Homogenization of Green and Neumann Functions

0.37 MB·English

by  Carlos E. Kenig

Periodic Homogenization
of Green and Neumann Functions
Carlos E. Kenig Fanghua Lin Zhongwei Shen
∗ † ‡
2
1
0
2

n Abstract
a
J For afamily of second-order elliptic operators with rapidly oscillating periodiccoef-

6 ficients, we study the asymptotic behavior of the Green and Neumann functions, using

Dirichlet and Neumann correctors. As a result we obtain asymptotic expansions of
]
P Poisson kernels and the Dirichlet-to-Neumann maps as well as near optimal conver-
A
gence rates in W1,p for solutions with Dirichlet or Neumann boundary conditions.
.
h
t
a
1 Introduction
m
[

The main purpose of this paper is to study the asymptotic behavior of the Green and
1
v Neumann functions for a family of elliptic operators with rapidly oscillating coefficients.
0
More precisely, consider
4
4
∂ x ∂
1 = div A(x/ε) = aαβ , ε > 0 (1.1)
1. Lε − ∇ −∂xi (cid:20) ij ε ∂xj(cid:21)
0 (cid:0) (cid:1) (cid:16) (cid:17)
2 (the summation convention is used throughout the paper). We will assume that A(y) =
1
(aαβ(y)) with 1 i,j d and 1 α,β m is real and satisfies the ellipticity condition
:
v ij ≤ ≤ ≤ ≤
i
X 1
µ ξ 2 aαβ(y)ξαξβ ξ 2 for y Rd and ξ = (ξα) Rdm, (1.2)
r | | ≤ ij i j ≤ µ| | ∈ i ∈
a
where µ > 0, and the periodicity condition
A(y +z) = A(y) for y Rd and z Zd. (1.3)
∈ ∈
We will also impose the smoothness condition
A(x) A(y) τ x y λ for some λ (0,1] and τ 0. (1.4)
| − | ≤ | − | ∈ ≥
Let G (x,y) = Gαβ(x,y) and N (x,y) = Nαβ(x,y) denote the Green and Neumann
ε ε ε ε
functions respectively, for in a bounded domain Ω, with pole at y. We are interested in
ε
(cid:0) L(cid:1) (cid:0) (cid:1)
∗Supported in part by NSF grant DMS-0968472
†Supported in part by NSF grant DMS-0700517
‡Supported in part by NSF grant DMS-0855294
1
the asymptotic behavior, as ε 0, of G (x,y), N (x,y), G (x,y) and N (x,y) as well
ε ε x ε x ε
→ ∇ ∇
as G (x,y) an N (x,y). We shall use G (x,y) and N (x,y) to denote the Green
x y ε x y ε 0 0
∇ ∇ ∇ ∇
and Neumann functions respectively, for the homogenized (effective) operator in Ω.
0
L
Let Pβ = x (0,...,1,...) with 1 in the βth position for 1 j d and 1 β m.
j j ≤ ≤ ≤ ≤
To state our main results, we need to introduce the matrix of Dirichlet correctors Φβ =
ε,j
Φ1β,...,Φmβ in Ω, defined by
ε,j ε,j
(cid:0) (cid:1)
(Φβ ) = 0 in Ω and Φβ = Pβ on ∂Ω, (1.5)
Lε ε,j ε,j j
as well as the matrix of Neumann correctors Ψβ = Ψ1β,...,Ψmβ in Ω, defined by
ε,j ε,j ε,j
(cid:0) (cid:1)
∂ ∂
(Ψβ ) = 0 in Ω and Ψβ = Pβ on ∂Ω. (1.6)
Lε ε,j ∂ν ε,j ∂ν j
ε 0
(cid:0) (cid:1) (cid:0) (cid:1)
Here ∂/∂ν denotes the conormal derivative associated with for ε 0.
ε ε
L ≥
The following are the main results of the paper.
