Holder estimate for non-uniform parabolic equations in highly heterogeneous media

(1)

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Nonlinear Analysis

journal homepage:www.elsevier.com/locate/na

Hölder estimate for non-uniform parabolic equations in highly

heterogeneous media

Li-Ming Yeh

Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 30050, Taiwan, ROC

a r t i c l e i n f o Article history: Received 7 October 2011 Accepted 26 January 2012 Communicated by S. Carl MSC: 35K10 35K20 35K65 35K67 Keywords:

Highly heterogeneous media Pseudo-differential operator Paramatrix Strict solution Infinitesimal generator Numerical range

a b s t r a c t

Uniform bound for the solutions of non-uniform parabolic equations in highly heteroge-neous media is concerned. The media considered are periodic and they consist of a con-nected high permeability sub-region and a disconcon-nected matrix block subset with low permeability. Parabolic equations with diffusion depending on the permeability of the me-dia have fast diffusion in the high permeability sub-region and slow diffusion in the low permeability subset, and they form non-uniform parabolic equations. Each medium is as-sociated with a positive numberϵ, denoting the size ratio of matrix blocks to the whole domain of the medium. Let the permeability ratio of the matrix block subset to the con-nected high permeability sub-region be of the orderϵ2τ _for_{τ ∈ (}₀_,₁]. It is proved that the Hölder norm of the solutions of the above non-uniform parabolic equations in the con-nected high permeability sub-region are bounded uniformly inϵ. One example also shows that the Hölder norm of the solutions in the disconnected subset may not be bounded uni-formly inϵ.

1. Introduction

Uniform Hölder estimate for the solutions of non-uniform parabolic equations in highly heterogeneous media is presented. The equations have many applications in multi-phase flows in porous media, the stress in composite materials, and so on (see [1–4] and references therein). The mediaΩ

⊂

_Rn

(

n

≥

2

)

contain a connected high permeability sub-region and a disconnected matrix block subset with low permeability. Let

∂

Ω denote the boundary ofΩ

, ϵ ∈ (

0

,

1

),

Ω

(

2

ϵ) ≡

{

x

∈

Ω

|

dist

(

x

, ∂

Ω

) >

2

ϵ}

, and Y

≡

(

0

,

1

)

n _{denote a cell consisting of a sub-domain Y}

mcompletely surrounded by another connected sub-domain Yf

(≡

Y

\

Ym

)

. The disconnected matrix block subset ofΩ isΩmϵ

≡ {

x

|

x

∈

ϵ(

Ym

+

j

) ⊂

Ω

(

2

ϵ)

for some j

∈

_Zn

_}

_{with boundary}

_∂

_Ωϵ

m, and the connected sub-region isΩfϵ

≡

Ω

\

Ω

ϵ

m. The non-uniform parabolic equations (see [4]) in

[

0

,

T

] ×

Ωare



∂

tUϵ

− ∇ ·

(

Λϵ_τ

∇

Uϵ

) =

Fϵ in

(

0

,

T

] ×

Ω

,

U_ϵ

=

0 on

(

0

,

T

] ×

∂

Ω

,

U_ϵ

=

U_ϵ,0 in

{

0

} ×

Ω

,

(1.1) where

τ ∈ (

0

, ∞),

Λϵ_τ

≡



_K_ϵ _inΩϵ f ϵ2τ_k

ϵ inΩ_mϵ (depending on the permeability ofΩ), and both Kϵ

,

kϵare positive smooth functions

inΩ.

E-mail address:[email protected].

(2)

Since

ϵ ∈ (

0

,

1

)

, equations in(1.1)are non-uniform parabolic equations with discontinuous coefficients. In [5], existence of solution in W2,1

p

([

0

,

T

] ×

Ω

)

space for uniform parabolic equations with discontinuous coefficients can be found. For non-uniform parabolic equations with smooth coefficients, existence of solution in C2,α

_([

₀

_,

_T

_{] ×}

_Ω

₎

_{space was studied in [}₆_{]. It} is also known that if F_ϵ

,

U_ϵ,0are smooth, a piecewise regular solution of(1.1)exists uniquely for each

ϵ

and, by the energy method, the H1norm of the parabolic solution of(1.1)in the connected high permeability sub-region is bounded uniformly in

ϵ

[2,7]. Hölder continuity of the parabolic solution of(1.1)in

[

0

,

T

] ×

Ωis proved for each

ϵ

[7], but the Hölder norm of the solution may go to infinity as

ϵ ↘

0. In [4], convergence of solution of(1.1)in L∞

_([

₀

_,

_T

_];

_L2

₍

_Ω

₎₎

_{space as}

_{ϵ ↘}

₀ was obtained. Many studies of the uniform estimate in

ϵ

for the solutions of the elliptic equations in heterogeneous media had been done [2,3,8–11], but not the case for parabolic equations. The existence of piecewise regular solutions for elliptic diffraction equations in Hilbert space was considered in [2,9]. The uniform Lipschitz estimate in

ϵ

for a Laplace equation in perforated domains was given in [11], and a uniform Lp_{estimate in}

_ϵ

_{of the same problem was considered in [}₁₀_{]. A Lipschitz} estimate for uniform elliptic equations was studied in [3]. Uniform Hölder and Lipschitz estimates in

ϵ

for uniform elliptic equations in periodic domains were obtained in [8].

This work is to present a uniform Hölder estimate in

ϵ

for the solutions of the non-uniform parabolic equations with discontinuous coefficients. More precisely, the Hölder norm of the non-uniform parabolic solutions in the connected high permeability sub-region is shown to be bounded uniformly in

ϵ

. However, the Hölder norm of the solutions in the disconnected subset may not be bounded uniformly in

ϵ

. This is due to the non-zero source in the disconnected subset. In Section2, we present one example to show that. Certainly this is different from usual uniform parabolic equation cases, in which solutions are regular in the whole time–space domains. From the proof, we can see that the results are established for complex-valued solutions. On the other hand, one also notes that a complex-valued solution of(1.1)with complex-valued coefficients may be discontinuous or even unbounded [12]. A similar case could be found in elliptic equations with complex-valued coefficients (see [13]). It seems that the techniques used here could be used to study more general systems of elliptic type and parabolic type, and this will be pursued later. Some related uniform regularity results in the case of elliptic systems can be seen in [14,15].

The rest of the work is organized as follows: Notation and main results are stated in Section2. The main results are proved in Section3based on semigroup theory and on uniform Hölder estimate in

ϵ

for non-uniform elliptic equations. To apply semigroup theory, an infinitesimal generator of an analytic semigroup from elliptic equations is required. So a W2,p_estimate for solutions of elliptic diffraction equations is derived in Section4. Two convergence results for solutions of non-uniform elliptic equations are shown in Section5. By results in Section5, a uniform Hölder estimate in

ϵ

for non-uniform elliptic solutions is proved in Section6.

2. Notation and main result

LetΩbe the closure of the domainΩ. Let Lp

(

Ω

)

(resp. Hk

(

Ω

),

Wk,p

(

Ω

)

) denote a complex Sobolev space with norm

∥ · ∥

_Lp₍_Ω₎(resp.

∥ · ∥

_Hk₍_Ω₎

, ∥ · ∥

_Wk,p₍_Ω₎), W₀1,p

(

Ω

) ≡ {ϕ ∈

W1,p

(

Ω

)|ϕ|

_∂_Ω

=

0

}

,

H₀1

(

Ω

) ≡

W₀1,2

(

Ω

),

C₀∞

(

Ω

)

be the set

containing all infinite differentiable functions with compact support inΩ

,

C

(

Ω

)

consist of all continuous functions inΩ with norm

∥ · ∥

_C₍_Ω₎

,

Cσ

(

Ω

)

(resp. C1,σ

₍

_Ω

₎

_{) denote a Hölder space with norm}

_{∥ · ∥}

C σ(Ω)(resp.

