simply-connected domain in Rn for n ≥ 3,
∂Ω be the boundary of Ω, Y ≡ (0, 1)n consist of a smooth sub-domain Ym completely
surrounded by another connected sub-domain Yf (≡ Y \ Ym), ǫ ∈ (0, 1], Ω(2ǫ) ≡
x ∈ Ω : dist(x, ∂Ω) ≥ 2ǫ, Ωǫm ≡ x : x ∈ ǫ(Ym + j) ⊂ Ω(2ǫ) for some j ∈ Zn be
a disconnected subset of Ω, Ωǫf (≡ Ω\Ωǫ
m) denote a connected sub-region of Ω, and
Kν,ǫ(x) ≡
1 if x ∈ Ωǫ
f
ν if x ∈ Ωǫm
for any ν, ǫ > 0.
The elliptic equation that we consider is−∇ · (Kω2,ǫ∇U +G) = F in Ω,
U = 0 on ∂Ω,(1.1)
where ω, ǫ ∈ (0, 1] and G,F are given functions. If G,F are bounded, a solution of
(1.1) in Hilbert space H1(Ω) exists uniquely for each ω, ǫ by Lax-Milgram Theorem
[12]. The L2 norm of the gradient of the solution of (1.1) in the connected sub-region
Ωǫf is bounded uniformly in ω, ǫ if G,F are small in Ωǫ
m. However, the L2 norm of
1
July 9, 2015
2 Elliptic and parabolic equations
the gradient of the solution of (1.1) in matrix blocks Ωǫm can be very large when ω
closes to 0. The parabolic equation that we consider is, for any ω, ǫ ∈ (0, 1],
∂tU −∇ · (Kω2,ǫ∇U) = F in Ω × (0, T ),
U = 0 on ∂Ω × (0, T ),
U(x, 0) = U0(x) in Ω.
(1.2)
If F,U0 are smooth, a solution of (1.2) in Hilbert space L2([0, T ];H1(Ω)) exists
uniquely for each ω, ǫ. The L2 norm of the gradient of the solution of (1.2) in
the connected sub-region Ωǫf × (0, T ) is bounded uniformly in ω, ǫ if F is small in
Ωǫm× (0, T ). However, the L2 norm of the gradient of the solution of (1.2) in matrix
blocks Ωǫm × (0, T ) can be very large when ω closes to 0. One also notes that for the
elliptic and the parabolic equations in periodic perforated domains, the H1 norm
of their solutions are bounded uniformly in ǫ.
There are some literatures related to this work. Lipschitz estimate andW 2,p esti-
mate for uniform elliptic equations with discontinuous coefficients had been proved
in [15, 18]. Uniform Holder, W 1,p, and Lipschitz estimates for uniform elliptic equa-
tions with Holder periodic coefficients were shown in [4, 5]. Uniform W 1,p estimate
for uniform elliptic equations with continuous periodic coefficients was considered
in [6] and the same problem with VMO periodic coefficients could be found in [22].
Uniform W 1,p estimate for the Laplace equation in periodic perforated domains was
considered in [19] and the same problem in Lipschitz estimate was studied in [21].
Uniform Holder, W 1,p, and Lipschitz estimates in ǫ for uniform parabolic equations
with oscillating periodic coefficients were obtained in [10]. For non-uniform ellip-
tic equations with smooth periodic coefficients, existence of C2,α solution could be
found in [13]. Uniform Holder estimate in ǫ for non-uniform parabolic equations
with discontinuous periodic coefficients was shown in [23].
Here we present uniform W 1,p estimate for the solutions of the non-uniform
elliptic and the non-uniform parabolic equations with Dirichlet boundary conditions
in fractured media. It is proved that the W 1,p norm of the elliptic and the parabolic
solutions in the high permeability sub-region Ωǫf are bounded uniformly in ω, ǫ.
However, the solutions in the low permeability subset may not be bounded uniformly
in ω, ǫ. For the elliptic and the parabolic equations in perforated domains, it is also
shown that the W 1,p norm of their solutions are bounded uniformly in ǫ. A three-
step compactness argument introduced in [4, 5] will be employed to obtain the
uniform estimate for non-uniform elliptic equations. Different from the approach in
[10], we apply semigroup theory and the uniform estimate results for non-uniform
elliptic equations to prove the uniform estimate for non-uniform parabolic equations.
The rest of this work is organized as follows: Notation and main results are stated
in section 2. In section 3, we present a priori estimates for some interface problems
and present some local uniform Lipschitz and local uniform W 1,p estimates in ω, ǫ
for the solutions of elliptic equations in fracture media. Proofs of the main results
are given in section 4. The proof of local uniform Lipschitz estimate for the solutions
July 9, 2015
Elliptic and parabolic equations 3
of elliptic equations in fracture media (claimed in section 3) is given in section 5.
2. Notation and main result
Let Ck,α denote the Holder space with norm ‖ · ‖Ck,α, W s,p the Sobolev space with
norm ‖ · ‖W s,p , and [ϕ]C0,α the Holder semi-norm of ϕ for k ≥ 0, α ∈ [0, 1], s ≥
−1, p ∈ [1,∞] (see [2, 12]). Lp = W 0,p and H1 = W 1,2. C∞0 (D) is the space
of infinitely differentiable functions with support in D and C∞per(R
n) is the space
of infinitely differentiable Y -periodic functions in Rn. W s,p0 (D) is the closure of
C∞0 (D) under the W s,p norm and W s,p
per(Rn) is the closure of C∞
per(Rn) under W s,p
norm and ‖ϕ‖W s,pper(Rn) ≡ ‖ϕ‖W s,p(Y ) for s ≥ 1, p ∈ [1,∞]. Am ≡ x : x ∈ Ym +
j for some j ∈ Zn and Af ≡ Rn \ Am. H1per(R
n) ≡ ϕ ∈ W 1,2per(R
n) :∫
Yfϕ(y)dy =
0 and H1per(Af ) ≡ ϕ|Af
: ϕ ∈ H1per(R
n). Let ‖ϕ1, · · · , ϕm‖B1 ≡ ‖ϕ1‖B1 + · · · +
‖ϕm‖B1 and ‖ϕ‖B1∩B2 ≡ ‖ϕ‖B1 + ‖ϕ‖B2 . Set rD = D/r−1 ≡ x : x/r ∈ D, D be
the closure of D, ∂D be the boundary of D, XD is the characteristic function on D,
and let Br(x) denote a ball centered at x with radius r. For any ϕ ∈ L1(D),
(ϕ)D ≡ −
∫
D
ϕ(y)dy ≡1
|D|
∫
D
ϕ(y)dy.
Kω,ν(x) ≡
1 if x ∈ νAf
ω if x ∈ νAm
for ω ∈ [0, 1], ν ∈ (0,∞). If ~n is an outward normal
vector on ∂Ym, we define, for any function ϕ in Y and x ∈ ∂Ym,
⌊ϕ⌋∂Ym(x) = ϕ,+(x) − ϕ,−(x) where ϕ,±(x) ≡ limt→0+
ϕ(x± t~n). (2.1)
Similarly, if ~nǫ is an outward normal vector on ∂Ωǫm, we define, for any function ϕ
in Ω and x ∈ ∂Ωǫm,
⌊ϕ⌋∂Ωǫm
(x) = ϕ,+(x) − ϕ,−(x) where ϕ,±(x) ≡ limt→0+
ϕ(x± t~nǫ).
Next we give two statements:
A1. Ω is a bounded smooth simply-connected domain in Rn for n ≥ 3,
A2. Ym is a smooth simply-connected sub-domain of Y .
A1–A2 will be assumed throughout this paper except in subsection 5.1. Our main
results are the following:
Theorem 2.1. Suppose A1–A2 and
A3. ω, ǫ ∈ (0, 1], p ∈ (1,∞), G ∈ Lp(Ω), F ∈W−1,p(Ω),
then a W 1,p(Ω) solution of (1.1) exists uniquely and satisfies
‖Kω/ǫ,ǫU,Kω,ǫ∇U‖Lp(Ω)
≤ c(‖K1/ω,ǫG‖Lp(Ω) + ‖F‖W−1,p(Ω) + ω−1‖F‖W−1,p(Ωǫ
m)
)for ω
ǫ ≤ 1,
‖U,Kω,ǫ∇U‖Lp(Ω)
≤ c(‖K1/ω,ǫG‖Lp(Ω) + ‖F‖W−1,p(Ω) + ω−1‖F‖W−1,p(Ωǫ
m)
)for ω
ǫ ≥ 1,
July 9, 2015
4 Elliptic and parabolic equations
where c is a constant independent of ω, ǫ.
Theorem 2.1 implies an analogous result for perforated domains.
Theorem 2.2. Suppose A1–A2 and
A4. ǫ ∈ (0, 1], p ∈ (1,∞), G ∈ Lp(Ωǫf ), F ∈W−1,p(Ω), ‖F‖W−1,p(Ωǫ
m) = 0,
then a W 1,p(Ωǫf ) solution of
−∇ · (∇U +G) = F in Ωǫf
(∇U +G) · ~nǫ = 0 on ∂Ωǫm
U = 0 on ∂Ω
(2.2)
exists uniquely and satisfies
‖U‖W 1,p(Ωǫf ) ≤ c(‖G‖Lp(Ωǫ
f ) + ‖F‖W−1,p(Ω)), (2.3)
where ~nǫ is a unit normal vector on ∂Ωǫm and c is a constant independent of ǫ.
For any ω, ǫ ∈ (0, 1] and p ∈ (1,∞), let us define
Bp ≡ϕ : ϕ ∈ W 1,p
0 (Ω) ∩W 2,p(Ωǫf ) ∩W 2,p(Ωǫ
m), ⌊Kω2,ǫ∇ϕ · ~nǫ⌋∂Ωǫm
= 0,
Dp ≡ϕ : ϕ ∈ W 2,p(Ωǫ
f ), ϕ|∂Ω = 0,∇ϕ · ~nǫ|∂Ωǫm
= 0,
where ~nǫ is a normal vector on ∂Ωǫm. By Lemma 3.4 [23], Bp with norm ‖ϕ‖Bp ≡
‖∇·(Kω2,ǫ∇ϕ)‖Lp(Ω) is a Banach space. If Bp denotes the closure of Bp in Lp space,
then Bp = Lp(Ω). Also note Dp with norm ‖ϕ‖Dp ≡ ‖∆ϕ‖Lp(Ωǫf ) is a Banach space.
The function spaces C([0, T ];B), Cσ([0, T ];B) for σ ∈ (0, 1] are defined as those in
pages 1, 3 [17].
Theorem 2.3. Suppose A1–A2 and
A5. ω, ǫ, σ ∈ (0, 1], p ∈ (n,∞), ǫ ≤ ω, F ∈ Cσ([0, T ];Lp(Ω)), U0 ∈ Bp,
then a C([0, T ];W 1,p(Ω)) solution of (1.2) exists uniquely and satisfies
‖U‖C1([0,T ];Lp(Ω)) + ‖Kω,ǫ∇U‖C([0,T ];Lp(Ω)) ≤ c(‖U0‖Bp + ‖F‖Cσ([0,T ];Lp(Ω))
),
where c is a constant independent of ω, ǫ.
Theorem 2.4. Suppose A1–A2 and
A6. ǫ, σ ∈ (0, 1], p ∈ (n,∞), F ∈ Cσ([0, T ];Lp(Ωǫf )), U0 ∈ Dp,
then a C([0, T ];W 1,p(Ωǫf )) solution of
∂tU − ∆U = F in Ωǫf × (0, T )
∇U · ~nǫ = 0 on ∂Ωǫm × (0, T )
U = 0 on ∂Ω × (0, T )
U(x, 0) = U0(x) in Ωǫf
(2.4)
July 9, 2015
Elliptic and parabolic equations 5
exists uniquely and satisfies
‖U‖C1([0,T ];Lp(Ωǫf)) + ‖∇U‖C([0,T ];Lp(Ωǫ
f)) ≤ c
(‖U0‖Dp + ‖F‖Cσ([0,T ];Lp(Ωǫ
f))
),
where ~nǫ is a normal vector on ∂Ωǫm and c is a constant independent of ǫ.
