High order correctors and two-scale expansions in stochastic homogenization

0.34 MB·

by  Yu Gu

HIGH ORDER CORRECTORS AND TWO-SCALE EXPANSIONS
IN STOCHASTIC HOMOGENIZATION
YUGU
Abstract. In this paper, we study high order correctors in stochastic ho-
6 mogenization. We consider elliptic equations in divergence form on Zd, with
1 the random coefficients constructed from i.i.d. random variables. We prove
0 moment bounds on the high order correctors and their gradients under di-
2 mensional constraints. It implies the existence of stationary correctors and
stationary gradients in high dimensions. As an application, we prove a two-
ct scaleexpansionofthesolutionstotherandom PDE,whichidentifiesthefirst
O andhigherorderrandom fluctuations inastrongsense.
Keywords: quantitative homogenization, high order corrector, two-scale ex-
3
pansion,random fluctuation.

]
P
A
. 1. Main result
h
at Quantitative stochastic homogenization has witnessed important progress in re-
m cent years,and a major contribution of the groundbreakingwork of Gloria-Otto is
to prove the high order moment estimates on the corrector [23, 25].
[
The result in the discrete setting can be described as follows. Let B be the set
3 of nearest neighbor edges in Zd, and e ,i = 1,...,d be the canonical basis of
v Zd. On a probability space (Ω, ,P), w{eihave a sequen}ce of i.i.d. non-degenerate
8
F
5 random conductances, denoted by ωe e B with ωe (δ,1) for some δ > 0. We
9 define a:Zd Rd d as a random d{iago}n∈al matrix fie∈ld such that
×
→
7
a(x)=diag(a (x),...,a (x))=diag(ω ,...,ω ).
0 1 d (x,x+e1) (x,x+ed)
1. The regularized corrector equation in the direction ξ Rd says
0 ∈
(1.1) (λ+ a(x) )φλ(x)= a(x)ξ, x Zd.
6 ∇∗ ∇ ξ −∇∗ ∈
1 Here the discrete gradient and divergence for f : Zd R and F : Zd Rd are
: → →
v defined as
i d
X
r ∇f =(∇1f,...,∇df), ∇∗F = ∇∗iFi,
a Xi=1
with
f(x)=f(x+e ) f(x), F =F (x e ) F (x).
∇i i − ∇∗i i i − i − i
It was shown in [23, Proposition 2.1] that when d > 3, for any p > 1, there exists
a constant C =C(d,δ,p,ξ)>0 such that
(1.2) φλ p 6C
h| ξ| i
uniformlyinλ>0,where denotesthe expectationon(Ω, ,P). Inparticular,it
h·i F
impliestheexistenceofastationarycorrectorwhend>3: thereexistsazero-mean
stationary random field φ solving
ξ
(1.3) a(x) φ (x)= a(x)ξ.
∗ ξ ∗
−∇ ∇ ∇
Thefirstgoalofthispaperistogobeyondthefirstordercorrectors,andpresent
aproofof (1.2)forhighordercorrectors. Asanapplicationofhighordercorrectors,
1
2 YUGU
we identity the first and higher order fluctuations in stochastic homogenization in
a strong sense that will be specified later.
1.1. High order correctors. Wewillstayinthesamesettingbutfurtherassume
ωe e B satisfies the log-Soblev inequality
{ } ∈
ζ2 1
(1.4) ζ2log 6 ∂ ζ 2
ζ2 ρ | e |
(cid:28) h i(cid:29) * e +
X
for some ρ > 0. Here ∂ is the weak derivative with respect to ω and ζ : Ω R
e e
→
is any function so that the r.h.s. of (1.4) makes sense. Through a formal two-scale
expansioninSection2.1,wedefinetheregularizedn thordercorrector,anddenote
−
them by ψλ. The following is our first main result.
n
Theorem 1.1. Fix any n > 2. When d > 2n 1, for any p > 1, there exists
−
C =C(d,δ,ρ,p)>0 such that
ψλ p 6C
h|∇ n| i
uniformly in λ>0. When d>2n+1, the same result holds for ψλ, i.e.,
n
ψλ p 6C
h| n| i
uniformly in λ>0. In particular, for the n th order corrector, it has a stationary
−
gradient when d>2n 1 and it is stationary when d>2n+1.