∥ · ∥

C1,σ₍_Ω₎), and

[

ϕ]

_{C σ}₍_Ω₎(resp.

[

ϕ]

_C1,σ₍_Ω₎) denote the Hölder semi-norm of

ϕ

(resp.

∇

ϕ

) for k

≥ −

1

,

p

∈ [

1

, ∞]

, and

σ ∈ (

0

,

1

]

[16,17]. If

ϕ

is a complex function,

ϕ

denotes its complex conjugate. If B1and B2are two Banach spaces,L

(

B1,B2)is the set of all bounded linear maps from B1to B2with norm

∥ · ∥

L(B1,B2). For any Banach space B, define

∥

ϕ1, ϕ2, . . . , ϕ

m

∥

B

≡ ∥

ϕ1

∥

B

+ ∥

ϕ2

∥

B

+ · · · + ∥

ϕ

m

∥

B,

denote its dual space by B′_{, and denote the pairing between B and its dual space B}′_by

_⟨·

_{, ·⟩}

B,B′. L∞

(

I

;

B

) ≡ {ϕ :

I

→

B

|

sup_t∈I

∥

ϕ(

t

)∥

B

< ∞}

. The function spaces C

(

I

;

B

),

Cσ

(

I

;

B

)

for

σ ∈ (

0

,

1

]

and an interval I

⊂

R are defined as those in pages 1, 3 [18]. Br

(

x

)

represents a ball centered at x with radius r. For any domain D

,

D is the closure of D

, ∂

D is the boundary of D

,

D

/

r

≡ {

x

|

rx

∈

D

}

, |

D

|

is the volume of D, andXDis the characteristic function on D. For any

ϕ ∈

L

1

₍

_B r

(

x

) ∩

Ω

)

,

(ϕ)

x,r

≡ −



Br(x)∩Ω

ϕ(

y

)

dy

≡

1

|

Br

(

x

) ∩

Ω

|



Br(x)∩Ω

ϕ(

y

)

dy

.

For any p

∈

(

1

, ∞), τ ∈ (

0

, ∞)

, and

ϵ ∈ (

0

,

1

)

,



Aϵ_τ

ϕ ≡ −∇ · (

Λϵ_τ

∇

ϕ),

Bp

(

Aϵ_τ

) ≡ ϕ ∈

W01,p

(

Ω

)| ϕ ∈

W 2,p

₍

_Ωϵ f

) ∩

W 2,p

₍

_Ωϵ m

),

Kϵ

∇

ϕ · ⃗

nϵ

|

∂Ωmϵ

=

ϵ

2τ_k ϵ

∇

ϕ · ⃗

nϵ

|

∂Ωmϵ



,

wheren

⃗

ϵis a normal vector on

∂

Ω_mϵ. It is not difficult to see that Bp

(

Aϵ_τ

)

with norm

∥

ϕ∥

Bp(Aϵτ)

≡ ∥

Aϵτ

ϕ∥

Lp₍_Ω₎is a normed

space. Let Bp

(

Aϵ_τ

)

denote the closure of Bp

(

Aϵ_τ

)

in Lpspace (we shall see Bp

(

Aϵ_τ

) =

Lp

(

Ω

)

fromLemma 3.4). For any

λ, ν >

0, we define

ˆ

K_λ,ν

(

x

) ≡

K_λ

(ν

x

)

and k

ˆ

_λ,ν

(

x

) ≡

k_λ

(ν

x

).

(2.1)

Let Ym

⊂

D

⊂

Y

=

Yf

∪

Ymsatisfy

(3)

We assume that there are

ϵ, σ ,

e

∈

(

0

,

1

), τ ∈ (

0

, ∞)

, and

δ, α, β >

0 such that A1. Ωand Ymare C1,edomains,

A2. K_ϵ

,

k_ϵ

∈

W1,∞

₍

_Ω

_),

_K

ϵ

,

kϵ

∈

(α, β), ∥ ˆ

Kϵ,ϵ

∥

W1,∞₍_Ω_/ϵ)is bounded independent of

ϵ

, and there is a set

{

α

_ϵ,j

∈

(α, β)|ϵ ∈

(

0

,

1

),

j

∈

_Zn

_}

_satisfying

∥ ˆ

K_ϵ,ϵ

−

α

_ϵ,_j

∥

_W1,∞₍₍_D_\_Y_m₊_j_)∩_Ω_/ϵ)

+ ∥ ˆ

kϵ,ϵ

−

α

ϵ,j

∥

W1,∞₍₍Ym+j)∩Ω/ϵ)

≤

c

α

ϵ,j

where c is small and depends on Ym, A3. F_ϵ

∈

Cσ

([

0

,

T

];

Ln+δ

₍

_Ω

₎₎

_,_Aϵ

τUϵ,0

−

Fϵ

|

t=0

∈

Bn+δ

(

Aϵ_τ

)

, Uϵ,0

∈

Bn+δ

(

Aϵ_τ

)

.

The main results are:

Theorem 2.1. Under A1–A3, the solution of(1.1)satisfies

∥

U_ϵ

∥

_C1_([₀_,_T_];_Ln+δ₍_Ω₎₎

+ ∥

U_ϵ

∥

_C_([₀_,_T];Bn+δ(Aϵτ))

≤

c

∥

Uϵ,0

∥

Bn+δ(Aϵτ)

+ ∥

Fϵ

∥

C σ([0,T];Ln+δ₍_Ω₎₎



,

where c is a constant independent of

ϵ, τ

.

Theorem 2.2. Under A1–A3 and

τ ∈ (

0

,

1

]

, the solution of(1.1)satisfies

∥

U_ϵ

∥

_C1_([₀_,_T_];_Ln+δ₍_Ω₎₎

+ ∥

U_ϵ

∥

_C_([₀_,_T_];_{C µ}₍_Ωϵ f))

+

sup j∈Zn ϵ(Ym+j)⊂Ωϵm

ϵ

τ

_∥

_U ϵ

∥

_C_([₀_,_T_];_{C µ}_(ϵ(_Y m+j)))

≤

c

∥

U_ϵ,₀

∥

_Bn_+δ₍_Aϵ_τ₎

+ ∥

F_ϵ

∥

_{C σ}_([₀_,_T_];_Ln+δ₍_Ω₎₎



,

(2.3)

ϵ

. Here

µ ∈ (

0

,

₂₍_nδ₊_δ)

)

is a constant depending on n

, δ, σ, α, β,

Yf

,

Ω. Besides, there is a

ν ∈ (

0

, µ)

such that

∥

U_ϵ

∥

_{C ν}_([₀_,_T_]×_Ωϵ

f)

≤

c

∥

Uϵ,0

∥

Bn+δ(Aϵτ)

+ ∥

Fϵ

∥

C σ([0,T];Ln+δ(Ω))



,

(2.4)

ϵ

.

In(2.3), we do not prove that the Hölder norm of the solution of(1.1)in the disconnected subset is bounded uniformly in

ϵ

. We now give one example to show that if the source F_ϵis not zero in the disconnected subset, it is really the case. Suppose

ϕ ∈

C∞

0

(

Rn

)

has support in Ym. Define, for

ϵ ∈ (

0

,

1

)

,

ϕ

ϵ

(

x

) ≡

ϕ



x

ϵ

−

j



if x

∈

ϵ(

Ym

+

j

) ⊂

Ω

(

2

ϵ)

for some j

∈

Zn

,

0 elsewhere

,

Φϵ

(

t

,

x

) ≡

e−t

ϕ

_ϵ

(

x

)

in Rn

.