Theorem 1.1. Let = div A(x/ε) with the matrix A(y) satisfying conditions (1.2),
ε
L − ∇
(1.3) and (1.4). Then for any x,y Ω,
(cid:0) ∈ (cid:1)
Cε
G (x,y) G (x,y) (1.7)
| ε − 0 | ≤ x y d 1
−
| − |
if Ω is a bounded C1,1 domain, and
∂ ∂ ∂ Cεln[ε 1 x y +2]
Gαβ(x,y) Φαγ(x) Gγβ(x,y) − | − | (1.8)
∂x ε − ∂x ε,j · ∂x 0 ≤ x y d
i i j
| − |
(cid:12) (cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9)(cid:12)
(cid:12) (cid:12)
if Ω is a bounded C2,η domain for some η (0,1), where C depends only on d, m, µ, τ, λ
∈
and Ω.
Theorem 1.2. Suppose that A(y) satisfies the same conditions as in Theorem 1.1. Also
assume that A = A, i.e., aαβ(y) = aβα(y) for 1 i,j d and 1 α,β m. Then for any
∗ ij ji ≤ ≤ ≤ ≤
x,y Ω,
∈
Cεln[ε 1 x y +2]
−
N (x,y) N (x,y) | − | (1.9)
ε 0
| − | ≤ x y d 1
−
| − |
if Ω is a bounded C1,1 domain, where C depends only on d, m, µ, τ, λ and Ω. Moreover, if
Ω is a bounded C2,η domain for some η (0,1),
∈
∂ ∂ ∂ C εtln[ε 1M +2]
Nαβ(x,y) Ψαγ(x) Nγβ(x,y) t − (1.10)
∂x ε − ∂x ε,j · ∂x 0 ≤ x y d 1+t
i i j −
| − |
(cid:12) (cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9)(cid:12)
(cid:12) (cid:12)
for any x,y Ω and t (0,1), where M = diam(Ω) and C depends only on d, m, µ, τ, λ,
t
∈ ∈
t and Ω.
A few remarks are in order.
2
Remark 1.3. In the case of a scalar equation (m = 1), the estimate (1.7) holds for bounded
measurable coefficients satisfying (1.2) and (1.3) (see Theorem 3.3).
Remark 1.4. The matrix of Dirichlet correctors Φβ was introduced in [2] to establish
ε,j
the boundary Lipschitz estimates for solutions with Dirichlet conditions, while the matrix
(cid:0) (cid:1)
of Neumann correctors Ψβ was introduced in [21] to establish the same estimates for
ε,β
solutions with Neumann(cid:0)boun(cid:1)dary conditions. It is known that kΦβε,j −PjβkL∞(Ω) ≤ Cε and
k∇Φβε,jkL∞(Ω)+k∇Ψβε,jkL∞(Ω) ≤ C. Under the condition Ψβε,j(x0) = Pjβ(x0) for some x0 ∈ Ω,
we also have kΨβε,j −PjβkL∞(Ω) ≤ Cεln[ε−1M +2] (see Propositions 2.4 and 2.5).
Remark 1.5. Estimates (1.7) and (1.9) in Theorems 1.1 and 1.2 allow us to establish O(ε)
estimates for u u (1 < p ) for solutions with Dirichlet or Neumann boundary
ε 0 Lp(Ω)
k − k ≤ ∞
conditions (see Theorems 3.4 and 4.5). More importantly, estimates (1.8) and (1.10) yield
near optimal convergence rates in W1,p(Ω) for any 1 < p < . In fact, let (u ) = F in Ω
ε ε
∞ L
and u = 0 on ∂Ω. Then
ε
∂uβ 1 1
kuε −u0 − Φβε,j −Pjβ ∂x0kW01,p(Ω) ≤ Cpε ln[ε−1M +2] 4|2−p|kFkLp(Ω). (1.11)
j
(cid:8) (cid:9) (cid:8) (cid:9)
In the case of Neumann boundary conditions we obtain
∂uβ
kuε −u0 − Ψβε,j −Pjβ ∂x0kW1,p(Ω) ≤ Ct,pεtkFkLp(Ω) (1.12)
j
(cid:8) (cid:9)
for any t (0,1), where (u ) = F in Ω, ∂uε = 0 on ∂Ω and F = u = 0. See
∈ Lε ε ∂νε Ω ∂Ω ε
subsections 3.2 and 4.2 for details. Let wε = uε −u0 −εχβj(x/ε)∂∂uxRβ0j, whereR(χβj(y)) denotes
the matrix of correctors for in Rd. Estimates (1.11) and (1.12) should be compared to
1
L
the well known O(ε1/2) estimate: wε H1(Ω) = O(ε1/2) (see e.g. [6]), and to the following
k k
estimate,
1/2
kwεkH1/2(Ω) + |∇wε(x)|2dist(x,∂Ω)dx ≤ CεkFkL2(Ω), (1.13)
(cid:26)ZΩ (cid:27)
proved in [20, Theorems 3.4 and 5.2]. Due to the presence of a boundary layer, both the
O(ε1/2) estimate and (1.13) are more or less sharp. The Dirichlet and Neumann correctors
are introduced precisely to deal with boundary layer phenomena in periodic homogenization.