−
Remark 1.2. The log-Soblev inequality (1.4) holds for i.i.d. random variables with
a continuous density, hence can not deal with distributions with atoms. A weaker
versionis presentedin [32, Equation(7)], coveringallpossible i.i.d. ensembles. We
believe that the approach in this paper can be applied with the weaker version
(with extra technicalities), and Theorem 1.1 holds for any i.i.d. ensemble. To keep
the presentation simple, we choose to work with (1.4).
The study of stochastic homogenization started from the early work of Kozlov
[30] and Papanicolaou-Varadhan [40], and revived recently from various quantita-
tiveperspectives[9,23,24,17,19,32,20,25,21,18,26,33,34,1,5,2,4]. Whilethe
first order correctors have been analyzed extensively due to their role in determin-
ing the effective coefficients and proving convergence in homogenization, the high
order correctors have been receiving less attention. Our interest in the high order
correctorscomesfromthecomparisonbetweenapointwisetwo-scaleexpansionand
a large scale central limit theorem derived for the solutions to the random PDE
x
(1.5) a( ) u (x)=f(x).
ε
−∇· ε ∇
The results in [28, 29] showed that when d>3, the first order correctorrepresents
the local fluctuation, which is measured by u (x) u (x) for fixed x Rd, but
ε ε
−h i ∈
does not suggest the global large scale fluctuation, which is measured weakly in
spaceby (u u )g withtestfunction g. We expect the surprisingphenomenon
ε ε
−h i
may be explained by high order correctors which only become visible in the weak
R
sense due to strong correlations; see a discussion in Section 5.
InSection2.1,wewillconstructhighordercorrectorsdirectlyfromaformaltwo-
scaleexpansion,andthereisaslightlydifferentwayofcharacterizingthehighorder
correctors as the “high order intrinsic polynomials” that come out of the Liouville
theorem: asthefirstordercorrectorφ isdefinedsothatξ x+φ (x)isa harmonic,
ξ ξ
· −
the high order correctorscorrectthe a harmonic high order polynomials to be
hom
−
a harmonic. It seems the two ways of construction are equivalent although we do
−
not attempt to prove it here. For our purpose, it is more convenient to directly
start from the formal expansion. For the recent breakthrough in the direction
HIGH ORDER CORRECTORS AND TWO-SCALE EXPANSIONS 3
of regularity theory of random operators in the continuous setting, we refer to
[1, 21, 5, 16, 2, 26, 4].
Our strategy of proving moment estimates in Theorem 1.1 follows [23], i.e., by
using a spectral gap inequality (see (2.16) below) and estimating the sensitivity
of the correctors to the individual conductance; see also the unpublished work of
Naddaf and Spencer [37]. A key quantity to control is ∂ ψλ, which describes the
e n
dependenceofthen thordercorrectorontheconductanceω . Itinvolvesthefirst
e
−
and second order derivatives of the Green’s function of a(x) , and we use the
∗
∇ ∇
p th momentestimates derivedin [32], which came fromthe log-Soblevinequality
−
together with the resultof Delmotte and Deuschel onthe lower order moment [12].
One ofthe difficulties is to obtainsomea prioriestimate onthe gradient ψλ. For
∇ n
the first order corrector, the bound on the second moment of φλ comes directly
∇ ξ
from (1.1) thanks to the divergence form of its r.h.s. source term. For high order
correctors,we provethat the source term canalso be expressedin divergenceform.
Once we have the p th moment estimates on ψλ, Theorem 1.1 follows from a
− ∇ n
straightforwardinduction argument.