Then we see thatΦ_ϵ

=

0 in

[

0

,

T

] ×

Ω_fϵandΦ_ϵhas support in

[

0

,

T

] ×

Ω_mϵ. If we set

τ =

K_ϵ

=

k_ϵ

=

1 inΛϵ_τ, thenΦ_ϵ satisfies



∂

tΦϵ

− ∇ ·

(

Λϵ1

∇

Φϵ

) =

fϵ in

(

0

,

T

] ×

Ω

,

Φϵ

=

0 on

(

0

,

T

] ×

∂

Ω

,

Φϵ

(

t

=

0

) = ϕ

ϵ inΩ

,

where f_ϵ

(

x

) ≡



−

e−t



1

ϕ



x

ϵ

−

j



+

ϕ



x

ϵ

−

j



if x

∈

ϵ(

Ym

+

j

) ⊂

Ω

(

2

ϵ)

for j

∈

Zn

,

0 elsewhere

.

Clearly, for any

δ >

0 and

ϵ, σ ∈ (

0

,

1

), ∥ϕ

_ϵ

∥

_Bn_+δ₍_Aϵ_τ₎

+ ∥

f_ϵ

∥

_{C σ}_([₀_,_T_];_Ln+δ₍_Ω₎₎is bounded uniformly in

ϵ

. But the Hölder norm

of the functionsΦ_ϵin the disconnected subsetΩ_mϵ is not be bounded uniformly in

ϵ

if the source function f_ϵ

̸=

0 inΩ_mϵ.

Remark 2.1. We recall an extension result from [19].

For 1

≤

p

< ∞

, there is a constant

γ (

Yf

,

p

)

and a linear continuous extension operatorΠϵ

:

W1,p

(

Ωfϵ

) →

W

1,p

₍

_Ω

₎

_such that (1) If

ϕ ∈

W1,p

(

Ω_fϵ

)

, then











Πϵ

ϕ = ϕ

inΩ_fϵalmost everywhere

,

∥

Π_ϵ

ϕ∥

_Lp₍_Ω₎

≤

γ (

Y_f

,

p

)∥ϕ∥

_Lp₍_Ωϵ f)

,

∥∇

Π_ϵ

ϕ∥

_Lp₍_Ω₎

≤

γ (

Y_f

,

p

)∥∇ϕ∥

_Lp₍_Ωϵ f)

,

∥

Π_ϵ

ϕ∥

_{C σ}₍_Ω₎

≤

γ (

Yf

,

p

)∥ϕ∥

C σ(Ωϵf) if

ϕ ∈

C σ

₍

_Ωϵ f

)

for

σ ∈ (

0

,

1

),

Πϵ

ϕ = ζ

inΩif

ϕ = ζ |

Ωfϵfor some linear function

ζ

inΩ

.

(4)

(2) If

ζ(

x

) ≡ ϕ(

sx

)

in B1(x0) ∩Ω_fϵ

/

s for any x0

∈

Ω

/

s and constant s

> ϵ

, thenΠϵ/s

ζ (

x

) =

Πϵ

ϕ(

sx

)

in B1/2(x0) ∩Ω

/

s. The Hölder estimate(2.5)4and the statement (2) are not written in [19], but can be seen from its proof.

From [4], we know that the solution U_ϵof(1.1)with

τ ∈ (

0

,

1

)

converges to a function U in L∞

([

0

,

T

];

L2

(

Ω

))

as

ϵ ↘

0, and the function U satisfies a heat equation. ByTheorem 2.2andRemark 2.1,

∥

Π_ϵU_ϵ

|

_Ωϵ

f

∥

C ν([0,T]×Ω)is bounded independent

of

ϵ

. It is not difficult to see that, for the solution U_ϵof(1.1)with

τ ∈ (

0

,

1

),

Π_ϵU_ϵ

|

_Ωϵ

f also converges to U in C

ν

_([

₀

_,

_T

_{] ×}

_Ω

₎

norm for some

ν ∈ (

0

,

1

)

as

ϵ ↘

0.

3. Proofs ofTheorems 2.1and2.2

Proofs ofTheorems 2.1and2.2are based on a sequence of lemmas. First we consider an interpolation result.

Lemma 3.1. If

ϕ ∈

Lq

₍

_Ω

_{) ∩}

_Cµ

₍

_Ω

₎

_{for any q}

_∈

₍

₁

_{, ∞)}

_and

_{µ ∈ (}

₀

_,

₁

₎

_{, then}

∥

ϕ∥

_{C ν}₍_Ω₎

≤

c

∥

ϕ∥

1−θ

Lq₍_Ω₎

∥

ϕ∥

θ_{C µ}₍_Ω₎

,

where

ν ∈ (

0

, µ), θ ∈ (

0

,

1

)

, and c is a constant depending on

ν,

n

,

q

, µ, θ,

Ω. Proof. By Proposition 1.1.3 [18],

ϕ

satisfies

∥

ϕ∥

_{C ν}₍_Ω₎

≤

c

∥

ϕ∥

1−θ1

C(Ω)

∥

ϕ∥

θC µ1(Ω)

,

(3.1)

where

ν ∈ (

0

, µ), θ1

∈

(

0

,

1

)

, and c is a constant depending on

ν, µ, θ1

. Fix x

∈

Ωand

δ >

0 to see

|

ϕ(

x

)| ≤



ϕ(

x

) − −



Bδ(x)∩Ω

ϕ(

y

)

dy



+



−



Bδ(x)∩Ω

ϕ(

y

)

dy



≤ [

ϕ]

_{C µ}₍_Ω₎

−



Bδ(x)∩Ω

|

x

−

y

|

µdy

+



−



Bδ(x)∩Ω

|

ϕ(

y

)|

qdy



1/q

≤

c1δµ

[

ϕ]

_{C µ}₍_Ω₎

+

c2δ−n/q

∥

ϕ∥

_Lq₍_Ω₎

,

(3.2)

where constants c1,c2depend on domainΩonly. Taking the minimum of the right hand side of(3.2)on

δ

, we obtain

|

ϕ(

x

)| ≤

c

(

n

,

q

, µ,

Ω

)∥ϕ∥

1−θ2

Lq₍_Ω₎

[

ϕ]

θ_{C µ}2₍_Ω₎

,

(3.3)

where

θ2

∈

(

0

,

1

)

and c depend on n

,

q

, µ,

Ω.(3.1)and(3.3)imply the lemma.

From the proof ofLemma 3.1, we also have

Lemma 3.2. If

ϕ ∈

Lq

₍

_Ωϵ

f

) ∩

Cµ

(

Ω

ϵ

f

)

for any q

∈

(

1

, ∞)

and

µ ∈ (

0

,

1

)

, then

∥

ϕ∥

_L∞₍_Ωϵ f)

≤

c

∥

ϕ∥

1−θ Lq₍_Ωϵ f)

[

ϕ]

θ C µ(Ωϵf)

,

where

θ ∈ (

0

,

1

)

and c is a constant depending on n

,

q

, µ,

Ymbut independent of

ϵ

. Consider the following elliptic problem:

−∇ ·

₍

_Λϵ_τ

∇

ϕ

_ϵ

) =

f_ϵ inΩ

,

ϕ

ϵ

=

0 on

∂

Ω

.

