July 9, 2015 Elliptic and parabolic equations in fractured media

Elliptic and parabolic equations in fractured media

Li-Ming Yeh

Department of Applied Mathematics

National Chiao Tung University, Hsinchu, 30050, Taiwan, R.O.C.

[email protected]

The elliptic and the parabolic equations with Dirichlet boundary conditions in frac- tured media are considered. The fractured media consist of a periodic connected high permeability sub-region and a periodic disconnected matrix block subset with low perme- ability. Letǫ∈(0,1] denote the size ratio of the matrix blocks to the whole domain and letω²∈(0,1] denote the permeability ratio of the disconnected subset to the connected sub-region. It is proved that theW^1,p norm of the elliptic and the parabolic solutions in the high permeability sub-region are bounded uniformly inω, ǫ. However, theW^1,p norm of the solutions in the low permeability subset may not be bounded uniformly in ω, ǫ. For the elliptic and the parabolic equations in periodic perforated domains, it is also shown that theW^1,pnorm of their solutions are bounded uniformly inǫ.

Keywords: fractured media, permeability, periodic perforated domain, VMO AMS Subject Classification: 35J05, 35J15, 35J25

1. Introduction

TheW^1,pestimates 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 inRⁿforn≥3,

∂Ω be the boundary of Ω,Y ≡(0,1)ⁿconsist of a smooth sub-domainYmcompletely surrounded by another connected sub-domain Yf (≡Y \Ym), ǫ∈ (0,1], Ω(2ǫ)≡ {x∈Ω :dist(x, ∂Ω)≥2ǫ}, Ω^ǫ_m≡ {x:x∈ǫ(Ym+j)⊂Ω(2ǫ) for somej∈Zⁿ}be a disconnected subset of Ω, Ω^ǫ_f (≡Ω\Ω^ǫ_m) denote a connected sub-region of Ω, and K_ν,ǫ(x)≡

(1 ifx∈Ω^ǫ_f

ν ifx∈Ω^ǫ_m for anyν, ǫ >0.

The elliptic equation that we consider is

(−∇ ·(K_ω²_,ǫ∇U+G) =F in Ω,

U = 0 on∂Ω, (1.1)

where ω, ǫ∈(0,1] andG, F are given functions. IfG, F are bounded, a solution of (1.1) in Hilbert spaceH¹(Ω) exists uniquely for eachω, ǫby Lax-Milgram Theorem [12]. TheL²norm of the gradient of the solution of (1.1) in the connected sub-region Ω^ǫ_f is bounded uniformly inω, ǫifG, F are small in Ω^ǫ_m. However, theL²norm of

1

the gradient of the solution of (1.1) in matrix blocks Ω^ǫ_mcan be very large whenω closes to 0. The parabolic equation that we consider is, for any ω, ǫ∈(0,1],



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

∂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 Ω^ǫ_f ×(0, T) is bounded uniformly in ω, ǫifF is small in Ω^ǫ_m×(0, T). However, theL²norm of the gradient of the solution of (1.2) in matrix blocks Ω^ǫ_m×(0, T) can be very large whenωcloses 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 inǫ.

There are some literatures related to this work. Lipschitz estimate andW^2,pesti- mate for uniform elliptic equations with discontinuous coefficients had been proved in [15, 18]. Uniform H¨older,W^1,p, and Lipschitz estimates for uniform elliptic equa- tions with H¨older periodic coefficients were shown in [4, 5]. UniformW^1,pestimate 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].

UniformW^1,pestimate for the Laplace equation in periodic perforated domains was considered in [19] and the same problem in Lipschitz estimate was studied in [21].

Uniform H¨older,W^1,p, and Lipschitz estimates inǫfor uniform parabolic equations with oscillating periodic coefficients were obtained in [10]. For non-uniform ellip- tic equations with smooth periodic coefficients, existence ofC^2,α solution could be found in [13]. Uniform H¨older 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 theW^1,pnorm of the elliptic and the parabolic solutions in the high permeability sub-region Ω^ǫ_f are bounded uniformly in ω, ǫ.

However, the solutions in the low permeability subset may not be bounded uniformly inω, ǫ. For the elliptic and the parabolic equations in perforated domains, it is also shown that theW^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 uniformW^1,p estimates inω, 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

of elliptic equations in fracture media (claimed in section 3) is given in section 5.

2. Notation and main result

LetC^k,α denote the H¨older space with normk · kC^k,α,W^s,p the Sobolev space with norm k · kW^s,p, and [ϕ]C^0,α the H¨older semi-norm of ϕ for k ≥ 0, α∈ [0,1], s≥

−1, p ∈ [1,∞] (see [2, 12]). L^p = W^0,p and H¹ = W^1,2. C₀^∞(D) is the space of infinitely differentiable functions with support in D and C_per^∞ (Rⁿ) is the space of infinitely differentiable Y-periodic functions in Rⁿ. W₀^s,p(D) is the closure of C₀^∞(D) under the W^s,p norm andW_per^s,p(Rⁿ) is the closure ofC_per^∞(Rⁿ) underW^s,p norm and kϕk_Wper^s,p_(Rⁿ₎ ≡ kϕkW^s,p(Y) fors ≥ 1,p∈ [1,∞]. Am ≡ {x :x∈ Ym+ j for some j∈Zⁿ}andAf ≡Rⁿ\ Am.H¹_per(Rⁿ)≡ {ϕ∈W_per^1,2(Rⁿ) :R