1.2. Two-scale expansions. As anapplicationofhighordercorrectors,we prove
an expansion of solutions to the random PDE (1.5). If homogenization is viewed
as a law of large numbers result, here we are looking for the next order random
fluctuations that may or may not lead to a central limit theorem. A classical two-
scale expansion indicates that the solutions to (1.5) take the form
(1.6) u =u +εu +ε2u +...,
ε 0 1 2
where u ,u ,u ,... are constructed by equating the power-like terms in ε upon
0 1 2
substituting (1.6)into(1.5). Since(1.6)isonlyaformalseries,itisaprioriunclear
whether u indeed representsthe “correct” firstorderfluctuation(it was shownnot
1
true when d = 1 [27]), and if it does, we need to understand in which sense (1.6)
holds. Thesecondgoalofthepaperistogiveananswertotheabovequestionsand
justify the formal two-scale expansion under appropriate dimensional constraint.
To avoid the effects from boundary layers, we work on the equation
x
(1.7) (α+∇∗εa(ε)∇ε)uε(x)=f(x),x∈εZd,
where α > 0 is a fixed constant and f(x) (Rd). The discrete gradient and
divergence are defined for g :εZd R and G∈ :Cεc∞Zd Rd as
→ →
g =( g,..., g) with g(x)=[g(x+εe ) g(x)]/ε,
ε ε,1 ε,d ε,i i
∇ ∇ ∇ ∇ −
and
d
G= G with G (x)=[G (x εe ) G (x)]/ε.
∇∗ε ∇∗ε,i i ∇∗ε,i i i − i − i
i=1
X
It is well-known that u converges in a certain sense to u solving
ε 0
(1.8) (α+ a )u (x)=f(x),x εZd,
∇∗ε hom∇ε 0 ∈
where the effective coefficient matrix is given by
a = a(I + φ) =a¯I with φ=[ φ ,..., φ ],
hom h d ∇ i d ∇ ∇ e1 ∇ ed
and φ is the first order corrector in the direction e . Our goal is to obtain the
ek k
first and higher order fluctuations in u u . By the formal two-scale expansion,
ε 0
→
the first order correction takes the form
d
x x
u (x, )= u (x)φ ( ),x εZd.
1 ε ∇ε,j 0 ej ε ∈
j=1
X
4 YUGU
Remark 1.3. Typically, one compare u with u¯ solving the equation in the contin-
ε 0
uous space:
(α a )u¯ (x)=f(x),x Rd.
hom 0
−∇· ∇ ∈
Inthispaper,wedonotanalyzeu u¯ ,whichisonlyanerrorduetodiscretization.
0 0
−
For functions f :εZd Ω Rd, we define the L2(εZd Ω) norm
× → ×
1
2
(1.9) f := εd f(x,ω)2 .
2,ε
k k h| | i
(cid:18) xX∈εZd (cid:19)
The following is our main result for the first order fluctuations.
Theorem 1.4. When d>3, we have
(1.10) u u εu =o(ε)
ε 0 1 2,ε
k − − k
as ε 0.
→
Remark 1.5. Since u = 0, Theorem 1.4 in particular implies that the determin-
1
h i
istic bias u u vanishes in the order of ε.
ε 0
h i−
Fortheerrorsquantifiedinthestrongsense,bythe normhere,Theorem1.4
2,ε
k·k
showsthatthe firstordercorrectionisgivenbythe firstordercorrectorthatcomes
from the formal expansion. This is consistent with the pointwise result in the con-
tinuous setting [28], and both results suggest that we do not expect a central limit
theorem for (u (x) u (x))/ε. It should be contrasted with the low dimensional
ε 0
−
case, e.g., when d = 1, it was shown in [8, 27] that (u (x) u (x))/√ε converges
ε 0
−
in distribution to a Gaussian process. For the errors quantified in the weak sense,
i.e., after taking a spatial average with a test function, central limit theorems are
obtainedfor the first ordercorrectoru andthe solution u [36, 35, 29, 15, 3]. Var-
1 ε
ious approximations to the effective coefficient matrix a also exhibit Gaussian
hom
fluctuations; see [38, 7, 41, 39, 22].