(3.4)

We have the following uniform a-priori estimates:

Lemma 3.3. If A1–A2 hold, then

(1) The solution of (3.4)satisfies, for p

∈

(

1

, ∞), τ ∈ (

0

, ∞)

, and

ϵ ∈ (

0

,

1

)

,

∥

ϕ

_ϵ

∥

_W1,p₍_Ω₎

+ ∥

ϕ

_ϵ

∥

_W2,p₍_Ωϵ

f)

+ ∥

ϕ

ϵ

∥

W2,p(Ωmϵ)

≤

cϵ,p

∥

fϵ

∥

Lp(Ω)

,

(3.5)

where c_ϵ,pis a constant independent of

ϕ

ϵ

,

fϵbut depending on

ϵ,

p

, τ

. (2) The solution of (3.4)satisfies, for any

δ >

0

, τ ∈ (

0

,

1

]

, and

ϵ ∈ (

0

,

1

)

,

∥

ϕ

_ϵ

∥

_{C µ}₍_Ωϵ f)

+

sup j∈Zn ϵ(Ym+j)⊂Ωϵm

ϵ

τ

_∥

_ϕ

ϵ

∥

_{C µ}_(ϵ(_Y m+j))

≤

c

∥

fϵ

∥

Ln+δ₍_Ω₎

,

(3.6)

ϵ

. Here

µ ∈ (

0

,

₂₍_nδ₊_δ)

)

is a constant depending on n

, δ, α, β,

Yf

,

Ω(see A2). The proof of(3.5)is given in Section4and the proof of(3.6)is in Section6.

(5)

Lemma 3.4. For any p

∈

(

1

, ∞), τ ∈ (

0

, ∞)

, and

ϵ ∈ (

0

,

1

)

, the set Bp

(

Aϵτ

)

is dense in Lp

(

Ω

)

and Bp

(

Aϵτ

)

with norm

∥

ϕ∥

_Bp₍_Aϵ_τ₎

≡ ∥

Aϵ_τ

ϕ∥

Lp₍_Ω₎is a Banach space.

Proof. DefineOϵ

≡ {

x

∈

Ω

|

dist

(

x

, ∂

Ω_fϵ

) ≥ ϵ

2

}

and letXOϵbe the characteristic function onOϵ. ThenXOϵconverges to 1 in measure (see page 91 [20]) on domainΩ as

ϵ ↘

0. For any

ϕ ∈

Lp

₍

_Ω

₎

_{, we have}

_ϕ

_X

Oϵ

∈

Lp

(

Ω

)

and

ϕ

XOϵ

=

0 in a neighborhood of

∂

Ω_fϵ. By the Lebesque dominant theorem and Proposition in page 92 [20], there is a subsequence of

ϕ

XOϵ (same notation for subsequence) converging to

ϕ

in Lp

(

Ω

)

as

ϵ ↘

0. So for any

δ >

0, there is a

ϵ0

such that

∥

ϕ − ϕ

XOϵ

∥

Lp₍_Ω₎

≤

δ/

2 as

ϵ < ϵ0

. From pages 147–148 [17], there is a mollifier

η

_δsuch that the convolution of

η

_δ and

ϕ

XOϵ(i.e.,

(ϕ

XOϵ) ∗ ηδ) for some

ϵ < ϵ0

satisfies

∥

ϕ

XOϵ

−

(ϕ

XOϵ

) ∗ η

δ

∥

Lp₍_Ω₎

≤

δ/

2 and

(ϕ

X_O_ϵ

) ∗ η

_δ

=

0 in some

neighborhood of

∂

Ω_fϵ. Clearly,

(ϕ

XOϵ) ∗ ηδ

∈

Bp

(

Aϵ_τ

)

and

∥

ϕ − (ϕ

XOϵ

) ∗ η

δ

∥

Lp₍_Ω₎

≤

δ

. So B_p

(

Aϵ_τ

)

is dense in Lp

(

Ω

)

. By

(3.5)inLemma 3.3, we see that Bp

(

Aϵ_τ

)

with norm

∥ · ∥

Bp(Aϵτ)is a Banach space.

∈

(

1

, ∞), τ ∈ (

0

, ∞)

, and

ϵ ∈ (

0

,

1

)

, the adjoint operator of Aϵ_τ

:

_B_p

(

Aϵ_τ

) ⊂

Lp

(

Ω

) →

Lp

(

Ω

)

is

Aϵ_τ

:

_B_q

(

Aϵ_τ

) ⊂

Lq

(

Ω

) →

Lq

(

Ω

)

, where1_p

+

1 q

=

1.

Proof. Fix a p

∈

(

1

, ∞), τ ∈ (

0

, ∞)

, and

ϵ ∈ (

0

,

1

)

, denote the adjoint ofAϵ_τ

:

_B_p

(

Aϵ_τ

) ⊂

Lp

₍

_Ω

_{) →}

_Lp

₍

_Ω

₎

_by_Aτ′

ϵ, and

assume1_p

+

1

q

=

1. Integration by parts yields

⟨

Aϵ_τ

ζ, η⟩

Lp₍_Ω_),_Lq₍_Ω₎

= ⟨

ζ,

Aϵ_τ

η⟩

_Lp₍_Ω_),_Lq₍_Ω₎ (3.7)

for every

ζ ∈

_Bp

(

Aϵ_τ

)

and

η ∈

Bq

(

Aϵ_τ

)

. See Section2for

⟨·

, ·⟩

Lp₍_Ω_),_Lq₍_Ω₎. Therefore B_q

(

Aϵ_τ

) ⊂

dom

(

Aτ

′

ϵ

)

(that is, the domain

ofAτ_ϵ′) andAϵ_τ

η =

A_ϵτ′

η

for

η ∈

_Bq

(

Aϵ_τ

)

. Let

η ∈

dom

(

Aτ_ϵ′

) ⊂

Lq

₍

_Ω

₎

_and

_{ϕ =}

_Aτ′

ϵ

(η)

. Then, by the definition of the adjoint operator, we have

⟨

Aϵ_τ

ζ, η⟩

Lp₍_Ω_),_Lq₍_Ω₎

= ⟨

ζ, ϕ⟩

_Lp₍_Ω_),_Lq₍_Ω₎ for all

ζ ∈

_B_p

(

Aϵ_τ

).

(3.8)

Since Bq

(

Aϵ_τ

)

is dense in Lq

(

Ω

)

byLemma 3.4, there is a sequence

η

s

∈

Bq

(

Aϵ_τ

)

such that

η

s

→

η

in Lq

(

Ω

)

as s

→ ∞

. By (3.7)and(3.8), lim s→∞

⟨

ζ,

Aϵ_τ

η

s

⟩

Lp₍_Ω_),_Lq₍_Ω₎

=

lim s→∞

⟨

Aϵ_τ

ζ , η

s

⟩

Lp₍_Ω_),_Lq₍_Ω₎

= ⟨

Aϵ_τ

ζ , η⟩

Lp₍_Ω_),_Lq₍_Ω₎

= ⟨

ζ, ϕ⟩

_Lp₍_Ω_),_Lq₍_Ω₎

.

Since Bp

(

Aϵ_τ

)

is dense in Lp

(

Ω

)

byLemma 3.4,A_τϵ

η

sconverges to

ϕ

weakly in Lq

(

Ω

)

as s

→ ∞

. By(3.5)inLemma 3.3, we see

η ∈

_Bq

(

Aϵ_τ

)

. So

ϕ =

Aϵ_τ

(η)

. Therefore, dom

(

Aτ

′

ϵ

) ⊂

Bq

(

Aϵ_τ

)

andAϵ_τ

=

Aτ ′

ϵ.