Yfϕ(y)dy= 0}andH¹_per(Af)≡ {ϕ|Af :ϕ∈ H¹_per(Rⁿ)}. Letkϕ1,· · · , ϕmk^B1≡ kϕ1k^B1+· · ·+ kϕmk^B1 andkϕk^B1∩B₂ ≡ kϕk^B1+kϕk^B2. SetrD=D/r⁻¹≡ {x:x/r∈D},Dbe the closure ofD,∂Dbe the boundary ofD,XDis the characteristic function onD, and letBr(x) denote a ball centered atxwith radiusr. For anyϕ∈L¹(D),

(ϕ)D ≡ − Z

D

ϕ(y)dy≡ 1

|D|

Z

D

ϕ(y)dy.

K_ω,ν(x)≡

(1 ifx∈νAf

ω ifx∈νAm

for ω ∈ [0,1], ν ∈ (0,∞). If~n is an outward normal vector on∂Ym, we define, for any functionϕinY andx∈∂Ym,

⌊ϕ⌋∂Ym(x) =ϕ,+(x)−ϕ,−(x) where ϕ,±(x)≡ lim

t→0⁺ϕ(x±t~n). (2.1) Similarly, if~n_ǫ is an outward normal vector on∂Ω^ǫ_m, we define, for any functionϕ in Ω and x∈∂Ω^ǫ_m,

⌊ϕ⌋∂Ω^ǫ_m(x) =ϕ,+(x)−ϕ,−(x) where ϕ,±(x)≡ lim

t→0⁺ϕ(x±t~n_ǫ).

Next we give two statements:

A1. Ω is a bounded smooth simply-connected domain inRⁿ forn≥3, A2. Ymis a smooth simply-connected sub-domain ofY.

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∈L^p(Ω),F ∈W^−1,p(Ω), then a W^1,p(Ω) solution of (1.1) exists uniquely and satisfies



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

kK_ω/ǫ,ǫU,K_ω,ǫ∇UkL^p(Ω)

≤c kK1/ω,ǫGkL^p(Ω)+kFkW^−1,p(Ω)+ω⁻¹kFkW^−1,p(Ω^ǫ_m)

for ^ω_ǫ ≤1, kU,K_ω,ǫ∇UkL^p(Ω)

≤c kK_1/ω,ǫGkL^p(Ω)+kFkW^−1,p(Ω)+ω⁻¹kFkW^−1,p(Ω^ǫ_m)

for ^ω_ǫ ≥1,

wherec is a constant independent ofω, 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∈L^p(Ω^ǫ_f),F ∈W^−1,p(Ω), kFk_W^−1,p_(Ω^ǫ_m)= 0, then aW^1,p(Ω^ǫ_f)solution of



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

−∇ ·(∇U+G) =F in Ω^ǫ_f (∇U+G)·~n_ǫ= 0 on ∂Ω^ǫ_m

U = 0 on ∂Ω

(2.2)

exists uniquely and satisfies

kUkW^1,p(Ω^ǫ_f)≤c(kGkL^p(Ω^ǫ_f)+kFkW^−1,p(Ω)), (2.3) where~n_ǫ is a unit normal vector on∂Ω^ǫ_mandc is a constant independent of ǫ.

For anyω, ǫ∈(0,1] andp∈(1,∞), let us define (B_p≡

ϕ: ϕ∈W₀^1,p(Ω)∩W^2,p(Ω^ǫ_f)∩W^2,p(Ω^ǫ_m),⌊K_ω2,ǫ∇ϕ·~n_ǫ⌋∂Ω^ǫ_m = 0 , D_p≡

ϕ: ϕ∈W^2,p(Ω^ǫ_f), ϕ|∂Ω= 0,∇ϕ·~nǫ|∂Ω^ǫ_m = 0 ,

where ~n_ǫ is a normal vector on∂Ω^ǫ_m. By Lemma 3.4 [23], B_p with normkϕk_Bp ≡ k∇ ·(K_ω2,ǫ∇ϕ)kL^p(Ω)is a Banach space. IfB_pdenotes the closure ofB_pinL^pspace, thenB_p=L^p(Ω). Also noteD_p with normkϕk_Dp≡ k∆ϕkL^p(Ω^ǫ_f)is a Banach space.

The function spacesC([0, T];B), C^σ([0, T];B) forσ∈(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∈C^σ([0, T];L^p(Ω)),U0∈B_p, then aC([0, T];W^1,p(Ω)) solution of (1.2) exists uniquely and satisfies

kUkC¹([0,T];L^p(Ω))+kKω,ǫ∇UkC([0,T];L^p(Ω)) ≤c kU0k_Bp+kFkC^σ([0,T];L^p(Ω))

,

wherec is a constant independent ofω, ǫ.