In higher dimensions, a higher order expansion similar to (1.10) can also be
obtained. To make sense of the expansion, the number of terms we are permitted
to include in the expansion depends on the dimension in light of Theorem 1.1.
Theorem 1.6. Fix anyn>2. When d>2n+1,thereexists u n and v n
{ k}k=1 { k}k=1
such that u is random with u =0, v is deterministic, and
k k k
h i
n n
u u εku εkv =o(εn)
ε 0 k k
− − −
(cid:13) k=1 k=1 (cid:13)2,ε
(cid:13) X X (cid:13)
as ε 0. (cid:13) (cid:13)
→ (cid:13) (cid:13)
Remark 1.7. Theorem 1.4 can be viewed as a special case of Theorem 1.6 when
n=1, in which case we have v 0.
1
≡
Remark 1.8. It can be seen from the proof of Theorem 1.6 that the k th order
−
random fluctuation u takes the form
k
x
u (x)= g (x)ϕ ( ),
k i i
ε
i
X
whereg is somederivativeofu andϕ is somezero-meanstationaryrandomfield.
i 0 i
Therefore,therandomfluctuationsinhomogenization,measuredinthestrongsense,
is a superposition of highly oscillatory random fields.
Using the high order correctors obtained in Theroem 1.1, the proof of Theo-
rem 1.4 and Theorem 1.6 mimics the periodic setting. It is well-known that in the
ideal periodicsetting(equationwithsmoothcoefficientsposedonthewholespace),
the formal two-scale expansion as in (1.6) indeed approximates the solution up to
HIGH ORDER CORRECTORS AND TWO-SCALE EXPANSIONS 5
arbitrary high order precision, so the message we want to convey here is that one
can still hope the expansion to be valid in the random setting, provided that we
havestationarycorrectors. We refer to [13, 14] for a carefuldealing with boundary
layers in the periodic setting.
Thepreviousworkonestimatingthesizeofu u includese.g. [42,10,11],with
ε 0
−
non-optimal exponents or optimal exponents in the small ellipticity regime. With
thestudy ofthefirstordercorrectors,[17,20,34]providesoptimalestimatesofthe
errorsize. Therecentpreprint[6]usessecondordercorrectorstoderivequantitative
estimates in the weak spatial norms. The main contribution of Theorem 1.4 and
Theorem1.6istoidentifythefirstandhigherorderfluctuationsinthestrongsense,
which seems to be the first result of this type.
The paper is organized as follows. In Section 2, we present the proof of Theo-
rem1.1. TheproofsofTheorem1.4andTheorem1.6areinSection3andSection4.
We leave some discussions to Section 5.
1.3. Notations.
For any e=(e,e¯) B and f :Zd R, we will write
• ∈ →
f(e)=f(e¯) f(e),
∇ −
and for any ξ Rd,
∈
ξ(e)=ξ e.
·
For x Zd, we define
• ∈
x = x +2.
| |∗ | |
Wewritea.bwhena6CbwiththeconstantC independentofthespatial
•
variables y,z and the edge variables b,e.
For x,y,c>0, we write
•
1
x.
yc
−
if for any δ >0, there exists C >0 such that
δ
1
x6C .
δyc δ
−
For f :Zd Zd R, x Zd and e,b B, we write
• × → ∈ ∈
f(x,e)=f(x,e¯) f(x,e),
∇ −
and
f(b,e)= f(¯b,e) f(b,e).
∇∇ ∇ −∇
We use to denote the Lp(Ω) norm. Recall that denote the
p 2,ε
• L2(εZd kΩ·k) norm. k·k
Weuse×Gλ, λ todenotethe Green’sfunctionofλ+ a ,λ+ onZd
∗ ∗
• and Gλ to dGenote the Green’s function of λ ∆ on R∇d. ∇ ∇ ∇
−
2. Moment bounds of high order correctors
Wefirstreviewthe classicaltwo-scaleexpansionofequationsindivergenceform,
then present a proof of Theorem 1.1.