Next we want to show

−

Aϵ_τ is an infinitesimal generator of an analytic semigroup. If so, by semigroup group theory, we can obtain the existence of the solutions of some time-dependent problems. For this purpose, we shall work on complex-valued functions in the next lemma.

∈

(

1

, ∞), τ ∈ (

0

, ∞)

, and

ϵ ∈ (

0

,

1

)

, the operator

−

Aϵ_τ is an infinitesimal generator of an analytic semigroup of contractions on Lp

₍

_Ω

₎

_and

∥

(λ +

Aϵ_τ

)

−1

∥

_L₍_Lp₍_Ω_),_Lp₍_Ω₎₎

≤

1

λ

for any

λ >

0

.

(3.9)

Moreover, there is a

θ ∈ (

0

, π/

2

)

independent of

ϵ, τ

such that

(1) The resolvent set

ρ(−

Aϵ_τ

)

of

−

Aϵ_τ(see page 8 [21]) satisfies

ρ(−

Aϵ_τ

) ⊃ ℜ(θ) ≡ {

z

∈

_C

| |

arg

(

z

)| < π − θ},

where arg

(

z

)

denotes the argument of the complex number z.

(2)

∥

(λ +

Aϵ_τ

)

−1

_∥

L(Lp₍_Ω_),_Lp₍_Ω₎₎

≤

1

cθ|λ|for any

λ ∈ ℜ(θ)

, where cθis a constant independent of

ϵ, τ

.

Proof. We assume p

∈

(

1

, ∞), τ ∈ (

0

, ∞)

, and

ϵ ∈ (

0

,

1

)

. The proof of this lemma includes three steps.

Step 1. Claim

λ +

Aϵ_τ

:

_B_p

(

Aϵ_τ

) ⊂

Lp

₍

_Ω

_{) →}

_Lp

₍

_Ω

₎

_{is injective for any}

_{λ >}

_{0. Let q}

₌

p

p−1. If

ϕ ∈

Bp

(

Aϵτ

)

, we define

ϕ

∗

≡ |

ϕ|

p−2

ϕ ∈

Lq

(

Ω

)

(

ϕ

is the complex conjugate of

ϕ

). Then

⟨

ϕ, ϕ

∗

⟩

Lp₍_Ω_),_Lq₍_Ω₎

= ∥

ϕ∥

p

Lp₍_Ω₎. Integration by parts yields

⟨

Aϵ_τ

ϕ, ϕ

∗

⟩

Lp₍_Ω_),_Lq₍_Ω₎

= −



Ω

∇ ·

(

Λϵ_τ

∇

ϕ)|ϕ|

p−2

ϕ

dx

=



ΩΛ ϵ τ

∇

ϕ∇(|ϕ|

p−2

ϕ)

dx

=



ΩΛ ϵ τ

|

ϕ|

p−2

∇

ϕ∇ϕ + ϕ∇ϕ∇|ϕ|

p−2



dx

.

(6)

Note

∇|

ϕ|

p−2

=

p−2 2

|

ϕ|

p−4

_{(ϕ∇ϕ + ϕ∇ϕ)}

_{. Denote}

_|

_ϕ|

(p−4)/2

_{ϕ∇ϕ ≡ ℓ +}

_i

_ω

_{. We find}

⟨

Aϵ_τ

ϕ, ϕ

∗

⟩

Lp₍_Ω_),_Lq₍_Ω₎

=



ΩΛ ϵ τ



(

p

−

1

)|ℓ|

2

+ |

ω|

2

+

i

(

p

−

2

)ℓ · ω

dx

,

where

|

ℓ|

(resp.

|

ω|

) is the length of the vector

ℓ

(resp.

ω

). So the real part of

⟨

A_τϵ

ϕ, ϕ

∗

⟩

Lp₍_Ω_),_Lq₍_Ω₎satisfies, by A2,

Re

⟨

Aϵ_τ

ϕ, ϕ

∗

⟩

Lp₍_Ω_),_Lq₍_Ω₎

≥

c_p_,α





Ωfϵ

|

ℓ|

2

+ |

ω|

2dx

+

ϵ

2τ



Ωmϵ

|

ℓ|

2

+ |

ω|

2dx



≥

0

,

(3.10)

where cp,αis a constant depending on p

, α

. The ratio of the imaginary part to the real part of

⟨

Aϵ_τ

ϕ, ϕ

∗

⟩

Lp₍_Ω_),_Lq₍_Ω₎then

satisfies, by A2,

|

Im

⟨

Aϵ_τ

ϕ, ϕ

∗

⟩

Lp₍_Ω_),_Lq₍_Ω₎

|

Re

⟨

Aϵ_τ

ϕ, ϕ

∗

⟩

Lp₍_Ω_),_Lq₍_Ω₎

|

≤

|

p

−

2

|

β





Ωfϵ

|

ℓ|

2

_{+ |}

_ω|

2_dx

₊

_ϵ

2τ



Ωmϵ

|

ℓ|

2

_{+ |}

_ω|

2_dx



2cp,α





Ωϵf

|

ℓ|

2

_{+ |}

ω|

2_dx

₊

ϵ

2τ



Ωmϵ

|

ℓ|

2

_{+ |}

ω|

2_dx



=

|

p

−

2

|

β

2cp,α

.

(3.11) From(3.10)it follows that, for any

λ >

0 and

ϕ ∈

_Bp

(

Aϵ_τ

)

,

λ∥ϕ∥

Lp₍_Ω₎

≤ ∥

(λ +

Aϵ_τ

)ϕ∥

_Lp₍_Ω₎

.

(3.12)

By(3.12),

λ +

Aϵ_τ

:

_B_p

(

_A_τϵ

) ⊂

Lp

₍

_Ω

_{) →}

_Lp

₍

_Ω

₎

_{is injective. So we prove the claim.}

Step 2. Claim

λ +

Aϵ_τ

:

_B_p

(

Aϵ_τ

) ⊂

Lp

₍

_Ω

_{) →}

_Lp

₍

_Ω

₎

_{is bijective for any}

_{λ >}

_{0. If}

_{η ∈}

_Lq

₍

_Ω

₎

_{for q}

₌

p

p−1 satisfies

⟨

(λ +

Aϵ_τ

)ϕ, η⟩

Lp₍_Ω_),_Lq₍_Ω₎

=

0 for all

ϕ ∈

_B_p

(

Aϵ_τ

)

, then

η

is in the domain of the adjoint operator

λ +

Aτ

′

ϵ (hereAτ

′

ϵ is

the adjoint operator ofAϵ_τ) of

λ +

Aϵ_τ. ByLemma 3.5,

⟨

ϕ, (λ +

Aϵ_τ

)η⟩

Lp₍_Ω_),_Lq₍_Ω₎

=

0 and

η ∈

_B_q

(

Aϵ_τ

)

. Since B_p

(

Aϵ_τ

)

is dense

on Lp

₍

_Ω

₎

_by_{Lemma 3.4}_,

_{(λ +}

_Aϵ

τ

)η =

0. Then(3.12), with p replaced by q, implies

η =

0. So the range of

λ +

Aϵτ is dense

in Lp

₍

_Ω

₎

_{. By}_(3.5)_,_Aϵ

τ

:

Bp

(

Aϵ_τ

) ⊂

Lp

(

Ω

) →

Lp

(

Ω

)

is a closed linear operator. It is not difficult to see that

λ +

Aϵ_τ is also a closed linear operator. Thus, the range of

λ +

Aϵ_τ is a closed set in Lp

(

Ω

)

. Since the range of

λ +

Aϵ_τ is dense and closed in

Lp

₍

_Ω

₎

_{, the range of}

_{λ +}

_Aϵ

τ is Lp

(

Ω

)

. So we prove the claim. Moreover, by(3.12),

∥

(λ +

Aϵ_τ

)

−1

∥

_L₍_Lp₍_Ω_),_Lp₍_Ω₎₎

≤

1

λ

for any

λ >

0

.