3. Preliminaries
From Theorem 2.1 [1], we know
Lemma 3.1. For p ∈ [1,∞) and ǫ ∈ (0, 1], there is a constant c(Yf , p) and a linear
continuous extension operator Pǫ : W 1,p(Ωǫf ) →W 1,p(Ω) such that if ϕ ∈W 1,p(Ωǫ
f ),
then
Pǫϕ = ϕ in Ωǫf ,
‖Pǫϕ‖Lp(Ω) ≤ c‖ϕ‖Lp(Ωǫf ),
‖∇Pǫϕ‖Lp(Ω) ≤ c‖∇ϕ‖Lp(Ωǫf ),
0 < d1 ≤ Pǫϕ ≤ d2 if 0 < d1 ≤ ϕ ≤ d2 for some constants d1, d2,
Pǫϕ = ζ in Ω if ϕ = ζ|Ωǫf
for some linear function ζ in Ω.
Moreover, if ζ(x) ≡ ϕ(rx) in B1(0)∩Ωǫf/r for any r > ǫ, then Pǫ/rζ(x) = Pǫϕ(rx)
in B1/2(0).
Remark 3.1. Tracing the proof of Theorem 7.25 [12], we know that if 0 ∈ ∂Ym and
p, ν ∈ [1,∞), there exist a constant c(Yf ) and a linear continuous extension operator
Pν : W 1,p(B1(0) ∩ νYf ) →W 1,p(B1(0)) such that, for any ϕ ∈ W 1,p(B1(0) ∩ νYf ),
Pνϕ = ϕ in B1(0) ∩ νYf ,
‖Pνϕ‖Lp(B1(0)) ≤ c‖ϕ‖Lp(B1(0)∩νYf ),
‖∇Pνϕ‖Lp(B1(0)) ≤ c‖∇ϕ‖Lp(B1(0)∩νYf ).
Lemma 3.2. Let ω ∈ (0, 1], ν ∈ (0,∞), ϕ ∈ H1(B1(0)), and Pνϕ|νAfdenote the
extension of ϕ|νAfon B1(0). There is a constant c independent of ω, ν such that
∥∥Kω,ν
(ϕ− (Pνϕ|νAf
)B1(0)
)∥∥L2(B1(0))
≤ c‖Kω,ν∇ϕ‖L2(B1(0)).
See section 2 for Kω,ν .
Proof. By Poincare inequality [12], Lemma 3.1, and Remark 3.1, the extension
function Pνϕ|νAf∈ H1(B1(0)) satisfies
∥∥Pνϕ|νAf− (Pνϕ|νAf
)B1(0)
∥∥L2(B1(0))
≤ c∥∥∇Pνϕ|νAf
∥∥L2(B1(0))
≤ c‖∇ϕ‖L2(B1(0)∩νAf ), (3.1)
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6 Elliptic and parabolic equations
where c is independent of ω, ν. (3.1), Lemma 3.1, Remark 3.1, and Poincare in-
equality imply∥∥Kω,ν(ϕ− (Pνϕ|νAf
)B1(0))∥∥
L2(B1(0))
≤∥∥Kω,ν(Pνϕ|νAf
− (Pνϕ|νAf)B1(0))
∥∥L2(B1(0))
+ω∥∥ϕ− Pνϕ|νAf
∥∥L2(B1(0)∩νAm)
≤ c‖∇ϕ‖L2(B1(0)∩νAf ) + cω∥∥∇ϕ−∇Pνϕ|νAf
∥∥L2(B1(0)∩νAm)
≤ c‖Kω,ν∇ϕ‖L2(B1(0)).
If 0 ∈ ∂Ω, by A1 and rotation, there is a smooth function Ψ : Rn−1 → R such
that
Ψ(0) = |∇Ψ(0)| = 0,
B1(0) ∩ Ω/r = B1(0) ∩ (x′, xn) ∈ Rn : rxn > Ψ(rx′) if r ∈ (0, 1].(3.2)
If r = 0, we define B1(0) ∩ Ω/r ≡ B1(0) ∩ (x′, xn) ∈ Rn : xn > 0. Set
Kν,ǫ,r ≡
1 in Ωǫ
f/r
ν in Ωǫm/r
for ν, ǫ, r ∈ (0, 1]. (3.3)
Similar to Lemma 3.2, we also have, by Poincare inequality [12], Lemma 3.1,
and Remark 3.1,
Lemma 3.3. If ω, ǫ, r ∈ (0, 1], 0 ∈ ∂Ω and ϕ ∈ H1(B2(0) ∩ Ω/r) with ϕ|∂Ω/r = 0,
there is a constant c independent of ω, ǫ, r such that
‖Kω,ǫ,rϕ‖L2(B1(0)∩Ω/r) ≤ c‖Kω,ǫ,r∇ϕ‖L2(B1(0)∩Ω/r).
3.1. Interface problem
Let Γ(x−y) denote the fundamental solution of the Laplace equation in Rn, see §6.2
[7]. Define a single-layer and a double-layer potentials as, for any smooth function
ϕ on the boundary ∂Ym of Ym,
S∂Ym(ϕ)(x) ≡
∫
∂Ym
Γ(x− y)ϕ(y)dy
L∂Ym(ϕ)(x) ≡
∫
∂Ym
∇yΓ(x− y)~ny ϕ(y)dyfor x ∈ ∂Ym,
where ~ny is the unit vector outward normal to ∂Ym. By tracing the argument of
Lemma 3.2 [23], we know
Lemma 3.4. For any p ∈ (1,∞) and s ∈ 0, 1, the linear operatorsS∂Ym : W s− 1
p ,p(∂Ym) →W s+1− 1p ,p(∂Ym)
L∂Ym : W s+1− 1p ,p(∂Ym) →W s+2− 1
p ,p(∂Ym)
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Elliptic and parabolic equations 7
are bounded; the operator I − ℓL∂Ym is continuously invertible in W s+1− 1p ,p(∂Ym)
for any ℓ ∈ [−2, 2]; and there is a constant c independent of ℓ so that
‖ϕ‖W
s+1− 1p
,p(∂Ym)
≤ c‖(I − ℓL∂Ym)(ϕ)‖W
s+1− 1p
,p(∂Ym)
for ϕ ∈W s+1− 1p ,p(∂Ym),
where I is the identity operator.
Let us introduce some notations:
∂Y is an open portion of the boundary ∂Y ,
D1,D2 are smooth domains satisfying Ym ⊂ D1 ⊂ D2 ⊂ Y ,
dist(Ym, ∂D1), dist(D1, ∂D2), dist(D2, ∂Y \ ∂Y ) > d0 > 0.
Lemma 3.5. Suppose ω ∈ (0, 1], any solution Φ of−∇ · (Kω2,1∇Φ + V ) = ζ in Y
Φ = 0 on ∂Y(3.4)
satisfies
‖Kωσ,1Φ‖W 1,p(D1\Ym)∩W 1,p(Ym) ≤ c(‖Φ‖L2(Yf )
+‖Kωσ−2,1V ‖Lp(Y ) + ‖Kωσ−2,1ζ‖W−1,p(Yf )∩W−1,p(Ym)
),
‖Φ‖W 2,p(D1\Ym)∩W 2,p(Ym) ≤ c(‖Φ‖L2(Yf )
+‖Kω−2,1V ‖W 1,p(Yf )∩W 1,p(Ym) + ‖Kω−2,1ζ‖Lp(Y )
),
(3.5)
where p ∈ [2,∞), σ ∈ 0, 1, and c is a constant independent of ω.
Proof. Define Iω,σ ≡ ‖Kωσ−2,1V ‖Lp(Y ) + ‖Kωσ−2,1ζ‖W−1,p(Yf )∩W−1,p(Ym) and let c
denote a constant independent of ω. First we prove (3.5)1.
Step 1: Assume V ∈ W 1,p0 (Yf ) ∩W 1,p
0 (Ym) and ζ ∈ Lp(Y ). Consider the fol-
lowing problem−∇ · (Kω2,1∇φ+ V ) = ζ in D2,
φ = 0 on ∂D2.(3.6)
The unique existence of a H1 solution of (3.6) is from Lax-Milgram Theorem [12].
By energy method, the solution satisfies
‖φ‖H1(D2\Ym) ≤ cIω,1. (3.7)
Let η ∈ C∞(D2\Ym), η ∈ [0, 1], η = 1 in D2\D1, η = 0 on ∂Ym, ‖η‖W 1,∞(D2\Ym) ≤
c. Multiply (3.6)1 by η to get−∇ · (∇(ηφ) − φ∇η + ηV ) = ηζ − (∇φ+ V )∇η in D2 \ Ym,
ηφ = 0 on ∂D2 ∪ ∂Ym.(3.8)
By (3.7)–(3.8), [8], Theorem 7.26 [12], and an iterative argument, we have
‖φ‖W 1,p(D2\D1) ≤ cIω,σ. (3.9)
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8 Elliptic and parabolic equations
Let ϕ in Ym satisfy−∇ · (ω2∇ϕ+ V ) = ζ in Ym,
ϕ = 0 on ∂Ym,(3.10)
and ϕ in D2 \ Ym satisfy−∇ · (∇ϕ+ V ) = ζ in D2 \ Ym,
ϕ = 0 on ∂(D2 \ Ym).(3.11)
By [8] again,
‖ϕ‖W 1,p(D2\Ym) + ωσ‖ϕ‖W 1,p(Ym) ≤ cIω,σ. (3.12)
If we define ψ ≡ φ− ϕ in D2, then (3.6) and (3.10)–(3.11) imply
∆ψ = 0 in D2 \ ∂Ym,
⌊ψ⌋∂Ym = 0 on ∂Ym,
⌊Kω2,1∇ψ⌋∂Ym · ~ny = F on ∂Ym,
ψ = 0 on ∂D2,
(3.13)
where ~ny is the unit vector outward normal to ∂Ym. See (2.1) for (3.13)2,3. Since
V ∈W 1,p0 (Yf ) ∩W 1,p
0 (Ym),
F ≡(ω2∇ϕ,− −∇ϕ,+
)· ~ny|∂Ym .
By (3.12),
‖F‖W
−1p
,p(∂Ym)
≤ cIω,σ. (3.14)
By Green’s formula, (3.13), and Theorem 6.5.1 [7],ψ/2 + L∂Ym(ψ) = S∂Ym(∇ψ,− · ~ny|∂Ym)
ψ/2 − L∂Ym(ψ) = −S∂Ym(∇ψ,+ · ~ny|∂Ym) + S∂D2(∂nyψ|∂D2)on ∂Ym,
where ∂nyψ|∂D2 is the normal derivative of ψ on ∂D2. So(I −
2(1 − ω2)
ω2 + 1L∂Ym
)ψ =
2
ω2 + 1
(S∂D2 (∂nyψ|∂D2) − S∂Ym(F)
)on ∂Ym. (3.15)
Then (3.9), (3.12)–(3.15), and Lemma 3.4 imply
‖ψ‖W
1− 1p
,p(∂Ym)
≤ c
(‖F‖
W−
1p
,p(∂Ym)
+ ‖∂nyψ‖W
−1p
,p(∂D2)
)≤ cIω,σ. (3.16)
(3.13) and (3.16) imply
‖ψ‖W 1,p(D2) ≤ cIω,σ.
Together with (3.12), we obtain
‖Kωσ,1φ‖W 1,p(D2\Ym)∩W 1,p(Ym) ≤ cIω,σ. (3.17)
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Elliptic and parabolic equations 9
Note W 1,p0 (Yf ) (resp. W 1,p
0 (Ym)) is dense in Lp(Yf ) (resp. Lp(Ym)) and Lp(Y )
is dense in W−1,p(Y ). By a limiting argument, we see that if V ∈ Lp(Y ) and
ζ ∈W−1,p(Y ), any solution of (3.6) satisfies (3.17).
Step 2: Let η be a smooth function satisfying η ∈ C∞0 (D2), η ∈ [0, 1], η = 1 in
D1, ‖η‖W 1,∞(D2) ≤ c. Multiply (3.4) by η to obtain−∇ · (Kω2,1∇(Φη) − Φ∇η + V η) = ζη − (∇Φ + V )∇η in D2,
Φη = 0 on ∂D2.