6 YUGU
2.1. Two-scale expansions. Consider the equation on Rd:
x
a( ) u (x)=f(x),
ε
−∇· ε ∇
where a : Rd Rd d is a stationary random field. We introduce the fast variable
×
→
y = x and write = + 1 . By further assuming
ε ∇ ∇x ε∇y
(2.1) u (x)=u (x)+εu (x,y)+ε2u (x,y)+...,
ε 0 1 2
we obtain
1 1
( + ) a(y)( + )(u +εu +...+εnu +...)=f.
x y x y 0 1 n
− ∇ ε∇ · ∇ ε∇
By matching the order of ε, we get equations satisfied by u .
n
When n=1,
(2.2) a(y) u + a(y) u =0.
y y 1 y x 0
∇ · ∇ ∇ · ∇
When n>2,
(2.3)
a(y) u + a(y) u + a(y) u + a(y) u
y y n y x n 1 x y n 1 x x n 2
∇ · ∇ ∇ · ∇ − ∇ · ∇ − ∇ · ∇ −
= a(y) u + a(y) u .
x y n 1 x x n 2
h∇ · ∇ − ∇ · ∇ − i
Note that in general, the corrector equations take the form
a(y) u (x,y)=F
y y n
∇ · ∇
and a solvability condition requires that the source satisfies F = 0, so we added
h i
some expectations to the r.h.s. of (2.3).
For (2.2), we can write
d
(2.4) u (x,y)= ∂ u (x)φ (y)= u (x) φ(y),
1 k 0 ek ∇ 0 ·
k=1
X
with φ=[φ ,...,φ ] and φ the first order corrector solving
e1 ed ek
a(y)( φ +e )=0.
∇y · ∇ ek k
The equation satisfied by the second order corrector takes the form
a(y) u = a u (x) a(y) φ(y) u (x)
y y 2 x hom x 0 x 0
(2.5) ∇ · ∇ ∇ · ∇ −∇ · ∇ ∇
a(y) u (x) a(y) 2u (x)φ(y),
x x 0 y 0
−∇ · ∇ −∇ · ∇
where 2u is the Hessian matrix of u and the matrix φ = [ φ ,..., φ ].
∇ 0 0 ∇ ∇ e1 ∇ ed
The r.h.s. of the above expression has mean zero since the homogenization matrix
a is given by
hom
a = a(y)(I + φ(y)) .
hom d
h ∇ i
By induction, it is clear that u is a linear combination of ∂ u , where α is a
n α 0
multi-index with α =n, so we write
| |
u (x,y)= φ (y)∂ u (x),
n α α 0
α
X
and those φ are the n th order correctors.
α
−
By (2.3), we further observe that φ with α =n is a linear combination of ψ ,
α n
| |
which solves the following three types of equations:
∂ (a (y)ψ (y)),
i ij n 1
−
(2.6) a(y) ψ (y)= a (y)∂ ψ (y) a (y)∂ ψ (y) ,
∇· ∇ n  ij j n−1 −h ij j n−1 i
a (y)ψ (y) a (y)ψ (y) .
 ij n 2 ij n 2
− −h − i

HIGH ORDER CORRECTORS AND TWO-SCALE EXPANSIONS 7
Herei,j =1,...,d,andwehavethefollowingcorrespondencebetweenther.h.s. of
(2.6) and (2.3) :
∂ (a (y)ψ (y)) a(y) u (x,y),
i ij n 1 y x n 1
− ←→∇ · ∇ −
a (y)∂ ψ (y) a (y)∂ ψ (y) a(y) u (x,y) a(y) u (x,y) ,
ij j n 1 ij j n 1 x y n 1 x y n 1
− −h − i←→∇ · ∇ − −h∇ · ∇ − i
a (y)ψ (y) a (y)ψ (y) a(y) u (x,y) a(y) u (x,y) .
ij n 2 ij n 2 x x n 2 x x n 2
− −h − i←→∇ · ∇ − −h∇ · ∇ − i
In other words,since (2.3) is a linear equation, we have decomposed it into finitely
many “small equations” written in the generic form of (2.6).