Remark 1.6. Our approach to Theorems 1.1 and 1.2 also leads to asymptotic estimates
of G (x,y) and N (x,y) (see subsection 3.3 and Remark 4.9). As a result we
x y ε x y ε
∇ ∇ ∇ ∇
obtain asymptotic expansions for (∂/∂x )( ) 1(∂/∂x ) and Λ , the Dirichlet-to-Neumann
i ε − j ε
L
map associated with .
ε
L
The asymptotic expansion of the fundamental solutions as well as the heat kernels for
in Rd has been studied, using the method of Bloch waves; see e.g. [26, 11] and their
ε
L
references(alsosee[24]forresultsobtainedbythemethodofG-convergence). Inthepresence
of boundary, the Bloch representation is no longer available. In a series of papers [2, 4,
3], M. Avellaneda and F. Lin introduced the compactness methods, which originated in
3
the regularity theory in the calculus of variations and minimal surfaces, to homogenization
problems. Inparticular, theyestablishedanasymptoticexpansionfor G (x,y)in[4],using
y ε
∇
Dirichlet correctors. As a result, it was proved in [4] that if Ω is C1,η for some η (0,1),
∈
P (x,y) = P (x,y)ω (y)+R (x,y) for x Ω and y ∂Ω, (1.14)
ε 0 ε ε
∈ ∈
where P (x,y) (ε 0) denotes the Poisson kernel for in Ω,
ε ε
≥ L
∂
ω (y) = Φ (y) n (y)h(y) n (y)n (y)a (y/ε) (1.15)
ε ∂n(y) ∗ε,k · k · i j ij
(cid:8) (cid:9)
and the remainder R (x,y) satisfies
ε
limsup R (x,y) : x E and y ∂Ω = 0 (1.16)
ε
ε 0 | | ∈ ∈
→
(cid:8) (cid:9)
for any compact subset E of Ω (the results are stated for the case m = 1; however, the
argument in [4] works equally well for elliptic systems). In (1.15) we have used Φ to
∗ε,k
denote the Dirichlet correctors for , the adjoint of . Also, h(y) = (aˆ n (y)n (y)) 1
L∗ε Lε ij i j −
and (aˆ ) is the (constant) coefficient matrix of . The expansion (1.14) was used in [4] to
ij 0
L
identify the limit, as ε 0, of solutions to a problem of exact boundary controllability for
the wave operator ∂2 +→ .
∂t2 Lε
Our Theorem 1.1 gives a much more refined estimate of R (x,y) in (1.14) (under the
ε
stronger condition ∂Ω C2,η). Indeed, it follows from the estimate (1.8) that
∈
Cεln[ε 1 x y +2]
−
R (x,y) | − | for any x Ω,y ∂Ω. (1.17)
| ε | ≤ x y d ∈ ∈
| − |
Besides its applications to boundary control problems, estimate (1.17) may also be used to
investigate the Dirichlet problem
(u ) = 0 in Ω and u (x) = f(x,x/ε) on ∂Ω, (1.18)
ε ε ε
L
where f(x,y) is 1-periodic in y. The Dirichlet problem (1.18) arises natually in the study
of boundary layer phenomena and higher-order convergence in periodic homogenization (see
e.g. [25, 1, 15, 14] and their references). Let v be the solution to
ε
(v ) = 0 in Ω and v = f(x,x/ε)ω (x) on ∂Ω, (1.19)
0 ε ε ε
L
where ω is given by (1.15). It follows from the estimate (1.17) that
ε
u v = O (ε[ln(1/ε)]2)1/p for any 1 p <
ε ε Lp(Ω)
k − k ≤ ∞
(see Theorem 3.9). This effectively(cid:0)reduces the as(cid:1)ymptotic problem (1.18) to the study of
convergence properties of ω on ∂Ω, under various geometric conditions on Ω. This line of
ε
research will be developed in a future work.