So we prove(3.9).

Step 3. Claim

−

Aϵ_τis an infinitesimal generator of an analytic semigroup on Lp

₍

_Ω

₎

_{. By Step 2,}_(3.5)_in_{Lemmas 3.3}_and_3.4_{, the} Hille–Yosida Theorem [21] implies that

−

Aϵ_τ is an infinitesimal generator of a C0-semigroup of contractions on Lp

(

Ω

)

. To prove that the semigroup generated by

−

Aϵ_τis analytic, we observe that, by(3.10)and(3.11), the numerical rangeN

(−

Aϵ_τ

)

of

−

Aϵ_τ(see page 12 [21] and Remark 3.2 in page 25 [22]) is contained in the set

Nθ1

≡ {

z

∈

C

| |

arg

(

z

)| > π − θ1

}

,

where

θ1

=

tan−1

(

|p_2c−2|β

p,α

) ∈ (

0

, π/

2

)

. Choosing

θ1

< θ < π/

2 and denoting

ℜ

(θ) ≡ {

z

∈

_C

| |

arg

(

z

)| < π − θ},

there is a constant c_θ

>

0 independent of

ϵ, τ

such that the distance from z

∈ ℜ

(θ)

toN

(−

A_τϵ

)

(i.e., dist

(

z

,

N

(−

Aϵ_τ

))

) satisfies

dist

(

z

,

N

(−

Aϵ_τ

)) ≥

c_θ

|

z

|

.

Since

λ >

0 is in the resolvent set

ρ(−

Aϵ_τ

)

of

−

Aϵ_τ by Step 2, Theorem 3.9 in page 12 [21] then implies

ℜ

(θ) ⊂ ρ(−

Aϵ_τ

)

and

∥

(λ +

Aϵ_τ

)

−1

∥

_L₍_Lp₍_Ω_),_Lp₍_Ω₎₎

≤

1

c_θ

|

λ|

for

λ ∈ ℜ(θ).

By(3.5)ofLemma 3.3and the energy method, 0

∈

ρ(−

A_τϵ

)

. By Theorem 5.2(c) in page 61 [21],

−

Aϵ_τ is an infinitesimal generator of an analytic semigroup on Lp

(

Ω

)

.

Proof of Theorem 2.1. Tracing the proofs of Proposition 2.1.1, Eq. (4.0.3), and Theorem 4.3.1 [18], and employing Lemma 3.6, we know

(7)

Let

δ, τ >

0

, σ ∈ (

0

,

1

),

F_ϵ

∈

Cσ

([

0

,

T

];

Ln+δ

(

Ω

)),

U_ϵ,0

∈

Bn+δ

(

Aϵ_τ

)

, andAϵ_τUϵ,0

−

Fϵ

(

t

=

0

) ∈

Bn+δ

(

Aϵ_τ

)

. A strict

solution U_ϵof (1.1)exists and there is a constant c independent of

ϵ, τ

such that

∥

U_ϵ

∥

_C1_([₀_,_T];Ln+δ₍_Ω₎₎

+ ∥

Uϵ

∥

C([0,T];Bn+δ(Aϵ_τ))

≤

c

∥

Uϵ,0

∥

Bn+δ(Aϵ_τ)

+ ∥

Fϵ

∥

_{C σ}_([₀_,_T_];_Ln+δ₍_Ω₎₎



.

(3.13) So we proveTheorem 2.1.

Proof of Theorem 2.2. By(1.1)and for each fixed t

∈

(

0

,

T

]

,

−∇ ·

₍

_Λϵ_τ

∇

U_ϵ

(

t

, ·)) =

F_ϵ

(

t

, ·) − ∂

tUϵ

(

t

, ·)

inΩ

,

U_ϵ

(

t

, ·) =

0 on

∂

Ω

.

(3.6)inLemma 3.3andTheorem 2.1then imply(2.3). ByRemark 2.1, we can extend the function U_ϵ

|

_Ωϵ

f

(

t

, ·)

toΩ. The

extended functionΠ_ϵU_ϵsatisfies, by(2.3),(3.13), andRemark 2.1,

∥

Π_ϵU_ϵ

∥

_C1_([₀_,_T_];_Ln+δ₍_Ω₎₎

+ ∥

Π_ϵU_ϵ

∥

_C_([₀_,_T_];_{C µ}₍_Ω₎₎

≤

c

∥

U_ϵ,0

∥

Bn+δ(Aϵ_τ)

+ ∥

Fϵ

∥

_{C σ}_([₀_,_T_];_Ln+δ₍_Ω₎₎



,

(3.14) where

µ ∈ (

0

,

1

)

and c is independent of

ϵ

.(2.4)follows from Proposition 1.1.4 [18],(3.14), andLemma 3.1. So we prove Theorem 2.2.

4. Proof of(3.5)ofLemma 3.3

LetΓ

(

x

−

y

)

denote the fundamental solution of the Laplace’s equation (see Section 6.2 [23]). Define the single-layer and the double-layer potentials as, for any smooth function

ϕ

on the boundary

∂

D of a bounded C1,edomain D,











E∂D

(ϕ)(

x

) ≡



∂D Γ

(

x

−

y

)ϕ(

y

)

d

σ

y T∂D

(ϕ)(

x

) ≡



∂D

∇

_yΓ

(

x

−

y

) · ⃗

ny

ϕ(

y

)

d

σ

y T_∂∗_D

(ϕ)(

x

) ≡



∂D

∇

_xΓ

(

x

−

y

) · ⃗

nx

ϕ(

y

)

d

σ

y for x

∈

∂

_D

,

where e

∈

(

0

,

1

)

andn

⃗

y(resp.

⃗

nx) is the unit vector outward normal to

∂

D at point y

∈

∂

D (resp. x

∈

∂

D).

∈

(

1

, ∞)

, the linear operators



E∂D

:

W 1−1_p_,p

_(∂

D

) →

W2− 1 p,p

(∂

D

)

T∂D

:

W 1−1_p,p

_(∂

D

) →

W2− 1 p,p

(∂

D

)

(4.1)

are bounded. The operator I

−

λ

T∂Dis continuously invertible in W

2−1_p,p

_(∂

D

)

for any p

∈

(

1

, ∞)

and

λ ∈ [−

2

,

2

]

, where I is

the identity operator. Furthermore, there is a constant c independent of

λ ∈ [−

2

,

2

]

so that

∥

ϕ∥

W2− 1 p,p_(∂ D)

≤

c

∥

(

I

−

λ

T∂D

)(ϕ)∥

W2− 1 p,p_(∂ D) for

ϕ ∈

W2−1p,p

(∂

D

).

(4.2)

Proof. Denote by OPS−₁_,1₀the pseudo-differential operator of order

−

1 (see page 38 [24]). Tracing the proof of Theorem 2.5 Chapter XI [24], we see that ifG

∈

OPS−₁_,10(∂_D

)

, thenGis a bounded linear operator from W1−1p,p

(∂

D

)

to W2− 1 p,p

(∂

D

)

. SinceE∂D

,

T∂D

∈

OPS −1

1,0(∂D

)

(see pages 87–93 [23]), we know thatE∂D

,

T∂Dare bounded operators from W

1−1_p,p

_(∂

D

)

to W2− 1 p,p

(∂

D

)

.