Theorem 2.4. Suppose A1–A2 and

A6. ǫ, σ∈(0,1],p∈(n,∞),F ∈C^σ([0, T];L^p(Ω^ǫ_f)),U0∈D_p, then aC([0, T];W^1,p(Ω^ǫ_f))solution of



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

∂tU−∆U =F in Ω^ǫ_f×(0, T)

∇U·~n_ǫ= 0 on ∂Ω^ǫ_m×(0, T) U = 0 on ∂Ω×(0, T) U(x,0) =U0(x) in Ω^ǫ_f

(2.4)

exists uniquely and satisfies

kUkC¹([0,T];L^p(Ω^ǫ_f))+k∇UkC([0,T];L^p(Ω^ǫ_f)) ≤c

kU0k_Dp+kFkC^σ([0,T];L^p(Ω^ǫ_f))

,

where~n_ǫ is a normal vector on∂Ω^ǫ_mandc is a constant independent of ǫ.

3. Preliminaries

From Theorem 2.1 [1], we know

Lemma 3.1. Forp∈[1,∞)andǫ∈(0,1], there is a constantc(Yf, p)and a linear continuous extension operatorPǫ:W^1,p(Ω^ǫ_f)→W^1,p(Ω)such that ifϕ∈W^1,p(Ω^ǫ_f), then



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Pǫϕ=ϕ inΩ^ǫ_f,

kPǫϕkL^p(Ω)≤ckϕkL^p(Ω^ǫ_f), k∇PǫϕkL^p(Ω)≤ck∇ϕkL^p(Ω^ǫ_f),

0< d1≤ Pǫϕ≤d2 if 0< d1≤ϕ≤d2 for some constantsd1, d2, Pǫϕ=ζ in Ωif ϕ=ζ|Ω^ǫ_f for some linear functionζ in Ω.

Moreover, if ζ(x)≡ϕ(rx)inB1(0)∩Ω^ǫ_f/r for anyr > ǫ, then Pǫ/rζ(x) =Pǫϕ(rx) in B_1/2(0).

Remark 3.1. Tracing the proof of Theorem 7.25 [12], we know that if 0∈∂Ymand p, ν∈[1,∞), there exist a constantc(Yf) and a linear continuous extension operator Pν:W^1,p(B1(0)∩νYf)→W^1,p(B1(0)) such that, for anyϕ∈W^1,p(B1(0)∩νYf),

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Pνϕ=ϕ in B1(0)∩νYf,

kPνϕkL^p(B1(0))≤ckϕkL^p(B1(0)∩νYf), k∇PνϕkL^p(B1(0))≤ck∇ϕkL^p(B1(0)∩νYf).

Lemma 3.2. Let ω ∈(0,1], ν ∈(0,∞),ϕ∈H¹(B1(0)), and Pνϕ|νAf denote the extension of ϕ|νAf onB1(0). There is a constantc independent ofω, ν such that

K_ω,ν ϕ−(Pνϕ|νAf)B1(0)

L²(B1(0))≤ckK_ω,ν∇ϕkL²(B1(0)). See section 2 for K_ω,ν.

Proof. By Poincar´e inequality [12], Lemma 3.1, and Remark 3.1, the extension functionPνϕ|νAf ∈H¹(B1(0)) satisfies

Pνϕ|νAf −(Pνϕ|νAf)B1(0)

L²(B1(0))

≤c∇Pνϕ|νAf

L²(B1(0))≤ck∇ϕkL²(B1(0)∩νAf), (3.1)

where c is independent of ω, ν. (3.1), Lemma 3.1, Remark 3.1, and Poincar´e in- equality imply

K_ω,ν(ϕ−(Pνϕ|νAf)B1(0))

L²(B1(0))

≤K_ω,ν(Pνϕ|νAf −(Pνϕ|νAf)B1(0))

L²(B1(0))

+ωϕ− Pνϕ|νAf

L²(B1(0)∩νAm)

≤ck∇ϕkL²(B1(0)∩νAf)+cω∇ϕ− ∇Pνϕ|νAf

L²(B1(0)∩νAm)

≤ckK_ω,ν∇ϕkL²(B1(0)).

If 0∈∂Ω, by A1 and rotation, there is a smooth function Ψ :Rⁿ⁻¹ →Rsuch that(

Ψ(0) =|∇Ψ(0)|= 0,

B1(0)∩Ω/r=B1(0)∩ {(x^′, xn)∈Rⁿ : rxn >Ψ(rx^′)} ifr∈(0,1]. (3.2) Ifr= 0, we defineB1(0)∩Ω/r≡B1(0)∩ {(x^′, xn)∈Rⁿ: xn>0}. Set

K˘ν,ǫ,r≡

(1 in Ω^ǫ_f/r

ν in Ω^ǫ_m/r forν, ǫ, r∈(0,1]. (3.3) Similar to Lemma 3.2, we also have, by Poincar´e inequality [12], Lemma 3.1, and Remark 3.1,

Lemma 3.3. Ifω, ǫ, r∈(0,1],0∈∂Ω andϕ∈H¹(B2(0)∩Ω/r) withϕ|_∂Ω/r = 0, there is a constantc independent ofω, ǫ, r such that

kK˘_ω,ǫ,rϕkL²(B1(0)∩Ω/r)≤ckK˘_ω,ǫ,r∇ϕkL²(B1(0)∩Ω/r).