We now describe the main ideas in the proof of Theorems 1.1 and 1.2. The basic tools
in our approach are representation formulas by Green and Neumann functions, uniform
estimates for Green functions established in [2],
G (x,y) + G (x,y) C x y 1 d and G (x,y) C x y d, (1.20)
x ε y ε − x y ε −
|∇ | |∇ | ≤ | − | |∇ ∇ | ≤ | − |
4
andthesameestimatesobtainedin[20]forNeumannfunctionsN (x,y). LetD = D(x ,r) =
ε r 0
B(x ,r) Ω and ∆ = ∆(x ,r) = B(x ,r) ∂Ω for some x Ω and 0 < r < r . First, to
0 r 0 0 0 0
∩ ∩ ∈
establish (1.7), we will show that if p > d,
kuε −u0kL∞(D1) ≤ C |uε −u0|dx+Cεk∇u0kL∞(D4) +Cpεk∇2u0kLp(D4), (1.21)
ZD4
where (u ) = (u ) in D and u = u on ∆ . This is done by considering w (x) =
ε ε 0 0 4 ε 0 4 ε
L L
u (x) u (x) εχ(x/ε) u and using the Green representation formula and the observation
ε 0 0
− − ∇
that (w ) = ε ∂ b (x/ε) ∂2u0 , where b (y) is a bounded periodic function. Estimate
Lε ε ∂xi ijk ∂xj∂xk ijk
(1.7) follows from (1(cid:16).21) by a more(cid:17)or less standard argument (see subsection 3.1). Next, we
show in subsection 3.2 that
C
k∇uε −(∇Φε)(∇u0)kL∞(Dr) ≤ rd+1 |uε −u0|dx+Cεr−1k∇u0kL∞(D4r) (1.22)
ZD4r
+Cεln[ε−1r+2]k∇2u0kL∞(D4r) +Cεrηk∇2u0kC0,η(D4r),
if (u ) = (u ) in D and u = u on ∆ . Estimate (1.8) follows easily from (1.22) by
ε ε 0 0 4r ε 0 4r
L L
taking u (x) = G (x,y ) and u (x) = G (x,y ). By repeating the argument, estimate (1.22)
ε ε 0 0 0 0
also gives an asymptotic expansion for G (x,y) (see Theorem 3.11). To prove (1.22),
x y ε
∇ ∇
we let
∂uβ
w = u (x) u (x) Φβ Pβ 0 (1.23)
ε ε − 0 − ε,j − j ∂x
j
(cid:8) (cid:9)
and represent w in D , using the Green function in D, where D is a C2,η domain such that
ε r
D D D .
3r 4r
⊂ ⊂
Although a bit more complicated, the proof of eTheorem e1.2 follows the same line of
argumeentasTheorem1.1. Insubsections4.1and4.2weestablishboundaryL andLipschitz
∞
estimates similar to (1.21) and (1.22) for u and u satisfying (u ) = (u ) in D and
ε 0 ε ε 0 0 4r
L L
∂uε = ∂u0 on ∆ . The results rely on the uniform Lp and Neumann function estimates
∂νε ∂ν0 4r
obtained in [23, 21] under the additional symmetry condition A = A.
∗
The rest of the paper is organized as follows. Section 2 contains some basic formulas
and estimates which are more or less known. The case of Dirichlet boundary conditions is
treated in Section 3, while Section 4 is devoted to the case of Neumann boundary conditions.
In Section 5 we prove two inequalities, which are used in subsection 4.3 and are of interest
in their own right, for the Dirichlet-to-Neumann map Λ .