Since D is a C1,edomain, bothT∂D

,

T

∗

∂Dare compact operators in L

p

_(∂

D

)

for p

∈

(

1

, ∞)

(see Corollary 2.2.14 [25]). For any

λ ∈

_{R, the dimensions of the kernels of I}

−

λ

_T_∂_Dand I

−

λ

_T∗

∂Dare same by Theorem 4.12 [26]. From Theorem 2.2.21 [25]

and Section 3.4 [27], there is a p0

∈

(

2

, ∞)

such that I

−

λ

T_∂∗_Dis continuously invertible in Lp

(∂

D

)

for any p

∈

(

1

,

p0)and

λ ∈ [−

2

,

2

]

. Since Lp

(∂

D

) ⊂

L2

(∂

D

)

for p

∈ [

2

, ∞),

I

−

λ

T_∂∗Dis injective for any p

∈ [

2

, ∞)

and

λ ∈ [−

2

,

2

]

. By Theorem

4.12 [26], I

−

λ

T∗

∂Dis continuously invertible for any p

∈ [

2

, ∞)

and

λ ∈ [−

2

,

2

]

. Again by Theorem 4.12 [26], we see that I

−

λ

T∂Dis also continuously invertible in L

p

_(∂

D

)

for p

∈

(

1

, ∞)

and

λ ∈ [−

2

,

2

]

. By(4.1)and inverse mapping theorem [28],

I

−

λ

T∂Dis continuously invertible in W

2−1_p,p

_(∂

D

)

for p

∈

(

1

, ∞)

and

λ ∈ [−

2

,

2

]

.

(4.2)is proved as follows. From above, we know thatT∂Dis a bounded linear operator in W

2−1_p,p

_(∂

D

)

and I

−

λ

T∂D

is continuously invertible in W2−1p,p

(∂

(8)

{

c_λ

,

d_λ

,

B_dλ

(λ)}

(depending on

λ

) satisfying











c_λ

,

d_λ

>

0

,

∥

(

I

−

λ

T∂D

)(ϕ)∥

W2− 1 p,p_(∂_D₎

≥

cλ

∥

ϕ∥

_W2−1p,p_(∂_D₎

,

∥

(

I

−

sT∂D

)(ϕ)∥

W2− 1 p,p_(∂_D₎

≥ ∥

(

I

−

λ

T∂D

)(ϕ)∥

_W2−1p,p_(∂_D₎

− |

s

−

λ| ∥

T∂D

(ϕ)∥

_W2−1p,p_(∂_D₎

≥

cλ 2

∥

ϕ∥

W2− 1 p,p_(∂ D) if s

∈

B_dλ

(λ) ⊂

R

.

Now we consider the open covering

{

B_dλ

(λ)}

_λ∈[−2,2] of

[−

2

,

2

]

. Since

[−

2

,

2

]

is a compact set, we can find a finite set

Z

⊂ [−

2

,

2

]

so that

{

B_dλ

(λ)}

_λ∈Zis also a covering of

[−

2

,

2

]

. Based on the finite sets

{

cλ

,

dλ

,

B_dλ

(λ)}

λ∈Z, we define c∗

=

min

{_cλ,_dλ,Bd_λ(λ)}λ∈Z c_λ

2

.

That is, c∗is the minimum value ofcλ₂ for

λ

in the finite setZ. If the c in(4.2)is taken to be c

=

1

/

c∗, we obtain(4.2).

Now we consider the following problem











−∇ ·

(

K

∇

Ψ_ϵ

) =

G_ϵ in Yf

,

−

ϵ

2τ

∇ ·

(

k

∇

ψ

_ϵ

) = ϵ

τg_ϵ in Ym

,

K

∇

Ψ_ϵ

· ⃗

ny

=

ϵ

2τk

∇

ψ

ϵ

· ⃗

ny on

∂

Ym

,

Ψϵ

=

ψ

_ϵ on

∂

Ym

,

(4.3)

where

τ ∈ (

0

, ∞), ϵ ∈ (

0

,

1

)

, andn

⃗

yis the unit vector outward normal to

∂

Ym. By D in(2.2), we define

D1

≡



x

∈

Yf

|

dist

(

x

, ∂

Yf

) >

1

4min

{

dist

(

Ym

, ∂

D

),

dist

(

D

, ∂

Y

)}



.

Then

∂

D

⊂

D1.

Lemma 4.2. Suppose

(1) K

,

k in Y satisfy

∥

K

−

d

∥

_W1,∞₍_Y

f)

+ ∥

k

−

d

∥

W1,∞(Ym)

≤

c0d where d

>

0 is a constant and c0

<

1

2is a small number

depending on Ym,

(2)

τ >

0

, ϖ ≡

min

{

2

,

p

}

for p

∈

(

1

, ∞), ∥

Ψ_ϵ

∥

_Lϖ₍_Y_f₎

+ ∥

G_ϵ

∥

_Lp₍_Y

f)

+ ∥

gϵ

∥

Lp(Ym)is bounded independently of

ϵ

, then any solution of (4.3)satisfies

∥

Ψ_ϵ

∥

_W2,p₍_D_\_Y m)

+

ϵ

τ

_∥

_ψ

ϵ

∥

_W2,p₍_Y_m₎

≤

c

,

(4.4)

ϵ, τ

.

Proof. Denote by c a constant independent of

ϵ, τ,

d. Consider(4.3)1in Yf. Theorem 8.8 and Theorem 9.11 [17] implies

d

∥

Ψ_ϵ

∥

_W2,p₍_D 1)

≤

c

.

(4.5) Let



ψ

ϵbe a solution of

−∇ ·

_(ϵ

2τ_d

_∇

_ψ

ϵ

+

ϵ

2τ

(

k

−

d

)∇ψ

_ϵ

) = ϵ

τg_ϵ in Ym

,



ψ

ϵ

|

∂Ym

=

0

,

(4.6) and



Ψϵa solution of







−∇ ·

(

d

∇

Ψ_ϵ

+

(

K

−

d

)∇

Ψ_ϵ

) =

G_ϵ in D

\

Ym

,



Ψϵ

|

_∂_Y_m

=

0

,



Ψϵ

−

Ψ_ϵ

|

_∂_D

=

0

.

(4.7)

Then, by(4.5)and Theorem 9.15 of [17],



d

∥

ψ

_ϵ

∥

_W2,p₍_Y_m₎

≤

c



ϵ

−τ

_{+ ∥}

₍

k

−

d

)∇ψ

_ϵ

∥

_W1,p₍_Y_m₎



,

d

∥

Ψ_ϵ

∥

_W2,p₍_D_\_Y m)

≤

c



1

+ ∥

(

K

−

d

)∇

Ψ_ϵ

∥

_W1,p₍_D_\_Y m)



.