3.1. Interface problem

Let Γ(x−y) denote the fundamental solution of the Laplace equation inRⁿ, see§6.2 [7]. Define a single-layer and a double-layer potentials as, for any smooth function ϕon the boundary∂Ym ofYm,



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

S∂Ym(ϕ)(x)≡ Z

∂Ym

Γ(x−y)ϕ(y)dy L∂Ym(ϕ)(x)≡

Z

∂Ym

∇yΓ(x−y)~n_y ϕ(y)dy

forx∈∂Ym,

where ~n_y 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,∞)ands∈ {0,1}, the linear operators (S∂Ym :W^s−1p,p(∂Ym)→W^s+1−p1,p(∂Ym)

L∂Ym :W^s+1−p1,p(∂Ym)→W^s+2−1p,p(∂Ym)

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 cindependent of ℓso that

kϕkW^s+1−1p,p(∂Ym)≤ck(I−ℓL∂Ym)(ϕ)k

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:

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∂Yf is an open portion of the boundary∂Y ,

D₁,D₂are smooth domains satisfying Ym⊂D₁⊂D₂⊂Y , dist(Ym, ∂D1), dist(D1, ∂D2), dist(D2, ∂Y \∂Yf)> d0>0.

Lemma 3.5. Supposeω∈(0,1], any solution Φof (−∇ ·(K_ω2,1∇Φ +V) =ζ in Y

Φ = 0 on ∂Yf (3.4)

satisfies

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

kK_ωσ,1Φk_W1,p(D₁\Ym)∩W^1,p(Ym)≤c kΦkL²(Yf)

+kK_ωσ−2,1VkL^p(Y)+kK_ωσ−2,1ζk_W^−1,p_(Yf)∩W^−1,p_(Ym) , kΦk_W2,p(D₁\Ym)∩W^2,p(Ym)≤c kΦkL²(Yf)

+kK_ω−2,1VkW^1,p(Yf)∩W^1,p(Ym)+kK_ω−2,1ζkL^p(Y)

,

(3.5)

where p∈[2,∞),σ∈ {0,1}, andc is a constant independent of ω.

Proof. DefineIω,σ ≡ kK_ωσ−2,1VkL^p(Y)+kK_ωσ−2,1ζkW^−1,p(Yf)∩W^−1,p(Ym)and letc denote a constant independent ofω. First we prove (3.5)1.

Step 1:Assume V ∈W₀^1,p(Yf)∩W₀^1,p(Ym) and ζ ∈L^p(Y). Consider the fol- lowing problem

(−∇ ·(K_ω2,1∇φ+V) =ζ inD₂,

φ= 0 on∂D2. (3.6)

The unique existence of aH¹ solution of (3.6) is from Lax-Milgram Theorem [12].

By energy method, the solution satisfies

kφk_H¹₍D₂\Ym)≤cIω,1. (3.7) Letη∈C^∞(D2\Ym),η∈[0,1],η= 1 inD₂\D₁,η= 0 on∂Ym,kηkW^1,∞(D₂\Ym)≤ c. Multiply (3.6)1 byηto 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

kφkW^1,p(D₂\D₁)≤cIω,σ. (3.9)

LetϕinYmsatisfy

(−∇ ·(ω²∇ϕ+V) =ζ inYm,

ϕ= 0 on∂Ym, (3.10)

andϕinD₂\Ymsatisfy

(−∇ ·(∇ϕ+V) =ζ in D₂\Ym,

ϕ= 0 on∂(D2\Ym). (3.11)

By [8] again,

kϕk_W1,p(D₂\Ym)+ω^σkϕkW^1,p(Ym)≤cIω,σ. (3.12) If we defineψ≡φ−ϕin D₂, then (3.6) and (3.10)–(3.11) imply

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∆ψ= 0 inD₂\∂Ym,

⌊ψ⌋∂Ym= 0 on∂Ym,

⌊K_ω2,1∇ψ⌋∂Ym·~n_y=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 ∈W₀^1,p(Yf)∩W₀^1,p(Ym),

F ≡ ω²∇ϕ,−− ∇ϕ,+

·~n_y|∂Ym. By (3.12),

kF kW^−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,−·~n_y|∂Ym)

ψ/2− L∂Ym(ψ) =−S∂Ym(∇ψ,+·~n_y|∂Ym) +S∂D₂(∂ⁿyψ|∂D₂) on∂Ym, where∂ⁿyψ|∂D₂ is the normal derivative ofψ on∂D2. So

I−2(1−ω²) ω²+ 1 L∂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

kψkW¹⁻

1 p,p

(∂Ym)≤c

kF kW⁻

1 p,p

(∂Ym)+k∂ⁿyψk

W⁻

1 p,p

(∂D₂)

≤cIω,σ. (3.16) (3.13) and (3.16) imply

kψkW^1,p(D₂)≤cIω,σ. Together with (3.12), we obtain

kK_ωσ,1φk_W1,p(D₂\Ym)∩W^1,p(Ym)≤cIω,σ. (3.17)

Note W₀^1,p(Yf) (resp. W₀^1,p(Ym)) is dense in L^p(Yf) (resp. L^p(Ym)) and L^p(Y) is dense in W^−1,p(Y). By a limiting argument, we see that if V ∈ L^p(Y) and ζ∈W^−1,p(Y), any solution of (3.6) satisfies (3.17).