By the result of Step 1, we have
‖Kωσ,1Φ‖W 1,p(D1\Ym)∩W 1,p(Ym) ≤ c(‖Φ‖Lp(D2\D1) + Iω,σ). (3.18)
Let η be another smooth function satisfying η ∈ C∞0 (Yf ), η ∈ [0, 1], η = 1 in D2\D1,
‖η‖W 1,∞(Y ) ≤ c. Multiply (3.4) by ηΦ and use energy method and Theorem 7.26
[12] to get
‖Φ‖Lp(D2\D1) ≤ c(‖Φ‖L2(Yf ) + Iω,σ).
Together with (3.18), we obtain (3.5)1. (3.5)2 are proved in a similar way as (3.5)1,
so we skip it.
By a similar argument as Lemma 3.5, we also have the following local estimate:
Lemma 3.6. If ω ∈ (0, 1], ν ∈ (1,∞), x0 ∈ ν∂Ym, and B1(x0) ⊂ νY , then any
solution Φ of
−∇ · (Kω2,ν∇Φ) = 0 in νY (3.19)
satisfies
‖Kωσ,νΦ‖W 2,p(B1/3(x0)∩νYf )∩W 2,p(B1/3(x0)∩νYm) ≤ c‖Kωσ,νΦ‖L2(B1(x0)), (3.20)
where p ∈ [2,∞), σ ∈ 0, 1, and c is a constant independent of ω, ν.
Proof. After translation, we assume x0 is the origin. For each ν > 1, we find a
smooth domain Dν such that
B1/2(x0) ∩ νYm ⊂ Dν ⊂ B1(x0) ∩ νYm and B1/2(x0) ∩ ν∂Ym ⊂ ∂Dν .
Since Dν is smooth, for any z ∈ ∂Dν there is a ball B(z) centered at z and there
is a smooth one-to-one mapping ϕz,ν of B(z) onto ϕz,ν(B(z)) ⊂ Rn satisfying
ϕz,ν(B(z) ∩Dν) ⊂ Rn+, ϕz,ν(B(z) ∩ ∂Dν) ⊂ ∂Rn
+, ϕz,ν(B(z) \ Dν) ⊂ Rn−. (3.21)
Here Rn+ ≡ x = (x1, · · · , xn) : xn > 0, ∂Rn
+ ≡ x : xn = 0,Rn− ≡ x : xn < 0.
Since ∂Dν is compact for each ν > 1, there exist a finite number ℓν of open balls
B(zi)ℓν
i=1 and one-to-one mappings ϕzi,νℓν
i=1 such that
zi ∈ ∂Dν for i ∈ 1, · · · , ℓν,
(3.21) holds for each ball B(zi) and i ∈ 1, · · · , ℓν,
∂Dν ⊂⋃ℓν
i=1 B(zi).
July 9, 2015
10 Elliptic and parabolic equations
Since Ym is smooth, it is possible to choose domains Dν for all ν > 1 such that
the number ℓν is bounded above by a constant independent of ν,
‖ϕz,ν‖C3,0(B(z)), ‖ϕ−1z,ν‖C3,0(ϕz,ν(B(z))) ≤ c∗,where c∗ is independent of ν, z.
By assumption x0 = 0 ∈ ν∂Ym, we define Kω2,ν and φ in Rn by
Kω2,ν ≡
ω2 in Dν ,
1 elsewhere,φ ≡
Φ in B1/2(x0),
0 elsewhere.
Let η ∈ C∞0 (B1/2(x0)) be a bell-shaped function satisfying η ∈ [0, 1], η = 1 in
B1/3(x0), ‖∇η‖W 1,∞(B1/2(x0)) ≤ c. Multiply (3.19) by η to get
−∇ ·
(Kω2,ν∇(ηφ) − Kω2,νφ∇η
)= −Kω2,ν∇φ∇η in B1(x0),
ηφ = 0 on ∂B1(x0).
Then we follow the argument of Step 1 of Lemma 3.5 to see that (3.20) holds.
Let X(j)ω,1 ∈ H1
per(Rn) for ω ∈ (0, 1] be a function satisfying
∇ · (Kω2,1(∇X(j)ω,1 + ~ej)) = 0 in Y , (3.22)
and let X(j)0,1 ∈ H1
per(Af )∩H1(Am) be a function satisfying X(j)0,1(x) = 0 in Am and
∇ · (∇X
(j)0,1 + ~ej) = 0 in Yf ,
(∇X(j)0,1 + ~ej) · ~ny = 0 on ∂Ym,
where ~ej , j = 1, · · · , n is the unit vector in the j-th coordinate direction, and ~ny
is a unit normal vector on ∂Ym. By Lax-Milgram Theorem [12], X(j)ω,1 for ω ∈ [0, 1]
and j = 1, · · · , n is uniquely solvable. By Theorem 6.30 [12] and (3.5)2 of Lemma
3.5,
‖X(j)ω,1‖W 2,p(Yf )∩W 2,p(Ym) ≤ c(n, Ym) for ω ∈ [0, 1], p ∈ [2,∞). (3.23)
Define Xω,1 ≡ (X(1)ω,1, · · · ,X
(n)ω,1) and Xω,ǫ(x) ≡ ǫXω,1(
xǫ ) for ω ∈ [0, 1], ǫ ∈ (0, 1].
Denote by Ξω for ω ∈ [0, 1] a n × n matrix function whose (i, j)-component is
∂iX(j)ω,1. By remark in pages 17-19, 94-95 [14],
Kω ≡
∫
Yf∪Ym
Kω2,1(I + Ξω)dy for ω ∈ [0, 1] (3.24)
is a symmetric positive definite matrix dependent only on ω. Here I is the identity
matrix. By (3.23), it is not difficult to see, for ω ∈ [0, 1],d3I ≤ Kω ≤ d4I where d3, d4 are positive constants,
Kω is a continuous function of ω.(3.25)
July 9, 2015
Elliptic and parabolic equations 11
3.2. Local Lipschitz and local Lp gradient estimates
We have the following Lipschitz estimate:
Lemma 3.7. Suppose ω, ǫ ∈ (0, 1], any solution of−∇ · (Kω2,ǫ∇Φ) = 0 in B1(0) ∩ Ω
Φ = 0 on B1(0) ∩ ∂Ω
satisfies
‖∇Φ‖L∞(B1/2(0)∩Ω) ≤ c‖Kω,ǫΦ‖L2(B1(0)∩Ω), (3.26)
where c is a constant independent of ω, ǫ.
Proof of Lemma 3.7 is given in section 5. Next is the local Lp gradient estimate:
Lemma 3.8. Let ω, ǫ, r ∈ (0, 1], p ∈ (2,∞), and either B2r(x0) ⊂ Ω or x0 ∈ ∂Ω.
Any solution of−∇ · (Kω2,ǫ∇Φ) = 0 in B2r(x0) ∩ Ω
Φ = 0 on B2r(x0) ∩ ∂Ω(3.27)
satisfies
(−
∫
Br/2(x0)
|Kω,ǫ∇Φ|pXΩ dx
)1/p
≤ c
(−
∫
Br(x0)
|Kω,ǫ∇Φ|2XΩ dx
)1/2
, (3.28)
where c is a constant independent of ω, ǫ, r, x0.
Proof. Let c denote a constant independent of ω, ǫ, r, x0.
Case I: For B2r(x0) ⊂ Ω case. By translation, we move x0 to the origin (that
is, x0 = 0 ∈ Ω). Let d ∈ R and ϕ(y) = Φ(ry). By (3.27), we know
−∇ · (Kω2,ǫ/r∇(ϕ+ d)) = 0 in B2(0).
If ǫ/r ≤ 1 (resp. ǫ/r > 1), Lemma 3.7 (resp. Theorem 9.11 [12] and Lemma 3.6)
implies
‖Kω,ǫ/r∇ϕ‖Lp(B1/2(0)) ≤ c‖Kω,ǫ/r(ϕ+ d)‖L2(B1(0)),
where c is also independent of d. By Lemma 3.2,
‖Kω,ǫ/r∇ϕ‖Lp(B1/2(0)) ≤ c‖Kω,ǫ/r∇ϕ‖L2(B1(0)).
Which implies (3.28).
Case II: For x0 ∈ ∂Ω case. Set x0 = 0 ∈ ∂Ω by translation and set ϕ(y) =
Φ(ry). By (3.3) and (3.27),−∇ · (Kω2,ǫ,r∇ϕ) = 0 in B2(0) ∩ ∂Ω/r,
ϕ = 0 on B2(0) ∩ ∂Ω/r.(3.29)
July 9, 2015
12 Elliptic and parabolic equations
If ǫ/r ≤ 1 (resp. ǫ/r > 1), Lemma 3.7 (resp. Theorem 9.13 [12]) implies that the ϕ
in (3.29) satisfies
‖Kω,ǫ,r∇ϕ‖Lp(B1/2(0)∩Ω/r) ≤ c‖Kω,ǫ,rϕ‖L2(B1(0)∩Ω/r).
By Lemma 3.3, we obtain
‖Kω,ǫ,r∇ϕ‖Lp(B1/2(0)∩Ω/r) ≤ c‖Kω,ǫ,r∇ϕ‖L2(B1(0)∩Ω/r).
Which implies (3.28).
4. Proof of main results
Proof of Theorem 2.1: Suppose ω, ǫ ∈ (0, 1], let us find U ∈ H1(Ω) satisfying−∇ · (Kω2,ǫ∇U + Kω,ǫG) = 0 in Ω,
U = 0 on ∂Ω.(4.1)
By Lax-Milgram Theorem [12], U exists uniquely if G ∈ L2(Ω). If we define T :
L2(Ω) → L2(Ω) by TG = Kω,ǫ∇U , then T is a linear and bounded operator on
L2(Ω) by energy method. Lemma 3.8 implies that the operator T satisfies (1.9) of
Theorem 1.3 [22] for any G ∈ Lp(Ω), p ∈ (2,∞). So T is a bounded and linear
operator in Lp(Ω) for p ∈ (2,∞) by Theorem 1.3 [22]. By Poincare inequality [12]
and Lemma 3.1, the solution of (4.1) satisfies
‖U‖Lp(Ωǫf ) ≤ ‖PǫU |Ωǫ
f‖Lp(Ω) ≤ c‖∇PǫU |Ωǫ
f‖Lp(Ω) ≤ c‖∇U‖Lp(Ωǫ
f ),
‖U‖Lp(Ωǫm) ≤ ‖U − PǫU |Ωǫ
f‖Lp(Ωǫ
m) + ‖PǫU |Ωǫf‖Lp(Ωǫ
m)
≤ cǫ‖∇U −∇PǫU |Ωǫf‖Lp(Ωǫ
m) + c‖U‖Lp(Ωǫf ),
where c is independent of ω, ǫ. Function PǫU |Ωǫf
above denotes the extension of
U |Ωǫf
on Ω. So we have
Lemma 4.1. Under A1–A2, if ω, ǫ ∈ (0, 1], p ∈ [2,∞), and G ∈ Lp(Ω), then a
W 1,p(Ω) solution U of (4.1) exists uniquely and‖Kω/ǫ,ǫU,Kω,ǫ∇U‖Lp(Ω) ≤ c‖G‖Lp(Ω) for ω
ǫ ≤ 1,
‖U,Kω,ǫ∇U‖Lp(Ω) ≤ c‖G‖Lp(Ω) for ωǫ ≥ 1,
where c is a constant independent of ω, ǫ.
By a duality argument, Poincare inequality [12], and Lemmas 3.1, 4.1, we have
Lemma 4.2. Under A1–A2, if ω, ǫ ∈ (0, 1], p ∈ (1, 2], and G ∈ Lp(Ω), then a
W 1,p(Ω) solution U of (4.1) exists uniquely and‖Kω/ǫ,ǫU,Kω,ǫ∇U‖Lp(Ω) ≤ c‖G‖Lp(Ω) for ω
ǫ ≤ 1,
‖U,Kω,ǫ∇U‖Lp(Ω) ≤ c‖G‖Lp(Ω) for ωǫ ≥ 1,
where c is a constant independent of ω, ǫ.