2.2. A proof of Theorem 1.1 by induction. Recalling that the coefficient
a(x) = diag(a (x),...,a (x)) is a diagonal matrix for x Zd, so given (2.6) in
1 d
the continuoussetting,weconsidercorrectorequationsonZ∈d ofthefollowingform:
(a (y)ψ (y)),
∇∗i i n−1
(2.7) a(y) ψ (y)= a (y) ψ (y) a (y) ψ (y) ,
∇∗ ∇ n  i ∇i n−1 −h i ∇i n−1 i
a (y)ψ (y) a (y)ψ (y) .
 i n 2 i n 2
− −h − i
For any λ > 0, we regularize (2.7) by adding a massive term and define ψλ as
 n
the unique solution to
(a (y)ψ (y)),
(2.8) (λ+ a(y) )ψλ(y)= a∇∗i(y)i ψ n−(1y) a (y) ψ (y) ,
∇∗ ∇ n  i ∇i n−1 −h i ∇i n−1 i
a (y)ψ (y) a (y)ψ (y) .
 i n 2 i n 2
− −h − i
To prove Theorem 1.1, we only analyze (2.8) when d > 2n 1. Since our proof is
 −
based on induction, we do not worry about the meaning of the r.h.s. of (2.8) for
the moment.
Letψ denotethe stationaryfirstordercorrectorsφ whend>3. Wemakethe
1 ei
statement I for n>1:
n
Statement I : when d>3, for any p>1, we have
1
1
(2.9) ∂ ψ (y) . ,
k e 1 kp y ed−1
| − |∗
and
1
(2.10) ∂ ψ (b) . ,
k e∇ 1 kp b ed
| − |∗
with the proportional constant independent of y Zd and b,e B.
∈ ∈
Statement I ,n>2: when d>2n 1, for any p>1, we have
n
−
1
(2.11) ∂ ψλ(y) . ,
k e n kp y e(d−n)−
| − |∗
and
1
(2.12) ∂ ψλ(b) . ,
k e∇ n kp b e(d−n+1)−
| − |∗
with the proportional constant independent of λ>0, y Zd and b,e B.
∈ ∈
The following result is the main result in this section.
Proposition 2.1. For any n>2, I +I implies I , where we take I as
n 2 n 1 n 0
− −
an empty statement.
First, we note that I holds. For the first order correctors ψ φ ,
1 1 ∈{ ek}k=1,...,d
∂ φλ (y)= Gλ(y,e)( φ +e )(e),
e ek −∇ ∇ ek k
and
∂ φλ (b)= Gλ(b,e)( φ +e )(e),
e∇ ek −∇∇ ∇ ek k
8 YUGU
where Gλ is the Green’s function of λ+ a . By [32, Theorem 1], we have
∗
∇ ∇
1 1
(2.13) Gλ(y,e) . and Gλ(b,e) . ,
k∇ kp |y−e|d∗−1 k∇∇ kp |b−e|d∗
which implies
1 1
(2.14) ∂ φ (y) . and ∂ φ (b) . .
k e ek kp |y−e|d∗−1 k e∇ ek kp |b−e|d∗
ByProposition2.1,I holdsforn>2,andthisimpliesTheorem1.1. First,the
n
log-Soblev inequality (1.4) implies the Spectral-Gap inequality:
(2.15) ζ 2 . ∂ ζ 2
e
h| | i * | | +
e
X
for zero-mean ζ, which can be generalized to control the 2p th moment for any
−
p>1 [19, Lemma 2]:
p
(2.16) ζ 2p . ∂ ζ 2 .
e
h| | i * | | ! +
e
X
Remark 2.2. Theresultpresentedin[19,Lemma2]isforadifferenttypeofvertical
derivative, but the same proof applies in our case.