Step 2:Letη be a smooth function satisfyingη∈C₀^∞(D2),η∈[0,1],η= 1 in D₁,kηkW^1,∞(D₂)≤c. Multiply (3.4) byη to obtain

(−∇ ·(K_ω2,1∇(Φη)−Φ∇η+V η) =ζη−(∇Φ +V)∇η inD₂,

Φη= 0 on∂D₂.

By the result of Step 1, we have

kK_ωσ,1Φk_W1,p(D₁\Ym)∩W^1,p(Ym)≤c(kΦkL^p(D₂\D₁)+Iω,σ). (3.18) Let ˜ηbe another smooth function satisfying ˜η∈C₀^∞(Yf), ˜η∈[0,1], ˜η= 1 inD₂\D1, kηk˜ W^1,∞(Y) ≤c. Multiply (3.4) by ˜ηΦ and use energy method and Theorem 7.26 [12] to get

kΦkL^p(D₂\D₁)≤c(kΦkL²(Yf)+Iω,σ).

Together with (3.18), we obtain (3.5)1. (3.5)2are 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 Φof

−∇ ·(K_ω2,ν∇Φ) = 0 inνY (3.19) satisfies

kK_ωσ,νΦkW^2,p(B_1/3(x0)∩νYf)∩W^2,p(B_1/3(x0)∩νYm)≤ckK_ωσ,νΦkL²(B1(x0)), (3.20) where p∈[2,∞),σ∈ {0,1}, andc is a constant independent of ω, ν.

Proof. After translation, we assume x0 is the origin. For each ν > 1, we find a smooth domainD_ν such that

B1/2(x0)∩νYm⊂D_ν ⊂B1(x0)∩νYm and B1/2(x0)∩ν∂Ym⊂∂Dν. Since D_ν is smooth, for anyz ∈∂Dν there is a ball B(z) centered at z and there is a smooth one-to-one mappingϕz,ν ofB(z) ontoϕz,ν(B(z))⊂Rⁿ satisfying

ϕz,ν(B(z)∩D_ν)⊂Rⁿ

+, ϕz,ν(B(z)∩∂D_ν)⊂∂Rⁿ

+, ϕz,ν(B(z)\D_ν)⊂Rⁿ

−. (3.21) Here Rⁿ

+ ≡ {x= (x1,· · ·, xn) :xn >0}, ∂Rⁿ

+≡ {x:xn = 0},Rⁿ

− ≡ {x: xn <0}.

Since ∂Dν is compact for eachν >1, there exist a finite numberℓν of open balls {B(zi)}^ℓ_i=1^ν and one-to-one mappings{ϕzi,ν}^ℓ_i=1^ν such that









zi∈∂Dν fori∈ {1,· · · , ℓν},

(3.21) holds for each ballB(zi) andi∈ {1,· · ·, ℓν},

∂D_ν⊂Sℓν

i=1B(zi).

SinceYmis smooth, it is possible to choose domainsD_ν for allν >1 such that (the numberℓν is bounded above by a constant independent ofν,

kϕz,νk_C3,0(B(z)),kϕ⁻¹_z,νk_C3,0(ϕz,ν(B(z)))≤c∗,where c∗ is independent ofν, z.

By assumptionx0= 0∈ν∂Ym, we defineKb_ω2,ν andφinRⁿ by b

K_ω2,ν≡

(ω² in D_ν,

1 elsewhere, φ≡

(Φ inB1/2(x0), 0 elsewhere.

Let η ∈ C₀^∞(B1/2(x0)) be a bell-shaped function satisfying η ∈ [0,1], η = 1 in B1/3(x0),k∇ηkW^1,∞(B1/2(x0))≤c. Multiply (3.19) byη to get





−∇ · b

K_ω2,ν∇(ηφ)−Kb_ω2,νφ∇η

=−Kb_ω2,ν∇φ∇η inB1(x0),

ηφ= 0 on∂B1(x0).

Then we follow the argument of Step 1 of Lemma 3.5 to see that (3.20) holds.

LetX^(j)

ω,1∈ H¹_per(Rⁿ) forω∈(0,1] be a function satisfying

∇ ·(K_ω2,1(∇X^(j)_ω,1+~ej)) = 0 inY , (3.22) and letX^(j)

0,1∈ H¹per(Af)∩H¹(Am) be a function satisfyingX^(j)

0,1(x) = 0 inAm and (∇ ·(∇X^(j)_0,1+~ej) = 0 in Yf,

(∇X^(j)

0,1+~ej)·~n_y= 0 on∂Ym,

where ~ej, j = 1,· · ·, n is the unit vector in the j-th coordinate direction, and ~n_y is a unit normal vector on∂Ym. By Lax-Milgram Theorem [12],X^(j)

ω,1 forω∈[0,1]

and j = 1,· · · , nis uniquely solvable. By Theorem 6.30 [12] and (3.5)2 of Lemma 3.5,

kX^(j)_ω,1kW^2,p(Yf)∩W^2,p(Ym)≤c(n, Ym) forω∈[0,1], p∈[2,∞). (3.23) Define X_ω,1 ≡ (X⁽¹⁾