(4.8)

Define

ψ

˘

_ϵ

≡

ψ

_ϵ

−



ψ

ϵin YmandΨ

˘

ϵ

≡

Ψϵ

−

Ψ



ϵin D

\

Ym.(4.3)and(4.6)–(4.7)imply











−

ϵ

2τ1

_ψ

˘

_ϵ

=

0 in Y_m

,

−

1_Ψ

˘

_ϵ

=

0 in D

\

Ym

,

˘

Ψϵ

|

_∂_Y_m

= ˘

ψ

_ϵ

|

_∂_Y_m

,

∇ ˘

Ψ_ϵ

· ⃗

ny

|

∂Ym

−

ϵ

2τ

_{∇ ˘}

_ψ

ϵ

· ⃗

ny

|

∂Ym

=

Fϵ

· ⃗

ny

/

d

,

˘

Ψϵ

|

_∂D

=

0

,

(4.9)

(9)

whereFϵ

=

(

d

−

K

)∇

Ψ_ϵ

−

ϵ

2τ

(

d

−

k

)∇ψ

_ϵ

−

d

∇



Ψϵ

+

ϵ

2τd

∇

ψ



ϵ. By(4.5),(4.8), and trace theorems in pages 240–241 [16],

∥

Fϵ

∥

_W1−1/p,p_(∂_Y m)

≤

c



1

+

ϵ

2τ

∥

(

k

−

d

)∇ψ

_ϵ

∥

_W1,p₍_Y m)

+ ∥

(

K

−

d

)∇

Ψϵ

∥

W1,p(D\Ym)



.

(4.10)

By Green’s formula,(4.9), and Theorem 6.5.1 [23], we see that



_˘

ψ

ϵ

/

2

+

T∂Ym

( ˘ψ

ϵ

) =

E∂Ym

(∂

ny

ψ

˘

ϵ

)

˘

Ψϵ

/

2

−

T∂Ym

( ˘

Ψϵ

) = −

E∂Ym

(∂

nyΨ

˘

ϵ

) +

E∂D

(∂

nyΨ

˘

ϵ

|

∂D

)

on

∂

Ym

,

where

∂

nyΨ

˘

ϵ

|

∂Dis the normal derivative ofΨ

˘

ϵon

∂

D. Therefore, by(4.9)4,

ϵ

2τ

₊

₁ 2

(

1

−

ϵ

2τ

)

˘

ψ

ϵ

−

T∂Ym

( ˘ψ

ϵ

) =

E∂D

(∂

nyΨ

˘

ϵ

|

∂D

)

1

−

ϵ

2τ

−

E∂Ym

(

Fϵ

· ⃗

ny

)

(

1

−

ϵ

2τ

)

_d on

∂

Ym

.

(4.11)

By(4.5),(4.8), and trace theorems in pages 240–241 [16],

d

∥

∂

_n_y_Ψ

˘

_ϵ

∥

_W1−1/p,p_(∂_D₎

≤

c



1

+ ∥

(

K

−

d

)∇

Ψ_ϵ

∥

_W1,p₍_D_\_Y_m₎



.

(4.12)

By(4.11)andLemma 4.1, we have

∥ ˘

ψ

_ϵ

∥

W2− 1 p,p_(∂_Y m)

≤

c



d−1

∥

Fϵ

∥

W1− 1 p,p_(∂_Y m)

+ ∥

∂

_n_yΨ

˘

_ϵ

∥

W1− 1 p,p_(∂_D₎



.

(4.13)

Eqs.(4.3)4,(4.8),(4.10),(4.12)and(4.13)imply

d

∥

Ψ_ϵ

∥

_W2,p₍_D_\_Y_m₎

+

ϵ

τd

∥

ψ

ϵ

∥

_W2,p₍_Y m)

≤

c



1

+

ϵ

τ

∥

(

k

−

d

)∇ψ

_ϵ

∥

_W1,p₍_Y m)

+ ∥

(

K

−

d

)∇

Ψϵ

∥

W1,p₍_D_\_Y_m₎



.

By assumption on K and k, we obtain(4.4).

Denote a portion of the boundary of Y by

∂1

Y

≡ {

y

∈

∂

Y

|

y

=

(

0

,

y2, . . . ,yn

)}

, and consider the following problem











−∇ ·

(

K

∇

Ψ_ϵ

) =

G_ϵ in Yf

,

−

ϵ

2τ

∇ ·

(

k

∇

ψ

_ϵ

) = ϵ

τg_ϵ in Ym

,

K

∇

Ψ_ϵ

· ⃗

ny

=

ϵ

2τk

∇

ψ

ϵ

· ⃗

ny on

∂

Ym

,

Ψϵ

=

ψ

ϵ on

∂

Ym

,

Ψϵ

=

Ψ_bϵ on

∂1

Y

,

(4.14)

where

τ ∈ (

0

, ∞)

andn

⃗

yis the unit vector outward normal to

∂

Ym. Let Ym

⊂



D

⊂

Y satisfy min

{

dist

(

Ym

, ∂



D

),

dist

(



D

, ∂

Y

\

∂1

Y

)} >

0 and

∂



D

∩

∂1

Y

̸= ∅

.

By an analogous argument asLemma 4.2, we also have

Lemma 4.3. Let

τ ∈ (

0

, ∞)

and

∥

K

−

d

∥

_W1,∞₍Yf)

+ ∥

k

−

d

∥

W1,∞(Ym)

≤

c0d where d

>

0 and c0

<

1

2 is a small number

depending on Ym. Any solution of (4.14)satisfies

∥

Ψ_ϵ

∥

_W2,p₍ D\Ym)

+

ϵ

τ

_∥

_ψ

ϵ

∥

_W2,p₍_Y_m₎

≤

c

∥

Ψ_ϵ

∥

_Lϖ₍_Y_f₎

+ ∥

G_ϵ

∥

_Lp₍_Y f)

+ ∥

gϵ

∥

Lp(Ym)

+ ∥

Ψbϵ

∥

W2,p₍_Y f)



,

where p

∈

(

1

, ∞), ϖ ≡

min

{

2

,

p

}

, and c is a constant independent of

ϵ, τ

.

Now we give the proof of(3.5)ofLemma 3.3. By partition of unity, A2, Theorem 8.8 and Theorem 9.11 [17],Lemmas 4.2 and4.3, we see that the solution of(3.4)satisfies, for fixed p

∈

(

1

, ∞), τ ∈ (

0

, ∞)

, and

ϵ ∈ (

0

,

1

)

,

∥

ϕ

_ϵ

∥

_W1,p₍_Ω₎

+ ∥

ϕ

_ϵ

∥

_W2,p₍_Ωϵ

f)

+ ∥

ϕ

ϵ

∥

W2,p(Ωmϵ)

≤

c

∥

fϵ

∥

Lp(Ω)

+ ∥

ϕ

ϵ

∥

Lϖ(Ωfϵ)



,

(4.15)

where

ϖ ≡

min

{

2

,

p

}

and c is a constant.

Now we consider the case p

∈ [

2

, ∞)

. The solution of(3.4)satisfies, by the energy method,

∥

ϕ

_ϵ

∥

_H1₍_Ω₎

≤

c

∥

f_ϵ

∥

_L2₍_Ω₎

,

where c is a constant. Together with(4.15), we see that(3.5)ofLemma 3.3holds for p

∈ [

2

, ∞)

. For any function

ζ ∈

Lr

(

Ω

)

with r

∈ [

2

, ∞)

, we obtain

η

_ϵby solving

−∇ ·

₍

_Λϵ_τ

∇

η

_ϵ

) = ζ

inΩ

,

η

ϵ

=

0 on

∂

Ω

.

(4.16)

We have proved that if r

∈ [

2

, ∞)

, the solution of(4.16)satisfies

∥

η

_ϵ

∥

_W1,r₍_Ω₎

+ ∥

η

_ϵ

∥

_W2,r₍_Ωϵ

f)

+ ∥

η

_ϵ

∥

_W2,r₍_Ωϵ