July 9, 2015
Elliptic and parabolic equations 13
If ω, ǫ ∈ (0, 1] and G,F ∈ L∞(Ω), then the H1(Ω) solution of−∇ · (Kω2,ǫ∇U) = F in Ω
U = 0 on ∂Ω(4.2)
and the H1(Ω) solution of−∇ · (Kω2,ǫ∇ϕ− Kω,ǫG) = 0 in Ω
ϕ = 0 on ∂Ω(4.3)
exist uniquely by Lax-Milgram Theorem [12]. Lemma 4.1 and Lemma 4.2 imply
that the solution of (4.3) satisfies‖Kω/ǫ,ǫϕ,Kω,ǫ∇ϕ‖Lr(Ω) ≤ c‖G‖Lr(Ω) for ω
ǫ ≤ 1,
‖ϕ,Kω,ǫ∇ϕ‖Lr(Ω) ≤ c‖G‖Lr(Ω) for ωǫ ≥ 1,
(4.4)
where r ∈ (1,∞) and c is a constant independent of ω, ǫ. Multiply (4.2) by the
solution of (4.3), multiply (4.3) by the solution of (4.2), integrate by part, as well
as employ (4.4), Lemma 3.1, and Holder inequality to get∫
Ω
Kω,ǫ∇U Gdx =
∫
Ω
ϕ Fdy =
∫
Ω
Pǫϕ|ΩǫfFdy +
∫
Ωǫm
(ϕ− Pǫϕ|Ωǫf)Fdy
≤ c‖G‖Lr(Ω)(‖F‖W−1,p(Ω) + ω−1‖F‖W−1,p(Ωǫm)),
where 1r + 1
p = 1 and c is independent of ω, ǫ. Since L∞(Ω) is dense in Lr(Ω) for
any r ∈ (1,∞), we obtain
‖Kω,ǫ∇U‖Lp(Ω) ≤ c(‖F‖W−1,p(Ω) + ω−1‖F‖W−1,p(Ωǫm)),
where 1r + 1
p = 1 and c is a constant independent of ω, ǫ. By Poincare inequality
[12] and Lemma 3.1, it is easy to see that‖Kω/ǫ,ǫU‖Lp(Ω) ≤ c(‖F‖W−1,p(Ω) + ω−1‖F‖W−1,p(Ωǫ
m)) for ωǫ ≤ 1,
‖U‖Lp(Ω) ≤ c(‖F‖W−1,p(Ω) + ω−1‖F‖W−1,p(Ωǫm)) for ω
ǫ ≥ 1,
where p ∈ (1,∞) and c is a constant independent of ω, ǫ. Together with Lemma
4.1 and Lemma 4.2, we see that Theorem 2.1 holds for G ∈ Lp(Ω), F ∈ L∞(Ω). If
G ∈ Lp(Ω), F ∈ W−1,p(Ω), Theorem 2.1 can be proved by a limiting argument.
Proof of Theorem 2.2: Suppose G,F ∈ Lp(Ωǫf ) for p ∈ (1,∞), let us do zero
extension for G,F . That is, set G ≡
G on Ωǫ
f
0 on Ωǫm
and F ≡
F on Ωǫ
f
0 on Ωǫm
. Then
G, F ∈ Lp(Ω). Let Uω,ǫ denote the solution of (1.1) with G,F replaced by G, F
above. By Theorem 2.1,‖Kω/ǫ,ǫUω,ǫ,Kω,ǫ∇Uω,ǫ‖Lp(Ω) ≤ c(‖G‖Lp(Ω) + ‖F‖W−1,p(Ω)) for ω
ǫ ≤ 1,
‖Uω,ǫ,Kω,ǫ∇Uω,ǫ‖Lp(Ω) ≤ c(‖G‖Lp(Ω) + ‖F‖W−1,p(Ω)) for ωǫ ≥ 1,
(4.5)
where c is independent of ω, ǫ. If we fix ǫ, then we see, by (4.5) and Lemma 3.1,
July 9, 2015
14 Elliptic and parabolic equations
• There is a subsequence of Uω,ǫ (same notation for subsequence) such that
Uω,ǫ|Ωǫf
converges weakly to U in W 1,p(Ωǫf ) as ω → 0.
• The limit function U satisfies (2.2) and (2.3).
So Theorem 2.2 is proved for G,F ∈ Lp(Ωǫf ), p ∈ (1,∞). For general case, Theorem
2.2 can be proved by a limiting argument.
Based on the above uniform results (that is, Theorem 2.1, Theorem 2.2), we then
apply semigroup theory to obtain the uniform estimates for parabolic equations.
Proof of Theorem 2.3: By A1–A2, A5 as well as by tracing the proof of
Theorem 2.1 [23], we know the solution of (1.2) exists uniquely and satisfies, for
p ∈ (n,∞),
‖U‖C1([0,T ];Lp(Ω)) + ‖U‖C([0,T ];Bp) ≤ c(‖U0‖Bp + ‖F‖Cσ([0,T ];Lp(Ω))
),
where c is a constant independent of ω, ǫ. (1.2) can be written as, for fixed t ∈ (0, T ],
−∇ · (Kω2,ǫ∇U(·, t)) = F (·, t) − ∂tU(·, t) in Ω,
U(·, t) = 0 on ∂Ω.
By Theorem 2.1, we see, for p ∈ (n,∞),
‖Kω,ǫ∇U(·, t)‖Lp(Ω) ≤ c(‖U0‖Bp + ‖F‖Cσ([0,T ];Lp(Ω))
),
where c is a constant independent of ω, ǫ. So Theorem 2.3 is proved.
Proof of Theorem 2.4: By A1–A2, A6 as well as by tracing the proof of
Theorem 2.1 [23], we know that the solution of (2.4) exists uniquely and satisfies,
for p ∈ (n,∞),
‖U‖C1([0,T ];Lp(Ωǫf )) + ‖∆U‖C([0,T ];Lp(Ωǫ
f )) ≤ c(‖U0‖Dp + ‖F‖Cσ([0,T ];Lp(Ωǫ
f ))
),
where c is a constant independent of ǫ. (2.4) can be written as, for fixed t ∈ (0, T ],
−∆U(·, t) = F (·, t) − ∂tU(·, t) in Ωǫf ,
∇U(·, t) · ~nǫ = 0 on ∂Ωǫm,
U(·, t) = 0 on ∂Ω.
By Theorem 2.2, we see, for p ∈ (n,∞),
‖∇U‖C([0,T ];Lp(Ωǫf )) ≤ c
(‖U0‖Dp + ‖F‖Cσ([0,T ];Lp(Ωǫ
f ))
),
where c is a constant independent of ǫ. So Theorem 2.4 is proved.
5. Local uniform Lipschitz estimate
We prove a local Lipschitz estimate, that is, Lemma 3.7. The idea of proof follows
from the arguments in [4]. We first derive local Holder estimate in subsection 5.1
and then derive local Lipschitz estimate in subsection 5.2.
July 9, 2015
Elliptic and parabolic equations 15
5.1. Holder estimate
An open set O ⊂ Rn with boundary ∂O is said to satisfy a uniform exterior ball
condition, if there exists a r > 0 with the following property: For each x ∈ ∂O,
there exists a point y = y(x) ∈ Rn such that Br(y) \ x ⊂ Rn \O and x ∈ ∂Br(y).
If O ⊂ Rn is a nonempty open bounded Lipschitz set and satisfy a uniform exterior
ball condition, then O is called a semiconvex domain.
In this subsection, we assume (1) A2 holds and (2) O is a semiconvex domain.
If 0 ∈ ∂O, by rotation, there is a Lipschitz function Ψ : Rn−1 → R such that
Ψ(0) = 0,
B1(0) ∩ O/r = B1(0) ∩ (x′, xn) ∈ Rn : rxn > Ψ(rx′) if r ∈ (0, 1].(5.1)
If r = 0, we define B1(0) ∩ O/r ≡ B1(0) ∩ (x′, xn) ∈ Rn : xn > 0. Similar to Ωǫf
and Ωǫm, one can also define analogous Oǫ
f and Oǫm. Define Kν,ǫ,r as
Kν,ǫ,r ≡
1 in Oǫ
f/r
ν in Oǫm/r
for ν, ǫ, r ∈ (0, 1].
Lemma 5.1. Let ω, ǫ, r ∈ (0, 1], ǫ ≤ r, either B2(0) ⊂ O/r or 0 ∈ ∂O/r, and ϕ be
a solution of−∇ · (Kω2,ǫ,r∇ϕ) = 0 in B2(0) ∩ O/r,
ϕ = 0 on B2(0) ∩ ∂O/r.
There is a constant c independent of ω, ǫ, r such that
‖ϕ‖H1(B1/2(0)∩O/r) ≤ c‖Kω,ǫ,rϕ‖L2(B2(0)∩O/r).
Proof. Let c denote a constant independent of ω, ǫ, r. By energy method,
‖Kω,ǫ,r∇ϕ‖L2(B1(0)∩O/r) ≤ c‖Kω,ǫ,rϕ‖L2(B2(0)∩O/r). (5.2)
For any z ∈ B1(0) ∩ O/r, we move z to the origin of the coordinate system by
translation and define
K(x) ≡ Kω2,ǫ,r(ǫrx)
ϕ(x) ≡ ϕ( ǫrx)
for x ∈ B1(z) ∩O/ǫ.
Then ϕ satisfies−∇ · (K∇ϕ) = 0 in B1(z) ∩ O/ǫ,
ϕ = 0 on B1(z) ∩ ∂O/ǫ.
By (3.5)1,
‖∇ϕ‖L2(B1/2(z)∩O/ǫ) ≤ c‖ϕ‖L2(B1(z)∩Oǫf /ǫ). (5.3)
By Poincare inequality [12], (5.3) implies
‖∇ϕ‖2L2(Bǫ/2r(z)∩O/r) ≤ c‖∇ϕ‖2
L2(Bǫ/r(z)∩Oǫf /r). (5.4)
July 9, 2015
16 Elliptic and parabolic equations
By covering B1(0) ∩ O/r with a finite number of balls of radius ǫ/2r, (5.4) implies
‖∇ϕ‖L2(B1/2(0)∩O/r) ≤ c‖∇ϕ‖L2(B1(0)∩Oǫf/r).
Together with (5.2) and Poincare inequality [12], we prove the lemma.
The rest of this subsection is to prove the following lemma.
Lemma 5.2. Suppose δ > 0 and ω, ǫ ∈ (0, 1], any solution of
−∇ · (Kω2,ǫ,1∇Φ) = 0 in B1(0) ∩ O
Φ = 0 on B1(0) ∩ ∂O
satisfies
‖Φ‖C0,µ(B1/2(0)∩O) ≤ c‖Kω,ǫ,1Φ‖L2(B1(0)∩O), (5.5)
where µ ≡ δn+δ and c is a constant independent of ω, ǫ.
The interior estimate of (5.5) is given in subsection 5.1.1 and the boundary
estimate of (5.5) is in subsection 5.1.2.
5.1.1. Interior Holder estimate
Assume B1(0) ⊂ O.
Lemma 5.3. For any δ > 0, there are constants θ1, θ2 ∈ (0, 1) with θ1 < θ22 and a
constant ǫ0 ∈ (0, 1) (depending on δ, θ2, Yf ) such that if
−∇ · (Kω2,ν,1∇ϕ) = 0 in B1(0),
‖Kω,ν,1ϕ‖L2(B1(0)) ≤ 1,(5.6)
then, for any ω ∈ (0, 1], ν ∈ (0, ǫ0], and θ ∈ [θ1, θ2],
−
∫
Bθ(0)
∣∣ϕ− (ϕ)Bθ(0)
∣∣2 dx ≤ θ2µ, (5.7)
where µ ≡ δn+δ .