ω,1,· · · ,X⁽ⁿ⁾

ω,1) and X_ω,ǫ(x) ≡ ǫX_ω,1(^x_ǫ) 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≡

Z

Yf∪Ym

K_ω2,1(I+ Ξω)dy forω∈[0,1] (3.24) is a symmetric positive definite matrix dependent only onω. HereI is the identity matrix. By (3.23), it is not difficult to see, for ω∈[0,1],

(d3I≤ Kω≤d4I whered3, d4 are positive constants,

Kω is a continuous function ofω. (3.25)

3.2. Local Lipschitz and local L^p gradient estimates We have the following Lipschitz estimate:

Lemma 3.7. Supposeω, ǫ∈(0,1], any solution of

(−∇ ·(Kω²,ǫ∇Φ) = 0 in B1(0)∩Ω

Φ = 0 on B1(0)∩∂Ω

satisfies

k∇ΦkL^∞(B_1/2(0)∩Ω)≤ckKω,ǫΦkL²(B1(0)∩Ω), (3.26) where cis a constant independent ofω, ǫ.

Proof of Lemma 3.7 is given in section 5. Next is the localL^pgradient 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

− Z

Br/2(x0)

|Kω,ǫ∇Φ|^pXΩ dx 1/p

≤c

− Z

Br(x0)

|Kω,ǫ∇Φ|²XΩ dx 1/2

, (3.28) where cis a constant independent ofω, ǫ, r, x0.

Proof. Letc denote a constant independent ofω, ǫ, r, x0.

Case I:ForB2r(x0)⊂Ω case. By translation, we movex0 to the origin (that is, x0= 0∈Ω). Letd∈Randϕ(y) = Φ(ry). By (3.27), we know

−∇ ·(K_ω²_,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

kK_ω,ǫ/r∇ϕkL^p(B1/2(0))≤ckK_ω,ǫ/r(ϕ+d)kL²(B1(0)), wherec is also independent ofd. By Lemma 3.2,

kKω,ǫ/r∇ϕkL^p(B_1/2(0))≤ckKω,ǫ/r∇ϕkL²(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),

(−∇ ·( ˘K_ω²_,ǫ,r∇ϕ) = 0 in B2(0)∩∂Ω/r,

ϕ= 0 onB2(0)∩∂Ω/r. (3.29)

Ifǫ/r≤1 (resp.ǫ/r >1), Lemma 3.7 (resp. Theorem 9.13 [12]) implies that theϕ in (3.29) satisfies

kK˘_ω,ǫ,r∇ϕkL^p(B1/2(0)∩Ω/r)≤ckK˘_ω,ǫ,rϕkL²(B1(0)∩Ω/r). By Lemma 3.3, we obtain

kK˘_ω,ǫ,r∇ϕkL^p(B1/2(0)∩Ω/r)≤ckK˘_ω,ǫ,r∇ϕkL²(B1(0)∩Ω/r). Which implies (3.28).

4. Proof of main results

Proof of Theorem 2.1:Supposeω, ǫ∈(0,1], let us find U ∈H¹(Ω) satisfying (−∇ ·(Kω²,ǫ∇U+K_ω,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 =K_ω,ǫ∇U, then T is a linear and bounded operator on L²(Ω) by energy method. Lemma 3.8 implies that the operatorT satisfies (1.9) of Theorem 1.3 [22] for any G ∈ L^p(Ω), p ∈ (2,∞). So T is a bounded and linear operator inL^p(Ω) forp∈(2,∞) by Theorem 1.3 [22]. By Poincar´e inequality [12]

and Lemma 3.1, the solution of (4.1) satisfies

kUkL^p(Ω^ǫ_f)≤ kPǫU|Ω^ǫ_fkL^p(Ω)≤ck∇PǫU|Ω^ǫ_fkL^p(Ω)≤ck∇UkL^p(Ω^ǫ_f), kUkL^p(Ω^ǫ_m)≤ kU− PǫU|Ω^ǫ_fkL^p(Ω^ǫ_m)+kPǫU|Ω^ǫ_fkL^p(Ω^ǫ_m)

≤cǫk∇U− ∇PǫU|Ω^ǫ_fkL^p(Ω^ǫ_m)+ckUkL^p(Ω^ǫ_f),

where c is independent of ω, ǫ. Function PǫU|Ω^ǫ_f above denotes the extension of U|Ω^ǫ_f on Ω. So we have

Lemma 4.1. Under A1–A2, if ω, ǫ ∈ (0,1], p ∈ [2,∞), and G ∈ L^p(Ω), then a W^1,p(Ω)solution U of (4.1) exists uniquely and

(kKω/ǫ,ǫU,Kω,ǫ∇UkL^p(Ω)≤ckGkL^p(Ω) for ^ω_ǫ ≤1, kU,K_ω,ǫ∇UkL^p(Ω)≤ckGkL^p(Ω) for ^ω_ǫ ≥1, wherec is a constant independent ofω, ǫ.