0
2 Preliminaries
Let = div(A(x/ε) ) with A(y) = aαβ(y) satisfying (1.2)-(1.3). Let χ(y) = χαβ(y)
Lε − ∇ ij j
denote the matrix of correctors for in Rd, where χβ(y) = (χ1β(y),...,χmβ(y)) is defined
L1 (cid:0) (cid:1) j j j (cid:0) (cid:1)
by the following cell problem:
(χβ) = (Pβ) in Rd,
L1 j −L1 j
(2.1)
χβ is periodic with respect to Zd and χβdy = 0,
 j j
ZY

5
foreach1 j dand1 β m. HereY = [0,1)d Rd/Zd andPβ(y) = y (0,...,1,...,0)
≤ ≤ ≤ ≤ ≃ j j
with 1 in the βth position. The homogenized operator is given by = div(Aˆ ), where
0
Aˆ = (aˆαβ) and L − ∇
ij
∂
aˆαβ = aαβ +aαγ χγβ dy. (2.2)
ij ij ik ∂y j
ZY (cid:20) k (cid:21)
(cid:16) (cid:17)
It is known that the constant matrix Aˆ is positive definite with an ellipticity constant de-
pending only on d, m and µ (see [6]).
Let
∂
bαβ(y) = aˆαβ aαβ(y) aαγ(y) χγβ . (2.3)
ij ij − ij − ik ∂y j
k
Since bαβdy = 0 and ∂ bαβ = 0 by (2.2) and (2.1), there(cid:0) exis(cid:1)ts Fαβ H1(Y) such that
Y ij ∂yi ij kij ∈
R (cid:0) (cid:1) ∂
bαβ = Fαβ and Fαβ = Fαβ. (2.4)
ij ∂y kij kij − ikj
k
(cid:8) (cid:9)
Remark 2.1. To see (2.4), onesolves ∆fαβ = bαβ inY with fαβ H1(Y)and fαβdy = 0,
ij ij ij ∈ Y ij
and let
∂fαβ ∂fαβ R
Fαβ(y) = ij kj
kij ∂y − ∂y
k i
(see e.g. [20]). Note that if A(y) is H¨older continuous, then χ and hence bαβ are H¨older
∇ ij
continuous. It follows that F is H¨older continuous. In particular, χ C1(Y) + F C1(Y) is
∇ k k k k
bounded by a constant depending only on d, m, µ, λ and τ. In the case of the scalar equation
(m = 1)withboundedmeasurablecoefficients, thecorrectorχisH¨oldercontinuousbytheDe
Giorgi -Nash estimates. This, together with Cacciopoli’s inequality and H¨older’s inequality,
implies that there exist t > 0 and C > 0, depending only on d and µ, such that
χ dy Crd 1+t for x Y and 0 < r < 1.
−
|∇ | ≤ ∈
ZB(x,r)
In view of (2.3) we obtain
b (y) dy Crd 1+t for x Y and 0 < r < 1. (2.5)
ij −
| | ≤ ∈
ZB(x,r)
Since ∆f = b in Y and f dy = 0,
ij ij Y ij
R b (y)
ij
k∇fijkL∞(Y) ≤ Ck∇fijkL2(Y) +Csup x| y d| 1 dy ≤ C(d,µ), (2.6)
x∈Y ZY | − | −
where we have used (2.5) to estimate the last integral in (2.6). It follows that F
kij
k k∞ ≤
C(d,µ).
Thefollowing propositionplays animportantroleinthispaper. Wementionthatformula
(2.8)withVεβ,j(x) = Pjβ(x)+εχβj(x/ε)isknownandmaybeusedtoshowthatkuε−u0kL2(Ω) ≤
Cε u0 H2(Ω), where ε(uε) = 0(u0) in Ω and uε = u0 on ∂Ω (see e.g. [20]). The proof of
k k L L
our main results on the first-order derivatives of Green and Neumann functions will rely on
(2.8) with the matrices of Dirichlet and Neumann correctors respectively in the place of the
functions Vβ .
ε,j
6
Proposition 2.2. Suppose that u H1(Ω),u H2(Ω) and (u ) = (u ) in Ω. Let
ε 0 ε ε 0 0
∈ ∈ L L
∂uβ
w (x) = u (x) u (x) Vβ (x) Pβ(x) 0, (2.7)
ε ε − 0 − ε,j − j · ∂x
j
(cid:8) (cid:9)
where Vβ = (V1β,...,Vmβ) H1(Ω) and (Vβ ) = 0 in Ω for each 1 j d and
ε,j ε,j ε,j ∈ Lε ε,j ≤ ≤
1 β m. Then
≤ ≤
∂ ∂2uγ
( (w ))α =ε Fαγ(x/ε) 0
Lε ε ∂x jik ∂x ∂x
i (cid:26) j k(cid:27)
∂ (cid:2) (cid:3) ∂2uγ
+ aαβ(x/ε) Vβγ(x) x δβγ 0 (2.8)
∂x ij ε,k − k ∂x ∂x
i (cid:26) j k(cid:27)
h i
∂ ∂2uγ
+aαβ(x/ε) Vβγ(x) x δβγ εχβγ(x/ε) 0 ,
ij ∂x ε,k − k − k ∂x ∂x
j i k
h i
where δβγ = 1 if β = γ, and zero otherwise.