Proof. Consider the following problem
−∇ · (Kω∗∇ϕ∗) = 0 in B2/3(0), (5.8)
where Kω∗for ω∗ ∈ [0, 1] is from (3.24). By Theorem 1.2 in page 70 [11] and (3.25),
there is a small θ ∈ (0, 2/3) such that
−
∫
Bθ(0)
∣∣ϕ∗ − (ϕ∗)Bθ(0)
∣∣2dx ≤ θ2µ′
−
∫
B2/3(0)
|ϕ∗|2dx, (5.9)
July 9, 2015
Elliptic and parabolic equations 17
for some µ′ ∈ (µ, 1). We choose θ1, θ2 ∈ (0, 2/3) such that θ1 < θ22 and (5.9) holds
if θ ∈ [θ1, θ2]. Now we claim (5.7). If not, there is a sequence ων , θν , ϕν satisfying
(5.6) and
ων → ω∗ ∈ [0, 1]
θν → θ∗ ∈ [θ1, θ2]
−
∫
Bθν (0)
∣∣ϕν − (ϕν)Bθν (0)
∣∣2 dx > θ2µν
as ν → 0. (5.10)
By Lemma 5.1 and tracing the proof of Theorem 2.3 [3], there is a subsequence
(same notation for subsequence) such thatϕν → ϕ∗ in L2(B2/3(0)) strongly
Kω2ν ,ν,1∇ϕν → Kω∗
∇ϕ∗ in L2(B2/3(0)) weaklyas ν → 0. (5.11)
Also the ϕ∗ above is a solution of (5.8). By (5.9)–(5.11), we conclude
θ2µ∗ = lim
ν→0θ2µ
ν ≤ limν→0
−
∫
Bθν (0)
∣∣ϕν − (ϕν)Bθν (0)
∣∣2 dx
= −
∫
Bθ∗(0)
|ϕ∗ − (ϕ∗)Bθ∗ (0)|2dx ≤ θ2µ′
∗ −
∫
B2/3(0)
|ϕ∗|2dx. (5.12)
But (5.12) is impossible if θ2 is small enough. Therefore, there is a ǫ0 such that
(5.7) holds for ν ≤ ǫ0.
Lemma 5.4. For any δ > 0, there are constants θ1, θ2 ∈ (0, 1) with θ1 < θ22 and a
constant ǫ0 ∈ (0, 1) (depending on δ, θ2, Yf ) such that if
−∇ · (Kω2,ǫ,1∇Φ) = 0 in B1(0), (5.13)
then, for any ω ∈ (0, 1], ǫ ∈ (0, ǫ0], θ ∈ [θ1, θ2], and k satisfying ǫ/θk ≤ ǫ0,
−
∫
Bθk (0)
∣∣∣Φ − (Φ)Bθk (0)
∣∣∣2
dx ≤ θ2kµ|Jω,ǫ|2, (5.14)
where µ ≡ δn+δ and Jω,ǫ ≡ ‖Kω,ǫ,1Φ‖L2(B1(0)).
Proof. The proof is done by induction on k. For k = 1, we define ϕ ≡ Φ/Jω,ǫ
and use Lemma 5.3 with ν = ǫ to obtain (5.14). Suppose (5.14) holds for some k
satisfying ǫ/θk ≤ ǫ0, then we define
ϕ(x) ≡ J−1ω,ǫ θ
−kµ(Φ(θkx) − (Φ)B
θk (0)
)in B1(0).
Then ϕ satisfies (5.6) with ν = ǫ/θk. By changing variable and employing Lemma
5.3, we obtain (5.14) with k + 1 in place of k.
Lemma 5.5. For any δ > 0, there is a constant ǫ∗ ∈ (0, 1) (depending on δ, Yf)
such that if ω ∈ (0, 1] and ǫ ∈ (0, ǫ∗], then any solution of (5.13) satisfies
[Φ]C0,µ(B1/2(0))≤ c‖Kω,ǫ,1Φ‖L2(B1(0)), (5.15)
July 9, 2015
18 Elliptic and parabolic equations
where c is a constant independent of ω, ǫ. See Lemma 5.4 for µ.
Proof. Let θ1, θ2, ǫ0, Jω,ǫ be same as those in Lemma 5.4 and define ǫ∗ ≡ ǫ0θ2/2.
Denote by c a constant independent of ω, ǫ. Because of θ1 < θ22, for any r ∈ [ǫ/ǫ0, θ2],
there are θ ∈ [θ1, θ2] and k ∈ N satisfying r = θk. Lemma 5.4 implies
−
∫
Br(0)
∣∣Φ − (Φ)Br(0)
∣∣2 dy ≤ cr2µ|Jω,ǫ|2 for r ∈ [ǫ/ǫ0, θ2]. (5.16)
Take r = 2ǫǫ0
and define
ϕ(y) ≡ J−1ω,ǫǫ
−µ(Φ(ǫy) − (Φ)B2ǫ/ǫ0
(0)
)in B2/ǫ0(0).
Then ϕ satisfies−∇ · (Kω2,1,1∇ϕ) = 0 in B2/ǫ0(0),
‖ϕ‖L2(B2/ǫ0(0)) ≤ c.
(3.5)1 of Lemma 3.5 implies [ϕ]C0,µ(B1/ǫ0(0)) ≤ c. Together with (5.16), then (5.16)
holds for r ∈ (0, θ2). Next we shift the origin of the coordinate system to any point
z ∈ B1/2(0) and repeat above argument to see that (5.16) with 0 replaced by any
z ∈ B1/2(0) also holds for r ∈ (0, θ2). By Theorem 1.2 in page 70 [11], we prove the
lemma.
Remark 5.1. Let ǫ∗ be same as that in Lemma 5.5. By (3.5)1 of Lemma 3.5,
we know that if ω ∈ (0, 1] and ǫ ∈ [ǫ∗, 1], any solution of (5.13) satisfies (5.15).
Together with Lemma 5.5, we conclude that any solution of (5.13) satisfies (5.15)
if ω, ǫ ∈ (0, 1].
5.1.2. Boundary Holder estimate
We assume 0 ∈ ∂O.
Lemma 5.6. For any δ > 0, there are constants θ1, θ2 ∈ (0, 1) with θ1 < θ22, and a
constant ǫ0 ∈ (0, 1) (depending on δ, θ2, Yf , ‖∇Ψ‖L∞(Rn−1)) such that if
−∇ · (Kω2,ǫ,r∇ϕ) = 0 in B1(0) ∩ O/r,
ϕ = 0 on B1(0) ∩ ∂O/r,
‖Kω,ǫ,rϕ‖L2(B1(0)∩O/r) ≤ 1,
(5.17)
then, for any ω, ǫ, r ∈ (0, 1], ǫ/r ≤ ǫ0, and θ ∈ [θ1, θ2],
−
∫
Bθ(0)∩O/r
|ϕ|2dx ≤ θ2µ, (5.18)
where µ ≡ δn+δ . See (5.1) for Ψ.
July 9, 2015
Elliptic and parabolic equations 19
Proof. Consider the following problem−∇ · (Kω∗
∇ϕ∗) = 0 in B2/3(0) ∩ O/r∗,
ϕ∗ = 0 on B2/3(0) ∩ ∂O/r∗,(5.19)
where Kω∗is from (3.24) and ω∗, r∗ ∈ [0, 1]. Note O/r∗ is a semiconvex domain
(see the beginning of subsection 5.1) as well as Kω∗is a symmetric positive definite
matrix and can be diagonalizable. By Theorem 4.5 [20], Theorem 7.26 [12], and
(3.25), there is a small θ ∈ (0, 2/3) such that
−
∫
Bθ(0)∩O/r∗
|ϕ∗|2dx ≤ θ2µ′
−
∫
B2/3(0)∩O/r∗
|ϕ∗|2dx, (5.20)
where µ′ ∈ (µ, 1). We choose θ1, θ2 ∈ (0, 2/3) such that θ1 < θ22 and (5.20) holds if
θ ∈ [θ1, θ2]. Now we claim (5.18). If not, there is a sequence ωǫ, rǫ, θǫ, ϕǫ satisfying
(5.17) and
ωǫ, rǫ → ω∗, r∗ ∈ [0, 1]
θǫ → θ∗ ∈ [θ1, θ2]
−
∫
Bθǫ(0)∩O/rǫ
|ϕǫ|2dx > θ2µ
ǫ
as ǫ/rǫ → 0. (5.21)
By Lemma 5.1 and tracing the proof of Theorem 2.3 [3], there is a subsequence
(same notation for subsequence) such thatϕǫ → ϕ∗ in L2(B2/3(0) ∩ O/r∗) strongly
Kω2ǫ ,ǫ,rǫ
∇ϕǫ → Kω∗∇ϕ∗ in L2(B2/3(0) ∩ O/r∗) weakly
as ǫ/rǫ → 0. (5.22)
Also the ϕ∗ above is a solution of (5.19). By (5.20)–(5.22), we conclude
θ2µ∗ = lim
ǫ/rǫ→0θ2µ
ǫ ≤ limǫ/rǫ→0
−
∫
Bθǫ(0)∩O/rǫ
|ϕǫ|2dx
= −
∫
Bθ∗(0)∩O/r∗
|ϕ∗|2dx ≤ θ2µ′
∗ −
∫
B2/3(0)∩O/r∗
|ϕ∗|2dx. (5.23)
But (5.23) is impossible if θ2 is small enough. So there is a ǫ0 such that (5.18) holds
for ǫ/r < ǫ0.
Lemma 5.7. For any δ > 0, there are constants θ1, θ2 ∈ (0, 1) with θ1 < θ22, and a
constant ǫ0 ∈ (0, 1) (depending on δ, θ2, Yf , ‖∇Ψ‖L∞(Rn−1)) such that if−∇ · (Kω2,ǫ,1∇Φ) = 0 in B1(0) ∩ O,
Φ = 0 on B1(0) ∩ ∂O,(5.24)
then, for any ω ∈ (0, 1], ǫ ∈ (0, ǫ0], θ ∈ [θ1, θ2], and k satisfying ǫ/θk ≤ ǫ0,
−
∫
Bθk (0)∩O
|Φ|2dx ≤ θ2kµ|Jω,ǫ|2, (5.25)
July 9, 2015
20 Elliptic and parabolic equations
where µ ≡ δn+δ and Jω,ǫ ≡ ‖Kω,ǫ,1Φ‖L2(B1(0)∩O).
Proof. The proof is done by induction on k. For k = 1, we set ϕ ≡ Φ/Jω,ǫ. Then
(5.25) is deduced from Lemma 5.6 with r = 1. Suppose (5.25) holds for some k
satisfying ǫ/θk ≤ ǫ0, then we define
ϕ(x) ≡ J−1ω,ǫθ
−kµΦ(θkx) in B1(0) ∩ O/θk.
Then ϕ satisfies (5.17) with r = θk. By changing variable and employing Lemma
5.6 with r = θk, we obtain (5.25) with k + 1 in place of k.
Lemma 5.8. For any δ > 0, there is a constant ǫ∗ ∈ (0, 1) (depending on δ, Yf ,
‖∇Ψ‖L∞(Rn−1)) such that if ω ∈ (0, 1] and ǫ ∈ (0, ǫ∗], then any solution of (5.24)
satisfies
[Φ]C0,µ(B1/2(0)∩O) ≤ c‖Kω,ǫ,1Φ‖L2(B1(0)∩O), (5.26)
where c is a constant independent of ω, ǫ. See Lemma 5.7 for µ.
Proof. Let θ1, θ2, ǫ0, Jω,ǫ be those in Lemma 5.7 and define ǫ∗ ≡ minǫ0θ2/3, ǫ∗
where ǫ∗ is the one in Lemma 5.5. Denote by c a constant independent of ω, ǫ. For
any x ∈ Bθ2/3(0) ∩ O, define ξ(x) ≡ |x − x0| where x0 ∈ ∂O satisfying |x − x0| =
miny∈∂O |x− y|. Then we have either case (1) ξ(x) > 2ǫ3ǫ0
or case (2) ξ(x) ≤ 2ǫ3ǫ0
.
Let us consider case (1). Because of θ1 < θ22, for any r ∈ [ǫ/ǫ0, θ2], there are
θ ∈ [θ1, θ2] and k ∈ N satisfying r = θk. Since ξ(x) ∈ [ 2ǫ3ǫ0, θ2
3 ], by Lemma 5.7,
−
∫
Br(x0)∩O
|Φ|2dy ≤ r2µ|Jω,ǫ|2 for r ∈ [32ξ(x), θ2].
So, for s ∈ [ ξ(x)2 , θ2
3 ],
−
∫
Bs(x)∩O
∣∣Φ − (Φ)Bs(x)∩O
∣∣2 dy ≤ cs2µ|Jω,ǫ|2. (5.27)
Next we move the origin of the coordinate system to x and define
ϕ(y) ≡ J−1ω,ǫξ
−µ(x)(Φ(ξ(x)y) − (Φ)Bξ(x)(x)
)in B1(x).