By a duality argument, Poincar´e 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 ∈ L^p(Ω), then a W^1,p(Ω)solution U of (4.1) exists uniquely and

(kKω/ǫ,ǫU,K_ω,ǫ∇UkL^p(Ω)≤ckGkL^p(Ω) for ^ω_ǫ ≤1, kU,K_ω,ǫ∇UkL^p(Ω)≤ckGkL^p(Ω) for ^ω_ǫ ≥1, wherec is a constant independent ofω, ǫ.

Ifω, ǫ∈(0,1] andG, F ∈L^∞(Ω), then theH¹(Ω) solution of (−∇ ·(Kω²,ǫ∇U) =F in Ω

U = 0 on∂Ω (4.2)

and theH¹(Ω) solution of

(−∇ ·(K_ω2,ǫ∇ϕ−K_ω,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

(kKω/ǫ,ǫϕ,K_ω,ǫ∇ϕkL^r(Ω)≤ckGkL^r(Ω) for ^ω_ǫ ≤1,

kϕ,K_ω,ǫ∇ϕkL^r(Ω)≤ckGkL^r(Ω) for ^ω_ǫ ≥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 H¨older inequality to get

Z

Ω

K_ω,ǫ∇U Gdx= Z

Ω

ϕ F dy= Z

Ω

Pǫϕ|Ω^ǫ_fF dy+ Z

Ω^ǫ_m

(ϕ− Pǫϕ|Ω^ǫ_f)F dy

≤ckGkL^r(Ω)(kFkW^−1,p(Ω)+ω⁻¹kFkW^−1,p(Ω^ǫ_m)),

where ¹_r +¹_p = 1 and c is independent of ω, ǫ. Since L^∞(Ω) is dense inL^r(Ω) for any r∈(1,∞), we obtain

kKω,ǫ∇UkL^p(Ω)≤c(kFkW^−1,p(Ω)+ω⁻¹kFkW^−1,p(Ω^ǫ_m)),

where ¹_r +¹_p = 1 and c is a constant independent of ω, ǫ. By Poincar´e inequality [12] and Lemma 3.1, it is easy to see that

(kKω/ǫ,ǫUkL^p(Ω)≤c(kFk_W^−1,p_(Ω)+ω⁻¹kFk_W^−1,p_(Ω^ǫ_m)) for ^ω_ǫ ≤1, kUkL^p(Ω)≤c(kFkW^−1,p(Ω)+ω⁻¹kFkW^−1,p(Ω^ǫ_m)) 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∈L^p(Ω), F ∈L^∞(Ω). If G∈L^p(Ω), F ∈W^−1,p(Ω), Theorem 2.1 can be proved by a limiting argument.

Proof of Theorem 2.2:SupposeG, F ∈L^p(Ω^ǫ_f) forp∈(1,∞), let us do zero extension for G, F. That is, set Ge ≡

(G on Ω^ǫ_f

0 on Ω^ǫ_m and Fe ≡

(F on Ω^ǫ_f

0 on Ω^ǫ_m. Then G,e Fe ∈ L^p(Ω). Let Uω,ǫ denote the solution of (1.1) with G, F replaced by G,e Fe above. By Theorem 2.1,

(kKω/ǫ,ǫUω,ǫ,K_ω,ǫ∇Uω,ǫkL^p(Ω)≤c(kGke L^p(Ω)+kFke W^−1,p(Ω)) for ^ω_ǫ ≤1, kUω,ǫ,K_ω,ǫ∇Uω,ǫkL^p(Ω)≤c(kGke L^p(Ω)+kFekW^−1,p(Ω)) for ^ω_ǫ ≥1, (4.5) wherec is independent ofω, ǫ. If we fixǫ, then we see, by (4.5) and Lemma 3.1,

• There is a subsequence of{Uω,ǫ}(same notation for subsequence) such that Uω,ǫ|Ω^ǫ_f converges weakly toU inW^1,p(Ω^ǫ_f) asω→0.

• The limit functionU satisfies (2.2) and (2.3).

So Theorem 2.2 is proved forG, F ∈L^p(Ω^ǫ_f), 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,∞),

kUkC¹([0,T];L^p(Ω))+kUkC([0,T];Bp)≤c kU0k_Bp+kFkC^σ([0,T];L^p(Ω))

,

wherecis a constant independent ofω, ǫ. (1.2) can be written as, for fixedt∈(0, T], (−∇ ·(Kω²,ǫ∇U(·, t)) =F(·, t)−∂tU(·, t) in Ω,

U(·, t) = 0 on∂Ω.

By Theorem 2.1, we see, forp∈(n,∞),

kKω,ǫ∇U(·, t)kL^p(Ω)≤c kU0k_Bp+kFkC^σ([0,T];L^p(Ω))

, wherec is a constant independent ofω, 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, forp∈(n,∞),

kUkC¹([0,T];L^p(Ω^ǫ_f))+k∆UkC([0,T];L^p(Ω^ǫ_f))≤c kU0k_Dp+kFkC^σ([0,T];L^p(Ω^ǫ_f))

, wherec is a constant independent ofǫ. (2.4) can be written as, for fixed t∈(0, T],









−∆U(·, t) =F(·, t)−∂tU(·, t) in Ω^ǫ_f,

∇U(·, t)·~n_ǫ= 0 on∂Ω^ǫ_m,

U(·, t) = 0 on∂Ω.