Proof. Note that
∂wβ x ∂uβ x ∂uβ x ∂ ∂uγ
aαβ(x/ε) ε = aαβ ε aαβ 0 aαβ Vβγ x δβγ 0
ij ∂x ij ε ∂x − ij ε ∂x − ij ε ∂x ε,k − k · ∂x
j j j j k
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) n o
x ∂2uγ
aαβ Vβγ x δβγ 0 .
− ij ε ε,k − k · ∂x ∂x
k j
(cid:16) (cid:17)n o
This, together with (u ) = (u ), gives
ε ε 0 0
L L
∂ ∂uβ ∂uγ
(w ) α = aˆαβ aαβ(x/ε) 0 + (Vγ Pγ) α 0
Lε ε − ∂x ij − ij ∂x Lε ε,k − k · ∂x
i ( j ) k
h i
(cid:8) (cid:9) (cid:8) (cid:9)
∂ ∂2uγ
+aαβ (x/ε) Vβγ x δβγ 0
ij ∂x ε,k − k · ∂x ∂x
j i k
n o
∂ ∂2uγ
+ aαβ(x/ε) Vβγ x δβγ 0 .
∂x ij ε,k − k · ∂x ∂x
i (cid:26) k j(cid:27)
h i
Since
Vγ Pγ = Pγ = εχγ(x/ε) ,
Lε ε,k − k −Lε k Lε k
we obtain (cid:0) (cid:1) (cid:0) (cid:1) (cid:8) (cid:9)
∂ ∂uβ
(w ) α = bαβ (x/ε) 0
Lε ε − ∂x ij ∂x
i ( j )
(cid:8) (cid:9)
∂ ∂χβγ ∂2uγ
+aαβ (x/ε) Vβγ(x) x δβγ k (x/ε) 0 (2.9)
ij ∂x ε,k − k − ∂x · ∂x ∂x
j ( j ) i k
∂ ∂2uγ
+ aαβ(x/ε) Vβγ x δβγ 0 ,
∂x ij ε,k − k · ∂x ∂x
i (cid:26) k j(cid:27)
h i
7
where bαβ(y) is defined by (2.3). In view of (2.4), we may re-write the first term in the right
ij
hand side of (2.9) as
∂ ∂ ∂uβ ∂ ∂2uβ
εFαβ(x/ε) 0 = ε Fαβ(x/ε) 0 .
−∂x ∂x kij · ∂x ∂x kij · ∂x ∂x
i ( k j ) i ( k j)
h i
The formula (2.8) now follows.
The next proposition will be used to handle the Neumann boundary condition (cf. [21]).
Proposition 2.3. Let w be given by (2.7). Suppose that ∂ Vβ = ∂ Pβ . Then
ε ∂νε ε,k ∂ν0 k
∂w α ∂u α ∂u α (cid:8) (cid:9) ∂2u(cid:8)γ (cid:9)
ε = ε 0 n aαβ(x/ε) Vβγ Pβγ 0 . (2.10)
∂ν ∂ν − ∂ν − i ij ε,k − k · ∂x ∂x
(cid:18) ε (cid:19) (cid:18) ε(cid:19) (cid:18) 0(cid:19) k j
(cid:8) (cid:9)
Proof. Note that
∂w α ∂u α ∂u α ∂ ∂uγ
ε = ε 0 n aαβ(x/ε) Vβγ Pβγ 0
∂ν ∂ν − ∂ν − i ij ∂x ε,k − k · ∂x
(cid:18) ε (cid:19) (cid:18) ε(cid:19) (cid:18) ε(cid:19) j k (2.11)
(cid:2) ∂2uγ (cid:3)
n aαβ(x/ε) Vβγ Pβγ 0 .