Then ϕ satisfies
−∇ · (Kω2,ǫ/ξ(x),1∇ϕ) = 0 in B1(x). (5.28)
Take s = ξ(x) < 1 in (5.27) to see ‖ϕ‖L2(B1(x)) ≤ c. Apply Remark 5.1 to (5.28) to
obtain [ϕ]C0,µ(B1/2(x)) ≤ c. Which implies
−
∫
Bs(x)
∣∣Φ − (Φ)Bs(x)
∣∣2 dy ≤ cs2µ|Jω,ǫ|2 for s < ξ(x)
2 . (5.29)
July 9, 2015
Elliptic and parabolic equations 21
Next we consider case (2). Because of θ1 < θ22 , for any r ∈ [ǫ/ǫ0, θ2], there are
θ ∈ [θ1, θ2] and k ∈ N satisfying r = θk. By Lemma 5.7,
−
∫
Br(x0)∩O
|Φ|2dy ≤ r2µ|Jω,ǫ|2 for r ∈ [ǫ/ǫ0, θ2]. (5.30)
This implies, for s ∈ [ ǫ3ǫ0, θ2
3 ],
−
∫
Bs(x)∩O
∣∣Φ − (Φ)Bs(x)∩O
∣∣2 dy ≤ cs2µ|Jω,ǫ|2. (5.31)
Again we move the origin of the coordinate system to x and define
ϕ(y) ≡ J−1ω,ǫǫ
−µ(Φ(ǫy) − (Φ)Bǫ/ǫ0
(x)∩O
)in B1/ǫ0(x) ∩ O/ǫ.
Then ϕ satisfies−∇ · (Kω2,ǫ,ǫ∇ϕ) = 0 in B1/ǫ0(x) ∩ O/ǫ,
ϕ = −J−1ω,ǫǫ
−µ(Φ)Bǫ/ǫ0(x)∩O on B1/ǫ0(x) ∩ ∂O/ǫ.
Let us take s = ǫ/ǫ0 in (5.31) to see ‖ϕ‖L2(B1/ǫ0(x)∩O/ǫ) ≤ c and take s = ǫ/ǫ0 in
(5.30) to see |J−1ω,ǫǫ
−µ(Φ)Bǫ/ǫ0(x)∩O| ≤ c. By (3.5)1 of Lemma 3.5,
[ϕ]C0,µ(B1/2ǫ0(x)∩O/ǫ) ≤ c. (5.32)
(5.32) implies that (5.31) holds for s ≤ ǫ2ǫ0
.
The Holder estimate of Φ follows from (5.27), (5.29), (5.31), (5.32), and Theorem
1.2 in page 70 [11].
Remark 5.2. Let ǫ∗ be same as that in Lemma 5.8. By (3.5)1 of Lemma 3.5, we
know that if ω ∈ (0, 1] and ǫ ∈ [ǫ∗, 1], any solution of (5.24) satisfies (5.26). Together
with Lemma 5.8, any solution of (5.24) satisfies (5.26) if ω, ǫ ∈ (0, 1].
Remark 5.1, Remark 5.2, and maximal principle imply Lemma 5.2.
5.2. Lipschitz estimate
A1–A2 are assumed and we show Lemma 3.7. The interior estimate of (3.26) is in
subsection 5.2.1 and the boundary estimate of (3.26) is in subsection 5.2.2.
5.2.1. Interior gradient estimate
We assume B1(0) ⊂ Ω.
Lemma 5.9. There exist θ, ǫ0 ∈ (0, 1) such that if−∇ · (Kω2,ν∇ϕ) = 0 in B1(0),
‖Kω,νϕ‖L2(B1(0)) ≤ 1,(5.33)
July 9, 2015
22 Elliptic and parabolic equations
then, for any ω ∈ (0, 1] and ν ∈ (0, ǫ0),
supBθ(0)
|ϕ(x) − ϕ(0) − (x+ Xω,ν(x))bω,ν | ≤ θ4/3, (5.34)
where bω,ν ≡ K−1ω −
∫
Bθ(0)
Kω2,ν∇ϕ dx and K−1ω is the inverse matrix of Kω. See
(3.23)–(3.24) for Xω,ν ,Kω.
Proof. Assume −∇ · (Kω∗∇ϕ∗) = 0 in B2/3(0), where Kω∗
for ω∗ ∈ [0, 1] is from
(3.24). If θ is small enough, by (3.25) and Taylor expansion,
supBθ(0)
∣∣ϕ∗(x) − ϕ∗(0) − x(∇ϕ∗)Bθ(0)
∣∣ ≤ θ3/2‖ϕ∗‖L2(B2/3(0)). (5.35)
Fix a small θ so that (5.35) holds and we prove (5.34) by contradiction. If not, there
is a sequence ων, ϕν satisfying (5.33) and, as ν → 0,ων → ω∗ ∈ [0, 1],
supBθ(0)
|ϕν(x) − ϕν(0) − (x+ Xω,ν(x))bω,ν | > θ4/3. (5.36)
By Lemma 5.1 and Lemma 5.2 and by tracing the proof of Theorem 2.3 [3], there
is a subsequence (same notation for subsequence) such thatϕν → ϕ∗ in C(B2/3(0))
Kω2ν ,ν∇ϕν → Kω∗
∇ϕ∗ in L2(B2/3(0)) weaklyas ν → 0. (5.37)
(5.37) implies that ϕ∗ satisfies −∇·(Kω∗∇ϕ∗) = 0 in B2/3(0). Together with (5.35),
(5.36), and (5.37), we get contradiction if θ is small. So we prove (5.34).
Lemma 5.10. There exist θ, ǫ0 ∈ (0, 1) such that if Φ satisfies
−∇ · (Kω2,ǫ∇Φ) = 0 in B1(0), (5.38)
then, for any ω ∈ (0, 1], ǫ ∈ (0, ǫ0), and k satisfying ǫ/θk ≤ ǫ0, there are constants
aω,ǫk , b
ω,ǫk satisfying|aω,ǫ
k | + |bω,ǫk | ≤ cJω,ǫ,
supB
θk (0)
|Φ(x) − Φ(0) − ǫaω,ǫk − (x + Xω,ǫ(x))b
ω,ǫk | ≤ θ4k/3Jω,ǫ,
(5.39)
where Jω,ǫ ≡ ‖Kω,ǫΦ‖L2(B1(0)) and c is independent of ω, ǫ.
Proof. If ϕ ≡ Φ/Jω,ǫ, then it satisfies (5.33) with ν = ǫ. By Lemma 5.9, we obtain
(5.39) for k = 1 where aω,ǫ1 = 0, b
ω,ǫ1 = K−1
ω −
∫
Bθ(0)
Kω2,ǫ∇Φ dx. If (5.39) holds for
some k satisfying ǫ/θk ≤ ǫ0, we define
ϕ(x) ≡Φ(θkx) − Φ(0) − ǫaω,ǫ
k −(θkx+ Xω,ǫ(θ
kx))b
ω,ǫk
θ4k/3Jω,ǫ
in B1(0).
July 9, 2015
Elliptic and parabolic equations 23
By induction and (3.22), we see ϕ satisfies (5.33) with ν = ǫ/θk. Apply Lemma 5.9
to obtain
supBθ(0)
∣∣∣ϕ(x) − ϕ(0) −(x+ Xω,ǫ/θk(x)
)bω,ǫ/θk
∣∣∣ ≤ θ4/3, (5.40)
where bω,ǫ/θk ≡ K−1ω −
∫
Bθ(0)
Kω2,ǫ/θk∇ϕ dx. (5.40) can be written as
supBθ(0)
∣∣∣Φ(θkx) − Φ(0) + ǫXω,1(0)bω,ǫk −
(θkx+ Xω,ǫ(θ
kx))b
ω,ǫk
−Jω,ǫθ4k/3
(x+ θ−kXω,ǫ(θ
kx))bω,ǫ/θk
∣∣∣ ≤ Jω,ǫθ4(k+1)/3. (5.41)
Define
aω,ǫk+1 ≡ −Xω,1(0)bω,ǫ
k and bω,ǫk+1 ≡ b
ω,ǫk + Jω,ǫθ
k/3bω,ǫ/θk . (5.42)
By energy method, we know that |bω,ǫ/θk | is bounded uniformly in ω, ǫ, k. So (5.39)1holds. Substituting (5.42) into (5.41) and changing variables, we obtain (5.39)2 for
k + 1 case.
Lemma 5.11. There exists ǫ0 ∈ (0, 1) such that if ω ∈ (0, 1] and ǫ ∈ (0, ǫ0), any
solution of (5.38) satisfies
‖∇Φ‖L∞(B1/2(0)) ≤ c‖Kω,ǫΦ‖L2(B1(0)), (5.43)
where c is a constant independent of ω, ǫ.
Proof. Let Jω,ǫ be that in Lemma 5.10; c denote a constant independent of ω, ǫ;
and k ∈ N satisfying ǫ/θk ≤ ǫ0 < ǫ/θk+1. By Lemma 5.10,
supB ǫ
ǫ0(0)
|Φ(x) − Φ(0) − ǫaω,ǫk − (x+ Xω,ǫ(x))b
ω,ǫk | ≤ c
∣∣ ǫǫ0
∣∣4/3Jω,ǫ.
Define
ϕ(x) ≡Φ(ǫx) − Φ(0) − ǫaω,ǫ
k −(ǫx+ Xω,ǫ(ǫx)
)b
ω,ǫk
ǫ4/3Jω,ǫ
in B 1ǫ0
(0).
Then ϕ satisfies−∇ · (Kω2,1∇ϕ) = 0 in B1/ǫ0(0),
‖ϕ‖L∞(B1/ǫ0(0)) ≤ c.
(3.5)2 of Lemma 3.5 implies
‖ϕ‖C1,0(B1/2ǫ0(0)∩Ωǫ
f /ǫ)∩C1,0(B1/2ǫ0(0)∩Ωǫ
m/ǫ) ≤ c. (5.44)
Since ∇ϕ(x) =∇Φ(ǫx)−(I+∇Xω,1(x))bω,ǫ
k
ǫ1/3Jω,ǫ, |∇Φ(ǫx)| ≤ cJω,ǫ for x ∈ B1/2ǫ0(0) by
(3.23), (5.44), and Lemma 5.10. We prove (5.43).
Remark 5.3. Let ǫ0 be same as that in Lemma 5.11. By (3.5)2 of Lemma 3.5, we
know that if ω ∈ (0, 1] and ǫ ∈ [ǫ∗, 1], any solution of (5.38) satisfies (5.43). Together
with Lemma 5.11, any solution of (5.38) satisfies (5.43) if ω, ǫ ∈ (0, 1].
July 9, 2015
24 Elliptic and parabolic equations
5.2.2. Boundary gradient estimate
Assume 0 ∈ ∂Ω and (3.2). Let Qd(0) ≡ Πni=1[−di, di] denote a cube, where d =
(d1, · · · , dn), di ∈ [3, 4]. Find a smooth cut-off function ρ ∈ C∞0 (Qd(0)) such that
ρ ∈ [0, 1] and ρ = 1 in Q2(0). For any ω, ǫ, r ∈ (0, 1] and ǫ ≤ r, we find W(n)ω,ǫ,r ∈
H1(Qd(0) ∩ Ω/r) by solving−∇ ·
(Kω2,ǫ,r
(∇W
(n)ω,ǫ,r + ~en
))= 0 in Qd(0) ∩ Ω/r,
W(n)ω,ǫ,r =
(1 − ρ
)X
(n)ω,ǫ/r on ∂(Qd(0) ∩ Ω/r),
(5.45)
where ~en is the unit vector in the n-th coordinate direction. See (3.22) for X(n)ω,ǫ/r
and (3.3) for Kω2,ǫ,r. We adjust the d of Qd(0) so that if ǫ(Ym + j) ⊂ Ωǫm for any
j ∈ Zn, then |Qd(0) ∩ ǫr (Y + j)| is either 0 or | ǫ
r |n. Define
D ≡ Qd(0) ∩ Ω/r,
D∗ ≡⋃
j∈Znǫr
(Ym+j)⊂Qd(0)∩Ωǫm/r
ǫr (Y + j). (5.46)
From (5.46)1, the definition of Ωǫm, and A1, we see
ρ = 0 on ∂D ∩ ∂D∗,
D \ D∗ ⊂ Ωǫf/r,
D \ D∗ ⊂ x ∈ D : dist(x, ∂Ω/r) ≤ c ǫr,
D is a simply-connected semiconvex domain,
(5.47)
where c is a constant independent of ǫ, r. See the beginning of subsection 5.1 for a
semiconvex domain. Let Gǫ,r(x, y) denote the Green’s function of−∇y ·
(Kω2,ǫ,r∇yGǫ,r(x, ·)
)= δ(x, ·) in D,
Gǫ,r(x, ·) = 0 on ∂D.(5.48)
By [16], Gǫ,r(x, ·) ∈W 1,1(D) exists uniquely. Next we give a local L∞ estimate.