By Theorem 2.2, we see, forp∈(n,∞),

k∇UkC([0,T];L^p(Ω^ǫ_f))≤c kU0k_Dp+kFkC^σ([0,T];L^p(Ω^ǫ_f))

,

wherec is a constant independent ofǫ. 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 H¨older estimate in subsection 5.1 and then derive local Lipschitz estimate in subsection 5.2.

5.1. H¨older 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 pointy=y(x)∈Rⁿ such thatBr(y)\ {x} ⊂Rⁿ\ Oandx∈∂Br(y).

IfO ⊂Rⁿis a nonempty open bounded Lipschitz set and satisfy a uniform exterior ball condition, thenOis called a semiconvex domain.

In this subsection, we assume (1) A2 holds and (2) Ois a semiconvex domain.

If 0∈∂O, by rotation, there is a Lipschitz functionΨ :e Rⁿ⁻¹→Rsuch that (Ψ(0) = 0,e

B1(0)∩ O/r=B1(0)∩ {(x^′, xn)∈Rⁿ: rxn >Ψ(rxe ^′)} ifr∈(0,1]. (5.1) Ifr= 0, we defineB1(0)∩ O/r≡B1(0)∩ {(x^′, xn)∈Rⁿ : xn>0}. Similar to Ω^ǫ_f and Ω^ǫ_m, one can also define analogousO_f^ǫ andO_m^ǫ . DefineKe_ν,ǫ,r as

Keν,ǫ,r≡

(1 inO^ǫ_f/r

ν inO^ǫ_m/r forν, ǫ, r∈(0,1].

Lemma 5.1. Letω, ǫ, r∈(0,1],ǫ≤r, either B2(0)⊂ O/r or0∈∂O/r, andϕbe a solution of

(−∇ ·(Ke_ω2,ǫ,r∇ϕ) = 0 in B2(0)∩ O/r,

ϕ= 0 on B2(0)∩∂O/r.

There is a constant c independent ofω, ǫ, r such that

kϕkH¹(B1/2(0)∩O/r)≤ckKeω,ǫ,rϕkL²(B2(0)∩O/r).

Proof. Letc denote a constant independent ofω, ǫ, r. By energy method,

kKeω,ǫ,r∇ϕkL²(B1(0)∩O/r)≤ckKeω,ǫ,rϕkL²(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)ˆ ≡Ke_ω²_,ǫ,r(_r^ǫx) ˆ

ϕ(x)≡ϕ(_r^ǫx) forx∈B1(z)∩ O/ǫ.

Then ˆϕsatisfies

(−∇ ·( ˆK∇ϕ) = 0ˆ inB1(z)∩ O/ǫ, ˆ

ϕ= 0 onB1(z)∩∂O/ǫ.

By (3.5)1,

k∇ϕkˆ L²(B1/2(z)∩O/ǫ)≤ckϕkˆ L²(B1(z)∩O^ǫ_f/ǫ). (5.3) By Poincar´e inequality [12], (5.3) implies

k∇ϕk²_L2(Bǫ/2r(z)∩O/r)≤ck∇ϕk²_L2(Bǫ/r(z)∩O_f^ǫ/r). (5.4)

By coveringB1(0)∩ O/r with a finite number of balls of radiusǫ/2r, (5.4) implies k∇ϕkL²(B1/2(0)∩O/r)≤ck∇ϕkL²(B1(0)∩O^ǫ_f/r).

Together with (5.2) and Poincar´e inequality [12], we prove the lemma.

The rest of this subsection is to prove the following lemma.

Lemma 5.2. Supposeδ >0 andω, ǫ∈(0,1], any solution of (−∇ ·(Ke_ω2,ǫ,1∇Φ) = 0 inB1(0)∩ O

Φ = 0 onB1(0)∩∂O

satisfies

kΦk_C0,µ(B1/2(0)∩O)≤ckKe_ω,ǫ,1ΦkL²(B1(0)∩O), (5.5) whereµ≡ _n+δ^δ andcis 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 H¨older estimate AssumeB1(0)⊂ O.

Lemma 5.3. For anyδ >0, there are constants θ1, θ2∈(0,1) withθ1< θ₂² and a constant ǫ0∈(0,1)(depending onδ, θ2, Yf) such that if

(−∇ ·(Ke_ω²_,ν,1∇ϕ) = 0 inB1(0),

kKe_ω,ν,1ϕkL²(B1(0))≤1, (5.6)

then, for any ω∈(0,1],ν ∈(0, ǫ0], andθ∈[θ1, θ2],

− Z

Bθ(0)

ϕ−(ϕ)Bθ(0)

²dx≤θ^2µ, (5.7)

whereµ≡ _n+δ^δ .

Proof. Consider the following problem

−∇ ·(Kω∗∇ϕ∗) = 0 inB2/3(0), (5.8) whereKω∗ forω∗∈[0,1] is from (3.24). By Theorem 1.2 in page 70 [11] and (3.25), there is a smallθ∈(0,2/3) such that

− Z

Bθ(0)

ϕ∗−(ϕ∗)Bθ(0)

²dx≤θ^2µ′− Z

B2/3(0)

|ϕ∗|²dx, (5.9)