− i ij ε,k − k · ∂x ∂x
j k
(cid:2) (cid:3)
α
Since ∂ Vβ = ∂ Pβ , the third term in the right hand side of (2.11) equals ∂u0 +
∂νε ε,k ∂ν0 k − ∂ν0
α
∂u0 . T(cid:8)his(cid:9)gives (2(cid:8).10).(cid:9) (cid:16) (cid:17)
∂νε
(cid:16) (cid:17)
The following two propositions provide the properties of the Dirichlet and Neumann
correctors needed in this paper.
Proposition 2.4. Let = div A(x/ε) with A(y) satisfying (1.2), (1.3) and (1.4). Let
ε
L − ∇
Φβ denote the matrix of Dirichlet correctors for in a C1,η domain Ω. Then
ε,j (cid:0) (cid:1) Lε
(cid:0) (cid:1)
Φβ(x) C, Φβ (x) Pβ(x) Cε (2.12)
|∇ j | ≤ | ε,j − j | ≤
and
ε
Φβ (x) Pβ(x) εχβ(x/ε) Cmin 1, (2.13)
∇ ε,j − j − j ≤ δ(x)
(cid:26) (cid:27)
(cid:12) (cid:8) (cid:9)(cid:12)
for any x Ω, whe(cid:12)re δ(x) = dist(x,∂Ω) and C depen(cid:12)ds only d, m, µ, λ, τ and Ω.
∈
Proof. The first estimate in (2.12) follows from the Lipschitz estimate in [2]. To see the
second estimate, let u (x) = Φβ (x) Pβ(x) εχβ(x/ε). Then (u ) = 0 in Ω and u (x) =
ε ε,j − j − j Lε ε ε
−εχβj(x/ε) forx ∈ ∂Ω. It againfollows from[2] that kuεkL∞(Ω) ≤ CkuεkL∞(∂Ω) ≤ Cε. Hence,
iknΦteβε,rjio−rPesjβtkimL∞a(tΩe)i≤n [C2]εa+ndkεχuβjεk∞L∞≤(Ω)Cε.CFεi,noanlley,onbotateintshatukε∇(xu)εkL∞C(Ωε)[δ≤(xC)].−1A. lTsoh,isbygitvhees
k k ≤ |∇ | ≤
the estimate (2.13).
8
Proposition 2.5. Let = div A(x/ε) with A(y) satisfying (1.2), (1.3), (1.4) and the
ε
L − ∇
symmetry condition A = A. Let Ψβ denote the matrix of Neumann correctors for in
∗ (cid:0) ε,j (cid:1) Lε
a C1,η domain Ω. Suppose Ψβ (x ) = Pβ(x ) for some x Ω. Then
ε,j 0(cid:0) (cid:1)j 0 0 ∈
Ψβ(x) C, Ψβ (x) Pβ(x) Cεln[ε 1M +2] (2.14)
|∇ j | ≤ | ε,j − j | ≤ −
and
ε
Ψβ (x) Pβ(x) εχβ(x/ε) Cmin 1, (2.15)
∇ ε,j − j − j ≤ δ(x)
(cid:26) (cid:27)
(cid:12) (cid:8) (cid:9)(cid:12)
for any x Ω, whe(cid:12)re M = diam(Ω) and C depends o(cid:12)nly d, m, µ, λ, τ and Ω.
∈
Proof. The estimate (2.15) as well as the first estimate in (2.14) was proved in [21]. To
prove the second estimate in (2.14), we let Hβ = Ψβ (x) Pβ(x) εχβ(x/ε). Since
ε,j ε,j − j − j
Hβ (x) Cmin 1,ε[δ(x)] 1 , by the Fundamental Theorem of Calculus, we may deduce
|∇ ε,j | ≤ −
that Hβ (x) Hβ (y) Cεln[ε 1M +2] for any x,y Ω. Since Hβ (x ) = ε χβ(x ) , we
| ε,j − ε,j (cid:0) | ≤ (cid:1)− ∈ | ε,j 0 | | j 0 |
obtain Hβ (x) Cεln[ε 1M +2] for any x Ω. This gives the desired estimate.
| ε,j | ≤ − ∈
3 Asymptotic behavior of Green functions
The goal of this section is to prove Theorem 1.1. We also establish several convergence
theorems for solutions with Dirichlet boundary conditions.