Lemma 5.12. If x∗ ∈ D, ω, ǫ, r ∈ (0, 1], ǫ ≤ r, and t > 0, any solution of−∇ · (Kω2,ǫ,r∇ϕ) = 0 in Bt(x
∗) ∩ D
ϕ = 0 on Bt(x∗) ∩ ∂D
(5.49)
satisfies
∣∣∣Kω,ǫ,rϕ∣∣∣ (x∗) ≤ c
∣∣∣∣ −∫
Bt(x∗)∩D
|Kω,ǫ,rϕ(y)|2dy
∣∣∣∣1/2
, (5.50)
for some constant c independent of ω, ǫ, r, x∗, t.
Proof. First we assume x∗ = 0 ∈ D and define ϕ(y) = ϕ(ty). Then (5.49) implies−∇ · (Kω2,ǫ,rt∇ϕ) = 0 in B1(0) ∩ D/t,
ϕ = 0 on B1(0) ∩ ∂D/t.
July 9, 2015
Elliptic and parabolic equations 25
Note ǫrt ≤ 1 or 1 < ǫ
rt . If ǫrt ≤ 1 (resp. 1 < ǫ
rt), Lemma 5.2 and (5.47)4 (resp.
Theorem 7.26 [12] and Lemma 3.6) imply
‖Kω,ǫ,rtϕ‖L∞(B1/4(0)∩D/t) ≤ c‖Kω,ǫ,rtϕ‖L2(B1(0)∩D/t), (5.51)
where c is a constant independent of ǫ, ω, r, t. By (5.51),
|Kω,ǫ,rϕ(0)| ≤ c
∣∣∣∣∫
B1(0)∩D/t
|Kω,ǫ,rtϕ(y)|2dy
∣∣∣∣12
≤ c
∣∣∣∣ −∫
Bt(0)∩D
|Kω,ǫ,rϕ(y)|2dy
∣∣∣∣12
.
So (5.50) holds for x∗ = 0 case. If x∗ 6= 0, we shift x∗ to the origin of the coordinate
system and repeat the above argument to obtain (5.50).
Lemma 5.13. Let ω, ǫ, r ∈ (0, 1], s ∈ (0, 1), ǫ ≤ r, and n ≥ 3. There is a constant
c independent of ω, ǫ, r, s such that, for any x, y ∈ D,
∣∣Gǫ,r(x, y)∣∣ ≤ c|x− y|2−nK1/ω,ǫ,r(x)K1/ω,ǫ,r(y),
|Gǫ,r(x, y)| ≤ c|ξr(x)|s|x− y|2−n−sK1/ω,ǫ,r(x)K1/ω,ǫ,r(y),
|Gǫ,r(x, y)| ≤ c|ξr(x)|s|ξr(y)|
s|x− y|2−n−2sK1/ω,ǫ,r(x)K1/ω,ǫ,r(y),
(5.52)
where ξr(x) (resp. ξr(y)) denotes the distance from x (resp. y) to the boundary
∂Ω/r.
Proof. Let c be a constant independent of ω, ǫ, r, s and set t ≡ |x− y|.
Proof of (5.52)1. Take F ∈ C∞0 (Bt/3(y) ∩D), and find ϕ ∈ H1(D) satisfying
−∇ · (Kω2,ǫ,r∇ϕ) = Kω,ǫ,rF in D,
ϕ = 0 on ∂D.
Note ϕ is solvable uniquely in H1(D). By Theorem 4.31 [2] and Theorem 2.1 [1],
‖Kω,ǫ,rϕ‖L
2nn−2 (D)
≤ c‖Kω,ǫ,r∇ϕ‖L2(D). (5.53)
By [16] and Lemma 5.12,
ϕ(x) =
∫
Bt/3(y)∩D
Gǫ,r(x, z)Kω,ǫ,r(z)F (z)dz,
|Kω,ǫ,rϕ(x)| ≤ c
∣∣∣∣ −∫
Bt/3(x)∩D
|Kω,ǫ,rϕ|2dz
∣∣∣∣12
≤ c
∣∣∣∣ −∫
Bt/3(x)∩D
∣∣Kω,ǫ,rϕ∣∣ 2n
n−2 dz
∣∣∣∣n−22n
.
(5.54)
(5.53)–(5.54) imply∣∣∣∣∫
Bt/3(y)∩D
Gǫ,r(x, z)Kω,ǫ,r(z)F (z)dz
∣∣∣∣
≤ cK1/ω,ǫ,r(x)
∣∣∣∣ −∫
Bt/3(x)∩D
∣∣Kω,ǫ,rϕ∣∣ 2n
n−2 dz
∣∣∣∣n−22n
≤ ct2−n
2 K1/ω,ǫ,r(x)‖Kω,ǫ,r∇ϕ‖L2(D) ≤ ct4−n
2 K1/ω,ǫ,r(x)‖F‖L2(Bt/3(y)∩D). (5.55)
July 9, 2015
26 Elliptic and parabolic equations
(5.55) and Lemma 5.12 imply
∣∣Gǫ,r(x, y)∣∣ ≤ cK1/ω,ǫ,r(y)
∣∣∣∣ −∫
Bt/3(y)∩D
|Kω,ǫ,r(z)Gǫ,r(x, z)|2dz
∣∣∣∣12
≤ ct2−nK1/ω,ǫ,r(x)K1/ω,ǫ,r(y).
So (5.52)1 is proved.
Proof of (5.52)2. By (5.52)1, it is enough to show (5.52)2 for the case ξr(x) < t/6.
By (5.52)1,
|Gǫ,r(x, y)| ≤ c|x− y|2−nK1/ω,ǫ,r(x)K1/ω,ǫ,r(y)
for all x satisfying |x− x| < t/3. Applying Lemma 3.6 and Lemma 5.2 to Gǫ,r(·, y)
in Bt/3(x) ∩ D, we obtain
|Gǫ,r(x, y)| ≤c|ξr(x)|
s
|x− y|n−2+sK1/ω,ǫ,r(x)K1/ω,ǫ,r(y) for x ∈ Bt/6(x) ∩ D.
(5.52)2 follows by setting x = x. (5.52)3 is obtained by (5.48), (5.52)2, Lemma 5.2,
and a similar argument as that for (5.52)2.
Lemma 5.14. Solution of (5.45) exists uniquely in H1(D). For any ω, ǫ, r ∈ (0, 1]
and ǫ ≤ r, there is a constant c independent of ω, ǫ, r such that
|W(n)ω,ǫ,r(x)| ≤ cǫ/r for x ∈ D.
Proof. Let c denote a constant independent of ω, ǫ, r.
Step 1. Unique existence of a solution of (5.45) in H1(D) is clear. If we define
Y(n)ω,ǫ,r ≡ W
(n)ω,ǫ,r − X
(n)ω,ǫ/r in D∗ (see (5.46)), then
−∇ ·
(Kω2,ǫ,r∇Y
(n)ω,ǫ,r
)= 0 in D∗,
Y(n)ω,ǫ,r = W
(n)ω,ǫ,r − X
(n)ω,ǫ/r on ∂D∗.
By Theorem 8.1 [12], (3.23), and (5.47)1,
supD∗
|W(n)ω,ǫ,r| ≤ cǫ/r + sup
∂D∗\∂D
|W(n)ω,ǫ,r|. (5.56)
Next we want to derive
|W(n)ω,ǫ,r(x)| ≤ cǫ/r for x ∈ D \ D∗. (5.57)
If so, together with (5.56), the lemma is proved.
Step 2. Define ν ≡ ǫ/r. Suppose G(x, y) is the Green’s function of−∇y ·
(Kω2,ǫ,ǫ∇G(x, ·)
)= δ(x, ·) in D/ν,
G(x, ·) = 0 on ∂D/ν,
it is easy to see
G(x, y) = νn−2Gǫ,r(νx, νy). (5.58)
July 9, 2015
Elliptic and parabolic equations 27
We claim, for any s ∈ (0, 1), there is a constant c independent of ǫ, ω, r, s such that
G(x, y) ≤ c|ξǫ(x)|s
|x−y|n−2+s K1/ω,ǫ,ǫ(x)K1/ω,ǫ,ǫ(y),
G(x, y) ≤ c|ξǫ(x)|s|ξǫ(y)|s
|x−y|n−2+2s K1/ω,ǫ,ǫ(x)K1/ω,ǫ,ǫ(y),
|∇yG(x, y)| ≤ c |ξǫ(x)|s
|x−y|n−1+s K1/ω,ǫ,ǫ(x)K1/ω,ǫ,ǫ(y) for |x− y| ≤ 1,
|∇yG(x, y)| ≤ c |ξǫ(x)|s|ξǫ(y)|s
|x−y|n−2+2s K1/ω,ǫ,ǫ(x)K1/ω,ǫ,ǫ(y) for |x− y| > 1,
(5.59)
where ξǫ(x) is the distance from x to the boundary ∂Ω/ǫ. By (5.58) and (5.52)2,
G(x, y) = νn−2Gǫ,r(νx, νy) ≤cνn−2|ξr(νx)|
s
νn−2+s|x− y|n−2+sK1/ω,ǫ,r(νx)K1/ω,ǫ,r(νy)
≤c|ξǫ(x)|
s
|x− y|n−2+sK1/ω,ǫ,ǫ(x)K1/ω,ǫ,ǫ(y). (5.60)
So we obtain (5.59)1. Similarly, by (5.52)3, we have (5.59)2. If t = |x − y| ≤ 1,
Lemma 3.6 and (5.60) imply
‖∇yG(x, ·)‖L∞(Bt/2(y)∩D/ν) ≤c
t‖G(x, ·)‖L∞(B3t/4(y)∩D/ν)
≤c|ξǫ(x)|
s
|x− y|n−1+sK1/ω,ǫ,ǫ(x)K1/ω,ǫ,ǫ(y).
So (5.59)3 holds. (5.59)4 follows from (3.5)2 and (5.59)2.
Step 3. We claim (5.57). The solution of (5.45) can be written as W(n)ω,ǫ,r =
X(n)ω,ǫ/r + U1 + U2, where U1 is the solution of
−∇ ·
(Kω2,ǫ,r∇U1
)= 0 in D,
U1 = −ρX(n)ω,ǫ/r on ∂D,
and U2 is the solution of
−∇ ·
(Kω2,ǫ,r(∇U2 + ∇X
(n)ω,ǫ/r + ~en)
)= 0 in D,
U2 = 0 on ∂D.