Now we apply (2.16) to ψλ, and use I to derive
n n
p 1 1
2p 2
ψλ(0) . ∂ ψλ(0)2 . ∂ ψλ(0) 2
k n k2p * | e n | ! + k e n k2p!
e e
X X
1
1 2
. .1
(2d 2n)
Xe |e|∗ − −!
when d>2n+1.
Similarly, we apply (2.16) to ψλ, and use I to derive
∇ n n
p 1 1
2p 2
ψλ(b) . ∂ ψλ(b)2 . ∂ ψλ(b) 2
k∇ n k2p * | e∇ n | ! + k e∇ n k2p!
e e
X X
1
1 2
. .1
Xe |b−e|∗(2d−2n+2)−!
when d>2n 1.
−
By the uniform bound on ψλ and ψλ(b) , it is standard to extract
k nk2p k∇ n k2p
a subsequence as λ 0 to get the existence of a stationary corrector ψ when
n
→
d > 2n+1 and the existence of a stationary gradient ψ when d > 2n 1. For
n
∇ −
the convenience of reader, we present the details in Appendix A.
Remark 2.3. It is clear that if ψλ(0) converges to ψ (0) weakly in Lp(Ω), then
n n
∂ ψλ(0) converges to ∂ ψ (0) weakly in Lp(Ω), thus the same estimates in I
e n e n n
holds for ψ when d>2n+1. The same discussion applies to ψ .
n n
∇
TheproofofProposition2.1isastraightforwardcalculationonceweassumethe
high moment bounds for ψλ:
∇ n
(2.17) I +I when d>2n 1, for any p>1, ψλ(b) .1.
n−2 n−1 ⇒ − k∇ n kp
We will leave the proof of (2.17) to the next section.
HIGH ORDER CORRECTORS AND TWO-SCALE EXPANSIONS 9
Proof of Proposition 2.1. By the previous discussion, when d>2n 1, we have
−
stationary (n 2) th and (n 1) th order corrector,so the equation of ψλ given
− − − − n
by (2.8) is well-defined:
(a (y)ψ (y)),
(λ+ a(y) )ψλ(y)= a∇∗i(y)i ψ n−(1y) a (y) ψ (y) ,
∇∗ ∇ n  i ∇i n−1 −h i ∇i n−1 i
a (y)ψ (y) a (y)ψ (y) .
 i n 2 i n 2
− −h − i
We take ∂ on both sides to obtain
e 
[∂ (a (y)ψ (y))],
(λ+∇∗a(y)∇)∂eψnλ(y)=(1y=e−1y=e¯)∇ψnλ(e)+ ∂∇e∗i(aie(y)i∇iψnn−−11(y)),
∂ (a (y)ψ (y)).
 e i n 2
−
By the Green’s function representation, we write

Gλ(y,z)[∂ (a (z)ψ (z))],
(2.18) ∂ ψλ(y)= Gλ(y,e) ψλ(e)+ zG∇λi(y,z)∂ (ae(z)i ψn−1(z)),
e n −∇ ∇ n  PzGλ(y,z)∂e(ai(z)∇ψi n(−z1)),
 Pz e i n−2
which implies that for any p>1:
 P
Gλ(y,z)[∂ (a (z)ψ (z))] ,
∂ ψλ(y) 6 Gλ(y,e) ψλ(e) + zkG∇λi(y,z)∂ (ae(z)i ψn−1(z)) k,p
k e n kp k∇ ∇ n kp  PzkGλ(y,z)∂e(ai(z)ψ∇i n(−z1)) .kp
 Pzk e i n−2 kp
Since ∂eai(z) = 0 for e 6= (z,z+ei),it iPs clear by In−2,In−1 and Remark 2.3
that
1
∂ (a (z)ψ (z)) . ,
k e i n−1 kp z e(d−n+1)−
| − |∗
1
∂ (a (z) ψ (z)) . ,
k e i ∇j n−1 kp z e(d−n+2)−
| − |∗
1
∂ (a (z)ψ (z)) . .