Let = div A(x/ε) with A(y) satisfying conditions (1.2), (1.3) and (1.4). Let
ε
L − ∇
G (x,y) denote the matrix of Green’s functions for in a bounded domain Ω. It follows
ε ε
(cid:0) (cid:1) L
from [2] that if Ω is C1,η for some η (0,1),
∈
C δ(x) δ(y) δ(x)δ(y)
G (x,y) min 1, , , ,
| ε | ≤ x y d 2 x y x y x y 2
 | − | − (cid:26) | − | | − | | − | (cid:27)
 C δ(y)
 |∇xGε(x,y)| ≤ x y d 1 min 1, x y ,
 | − | − (cid:26) | − |(cid:27) (3.1)

 C δ(x)
 G (x,y) min 1, ,
|∇y ε | ≤ x y d 1 x y
| − | − (cid:26) | − |(cid:27)
 C

|∇x∇yGε(x,y)| ≤ x y d

 | − |



foranyx,y Ω, whereδ(x) = dist(x,∂Ω). Theseestimates, whicharewellknownforsecond-
∈
order elliptic operators with constant coefficients, play an essential role in our approach to
Theorem 1.1.
3.1 L estimates
∞
In this subsection we give the proof of the estimate (1.7). As a corollary of (1.7), we also
establish an O(ε) estimate for u u for any p > 1 (see Theorem 3.4). Throughout
ε 0 Lp(Ω)
k − k
9
thesubsection wewill assume thatA(y)satisfies conditions (1.2)-(1.3) andinthecase m > 1,
A(y) is H¨older continuous. Let
D = D (x ,r) = B(x ,r) Ω and ∆ = ∆(x ,r) = B(x ,r) ∂Ω
r r 0 0 r 0 0
∩ ∩
for some x Ω and 0 < r < r .
0 0
∈
Lemma 3.1. Assume that Ω is Lipschitz if m = 1, and C1,η for some η (0,1) if m > 1.
∈
Then
C
kuεkL∞(Dr) ≤ CkfkL∞(∆3r) + rd |uε|dx, (3.2)
ZD3r
where (u ) = 0 in D and u = f on ∆ .
ε ε 3r ε 3r
L
Proof. By rescaling we may assume that r = 1. The estimate is well known in the case
m = 1 and follows from the maximum principle and De Giorgi -Nash estimate. If m > 1
and f = 0, estimate (3.2) follows directly from [2, Lemma 12]. To treat the general case,
consider (v ) = 0 in D with the Dirichlet data v = f on ∂D ∂Ω and v = 0 on ∂D ∂Ω,
ε ε ε ε
L ∩ \
where D is a C1,η domain such that D2 ⊂ D ⊂ D3. Note that kvεkL∞(D2) ≤ CkfkL∞(∆3) by
[2, Theorem 3], and u e v may be handled by [2, Lemma 12e], as before. e
ε ε
−
e e
The next lemma provides a boundary L estimate.
∞
Lemma 3.2. Assume that Ω satisfies the same assumption as in Lemma 3.1. Let u
ε
∈
H1(D ) and u W2,p(D ) for some d < p . Suppose that
4r 0 4r
∈ ≤ ∞
(u ) = (u ) in D and u = u on ∆ .
ε ε 0 0 4r ε 0 4r
L L
Then,
C
kuε −u0kL∞(Dr) ≤ rd |uε−u0|dx+Cεk∇u0kL∞(D4r)
ZD4r (3.3)
+Cpεr1−pd k∇2u0kLp(D4r).
Proof. Note that if (u ) = F, then (v) = F , where v(x) = r 2u (rx) and F (x) =
ε ε ε/r 1 − ε 1
L L
F(rx). Thus, by rescaling, it suffices to consider the case r = 1. To this end we choose
a domain D, which is C1,η for m > 1 and Lipschitz for m = 1, such that D D D .
3 4
⊂ ⊂
Consider
e w = u u εχβ(x/ε) ∂uβ0 = w(1) +w(2) in D, e
ε ε − 0 − j ∂x ε ε
j
where e
(w(1)) = (w ) in D and w(1) H1(D) (3.4)
Lε ε Lε ε ε ∈ 0
and
e e
(w(2)) = 0 in D and w(2) = w on ∂D. (3.5)
Lε ε ε ε
e e
10