By (3.23) and maximal principle [12], we see
‖X(n)ω,ǫ/r‖L∞(D) + ‖U1‖L∞(D) ≤ cǫ/r. (5.61)
Set ν ≡ ǫ/r and D/ν(ℓ) ≡ x ∈ D/ν : dist(x, ∂Ω/ǫ) ≤ ℓ for ℓ > 2. By (5.47)3,
there is a ℓ∗ so that (D \ D∗)/ν ⊂ D/ν(ℓ∗). Find η ∈ C∞(D/ν) with support in
˜D/ν(ℓ∗ + 1) so that η ∈ [0, 1], η = 1 in D/ν(ℓ∗), and ‖∇η‖L∞(D/ν) ≤ c. By (3.23),
July 9, 2015
28 Elliptic and parabolic equations
(5.47)2, and (5.59), for any x ∈ (D \ D∗)/ν and s ∈ (12 , 1),
U2(νx) = −
∫
D/ν
∇yG(x, y)Kω2,ǫ,ǫν(∇X(n)ω,1(y) + ~en)dy
= −
∫
D/ν
∇yG(x, y)(1 − η(y))Kω2,ǫ,ǫν(∇X(n)ω,1(y) + ~en)dy
−
∫
D/ν
∇yG(x, y)η(y)Kω2,ǫ,ǫν(∇X(n)ω,1(y) + ~en)dy
= −
∫
˜D/ν(ℓ∗+1)
G(x, y)∇η(y)Kω2,ǫ,ǫν(∇X(n)ω,1(y) + ~en)dy
−
∫
˜D/ν(ℓ∗+1)
∇yG(x, y)η(y)Kω2,ǫ,ǫν(∇X(n)ω,1(y) + ~en)dy
≤ c
∫
˜D/ν(ℓ∗+1)∩|x−y|≤1
ν|ξǫ(x)|s
|x− y|n−1+sdy
+c
∫
˜D/ν(ℓ∗+1)∩|x−y|>1
ν|ξǫ(x)|s
|x− y|n−2+2sdy ≤ cǫ/r. (5.62)
(5.62) implies ‖U2‖L∞(D\D∗) ≤ cǫ/r. Together with (5.61), we obtain (5.57). So we
prove the lemma.
Lemma 5.15. Let θ, ǫ0 be those in Lemma 5.9. There exist constants θ, ǫ0 ∈ (0, 1)
satisfying θ < θ, ǫ0 < ǫ0 such that if−∇ · (Kω2,ǫ,r∇ϕ) = 0 in B1(0) ∩ Ω/r,
ϕ = ϕb on B1(0) ∩ ∂Ω/r,(5.63)
and ifϕb(0) = ∂Tϕb(0) = 0,
‖Kω,ǫ,rϕ‖L2(B1(0)∩Ω/r), [ϕb]C1,α(B1(0)∩Ω/r) ≤ 1,
then, for any ω, ǫ, r ∈ (0, 1] and ǫ/r ≤ ǫ0,
supBθ(0)∩Ω/r
∣∣∣ϕ(x) −(xn + W(n)
ω,ǫ,r(x))dω,ǫ,r
∣∣∣ ≤ θ1+τ ,
where α ∈ (0, 1), τ = α2 , ∂Tϕb(0) is the tangential derivative of ϕb at 0, dω,ǫ,r is
the n-th component of K−1ω −
∫
Bθ(0)∩Ω/r
Kω2,ǫ,r∇ϕ dx, and K−1ω is the inverse matrix
of Kω (see (3.24)).
Proof. The proof is similar to that of Lemma 5.9. Let r∗, ω∗ ∈ [0, 1], (ϕ∗, ϕb,∗)
satisfy−∇ · (Kω∗
∇ϕ∗) = 0 in B2/3(0) ∩ Ω/r∗,
ϕ∗ = ϕb,∗ on B2/3(0) ∩ ∂Ω/r∗,
July 9, 2015
Elliptic and parabolic equations 29
and ϕb,∗ is smooth with ϕb,∗(0) = ∂Tϕb,∗(0) = 0. By (3.25) and Taylor expansion,
there exist θ ∈ (0, 23 ) and τ ′ satisfying τ < τ ′ < α such that
supBθ(0)∩Ω/r∗
∣∣ϕ∗ − xn(∂nϕ∗)Bθ(0)∩Ω/r∗
∣∣
≤ θ1+τ ′(‖ϕ∗‖L∞(B2/3(0)∩Ω/r∗) + [ϕb,∗]C1,α(B2/3(0)∩Ω/r∗)
). (5.64)
If we fix a small θ ∈ (0, 1) so that (5.64) holds, the conclusion will follow by con-
tradiction provided we prove limǫ/r→0 ‖W(n)ω,ǫ,r‖L∞(B1(0)∩Ω/r) = 0. But that is the
result of Lemma 5.14. So we prove this lemma.
Lemma 5.16. θ, ǫ0, τ are same as those in Lemma 5.15. If−∇ · (Kω2,ǫ∇Φ) = 0 in B1(0) ∩ Ω,
Φ = 0 on B1(0) ∩ ∂Ω,(5.65)
then, for any ω ∈ (0, 1], ǫ ∈ (0, ǫ0), and k satisfying ǫ/θk ≤ ǫ0, there is a constant
dω,ǫk satisfying
|dω,ǫk | ≤ cJω,ǫ,
supB
θk (0)∩Ω
∣∣∣∣Φ −k−1∑
j=0
θτj(xn + θjW
(n)
ω,ǫ,θj(θ−jx)
)d
ω,ǫj
∣∣∣∣ ≤ θk(1+τ)Jω,ǫ,(5.66)
where Jω,ǫ ≡ ‖Kω,ǫΦ‖L2(B1(0)∩Ω) and c is a constant independent of ω, ǫ.
Proof. This is done by induction on k. When k = 1, (5.66) holds by Lemma 5.15
with r = 1. dω,ǫ0 is the n-th component of K−1
ω −
∫
Bθ(0)∩Ω
Kω2,ǫ∇Φdx. Suppose (5.66)
holds for some k satisfying ǫ/θk ≤ ǫ0, we define, in B1(0) ∩ Ω/θk,
ϕ(x) ≡ J−1ω,ǫθ
−k(1+τ)
(Φ(θkx) −
k−1∑
j=0
θτj(θkxn + θjW
(n)
ω,ǫ,θj(θk−jx)
)d
ω,ǫj
),
ϕb(x) ≡ −J−1ω,ǫθ
−k(1+τ)k−1∑
j=0
θτj θkdω,ǫj xn.
Then those functions satisfy (5.63) with r = θk. Note [ϕb]C1,α(B1(0)∩Ω/θk)= 0. By
induction,
max‖ϕ‖L∞(B1(0)∩Ω/θk), [ϕb]C1,α(B1(0)∩Ω/θk)
≤ 1. (5.67)
Apply Lemma 5.15,
supBθ(0)∩Ω/θk
∣∣∣∣ϕ(x) −(xn + W
(n)
ω,ǫ,θk(x)
)dω,ǫ,θk
∣∣∣∣ ≤ θ1+τ , (5.68)
July 9, 2015
30 Elliptic and parabolic equations
where dω,ǫ,θk is the n-th component of K−1ω −
∫
Bθ(0)∩Ω/θk
Kω2,ǫ,θk∇ϕ dx. By energy
method and (5.67), |dω,ǫ,θk | is bounded uniformly in ω, ǫ, θk. Rewrite (5.68) in terms
of Φ in Bθk+1(0) to obtain
supB
θk+1 (0)∩Ω
∣∣∣∣Φ(x) −
k−1∑
j=0
θτj(xn + θjW
(n)
ω,ǫ,θj(θ−jx)
)d
ω,ǫj
−θkτ Jω,ǫ
(xn + θkW
(n)
ω,ǫ,θk(θ−kx)
)dω,ǫ,θk
∣∣∣∣ ≤ θ(k+1)(1+τ)Jω,ǫ.
If dω,ǫk ≡ Jω,ǫdω,ǫ,θk, we conclude that (5.66) holds for k + 1.
Lemma 5.17. Let ǫ0 be same as that in Lemma 5.16. Suppose ω ∈ (0, 1] and
ǫ ∈ (0, ǫ0), any solution of (5.65) satisfies
‖∇Φ‖L∞(B1/2(0)∩Ω) ≤ c‖Kω,ǫΦ‖L2(B1(0)∩Ω), (5.69)
where c is a constant independent of ω, ǫ.
Proof. By (3.2), we have a local coordinate x = (x′, xn) so that
B1(0) ∩ Ω =(x′, xn) ∈ Ω : |x′|2 + |xn|
2 < 1,Ψ(x′) < xn
.
To obtain the Lipschitz estimate in (5.69), it is suffice to show
sup(0,xn)∈B1/2(0)∩Ω
|∇Φ(0, xn)| ≤ c‖Kω,ǫΦ‖L2(B1(0)∩Ω). (5.70)
The reason is that one can repeat the same argument by varying the origin along
the boundary B1(0) ∩ ∂Ω and by adjusting the constant c to obtain the estimate.
Let θ, Jω,ǫ, τ are same as those in Lemma 5.16, let c be a constant independent
of ω, ǫ, and let k satisfy ǫ/θk ≤ ǫ0 < ǫ/θk+1. For any x ≡ (0, xn) ∈ B1/2(0) ∩ Ω, we
have either case (1): 12 θ
ℓ < xn ≤ 12 θ
ℓ−1 for 1 ≤ ℓ ≤ k or case (2): 0 ≤ xn ≤ 12 θ
k.
For case (1): By Lemma 5.16, we have
supB
θℓ−1 (0)∩Ω
∣∣∣∣Φ(y) −
ℓ−2∑
j=0
θτj(yn + θjW
(n)
ω,ǫ,θj(θ−jy)
)d
ω,ǫj
∣∣∣∣ ≤ cθℓ(1+τ)Jω,ǫ. (5.71)
Hence, by Lemma 5.14 and (5.71),
supB
θℓ−1 (0)∩Ω
|Φ| ≤ cJω,ǫ
(θℓ(1+τ) + (ξ(x) + ǫ)
ℓ−2∑
j=0
θτj
)≤ cξ(x)Jω,ǫ. (5.72)
Here we use ξ(x) ≡ |x−x0| where x0 ∈ ∂Ω so that |x−x0| = miny∈∂Ω |x− y|. Note
ǫ ≤ ǫ0θk ≤ 2ǫ0
12 θ
ℓ ≤ 2ǫ0xn ≤ cǫ0ξ(x). By (5.72),
supBξ(x)/2(x)
|Φ| ≤ cξ(x)Jω,ǫ. (5.73)
July 9, 2015
Elliptic and parabolic equations 31
Then we move the origin of the coordinate system to x and define
ϕ(y) ≡ ξ(x)−1J−1ω,ǫΦ(ξ(x)y).
By (5.73),
‖ϕ‖L∞(B1/2(x)) ≤ c.
Function ϕ satisfies
−∇ · (Kω2,ǫ/ξ(x)∇ϕ) = 0 in B1/2(x).
By Lemma 5.11,
‖∇Φ‖L∞(Bξ(x)/4(x)) ≤ cJω,ǫ.
This proves (5.70) for case (1).
For case (2): Apply Lemma 5.16 to obtain
supB
θk (0)∩Ω
∣∣∣∣Φ(y) −
k−1∑
j=0
θτj(yn + θjW
(n)
ω,ǫ,θj(θ−jy)
)d
ω,ǫj
∣∣∣∣ ≤ cJω,ǫθk(1+τ).
By Lemma 5.14,
supB
θk (0)∩Ω
|Φ| ≤ cǫJω,ǫ. (5.74)
Define ϕ(y) ≡ ǫ−1J−1ω,ǫΦ(ǫy). By (5.74),
‖ϕ‖L∞(B1(0)∩Ω/ǫ) ≤ c.
Function ϕ satisfies−∇ · (Kω2,ǫ,ǫ∇ϕ) = 0 in B1(0) ∩ Ω/ǫ,
ϕ = 0 on B1(0) ∩ ∂Ω/ǫ.
(3.5)2 of Lemma 3.5 implies ‖ϕ‖C1,0(B1(0)∩Ωǫf /ǫ)∩C1,0(B1(0)∩Ωǫ
m/ǫ) ≤ c. This gives the
proof of (5.70) for case (2).
Remark 5.4. By Lemma 3.5 and Lemma 5.17, we know that if ǫ, ω ∈ (0, 1], any
solution of (5.65) satisfies ‖∇Φ‖L∞(B1/2(0)∩Ω) ≤ c‖Kω,ǫΦ‖L2(B1(0)∩Ω), where c is a
constant independent of ω, ǫ.
Lemma 3.7 is direct result of Remark 5.3 and Remark 5.4.
Acknowledgement
The author would like to thank the anonymous referee’s valuable suggestions for
improving the presentation of this paper. This research is supported by the grant
number NSC 102-2115-M-009 -014 from the research program of National Science
Council of Taiwan.
July 9, 2015
32 Elliptic and parabolic equations
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