k e i n−2 kp z e(d−n+2)−
| − |∗
Notethatwhenn=2,ψ (z)isaconstantascanbeseenfrom(2.5),sotheabove
n 2
−
estimates still holds. An application of Hölder inequality together with (2.13) and
(2.17) leads to
1 1 ,
k∂eψnλ(y)kp . |y−1e|d∗−1 + PPzzz |||yyy−−−zzz11|||d∗d∗d∗−−−122|||zzz−−−eee|||∗∗∗(((ddd11−−−nnn+++122)))−−−,.
By the discrete convolution inequality [29, LPemma A.6], we obtain
1
(2.19) ∂ ψλ(y) . .
k e n kp y e(d−n)−
| − |∗
By (2.18), we also have
(2.20)
Gλ(b,z)[∂ (a (z)ψ (z))],
∂ ψλ(b)= Gλ(b,e) ψλ(e)+ z∇G∇λi(b,z)∂ (ae(z)i ψn−1(z)),
e∇ n −∇∇ ∇ n  Pz∇Gλ(b,z)∂e(ai(z)∇ψi n(−z1)),
 Pz∇ e i n−2
so the same discussion as above gives
 P
1
(2.21) ∂ ψλ(b) . .
k e∇ n kp b e(d−n+1)−
| − |∗
The statement I consists of (2.19) and (2.21), so the proof of Proposition 2.1 is
n
complete.
10 YUGU
Remark 2.4. Thereasonwechoosetohave(d n) and(d n+1) astheexponents
− − − −
in (2.11) and (2.12) is due to the following discrete convolution inequality: for any
α (0,d),
∈
1 1 log y 1
(2.22) . | |∗ . .
z α y z d y α y α−
Xz | |∗ | − |∗ | |∗ | |∗
As can be seen from (2.20), the term Gλ(b,z) produces a factor of b z d,
i −
and we use “ ” to absorb the logarithm∇i∇c factor when applying (2.22). | − |∗
−
2.3. Estimates on gradient of correctors: proof of (2.17). To prove
ψλ(b) .1
k∇ n kp
for any p>1, we first consider p=2, then derive it for any p>2.
2.3.1. p=2. We write the equation (2.8) as
(a (y)ψ (y)),
(2.23) (λ+∇∗a(y)∇)ψnλ(y)=F(y)= a∇i∗i(y)∇i iψn−n−1(1y)−hai(y)∇iψn−1(y)i,
a (y)ψ (y) a (y)ψ (y) .
 i n 2 i n 2
− −h − i
By I +I , when d>2n 1, F is a zero-meanstationary randomfield, and
n−2 n−1 − 
ψ + ψ + ψ .1
n 1 q n 1 q n 2 q
k − k k∇ − k k − k
for any q > 1. We claim that there exists a stationary random field Ψ such that
Ψ(0) Lq(Ω) for any q >1 and
∈
(2.24) Ψ=F.
∗
∇
Using the fact a >δ >0, we conclude from (2.24) that
i
ψλ . Ψ .1.
k∇ nk2 k k2
For F = (a (y)ψ (y)), we only need to choose
∇∗i i n−1
Ψ=Ψ e with Ψ =a ψ .
i i i i n 1
−
For the other two cases, we consider the equation with k =1,...,d
(2.25) (λ+ )Ψλ = F,
∇∗∇ k ∇k
and by I +I , we have
n 2 n 1
− −
1
k∂eΨλk(y)kq 6Xz k∇∗kGλ(y,z)∂eF(z)kq .Xz |y−z|d∗−1k∂eF(z)kq
1 1
.
Xz |y−z|d∗−1|z−e|∗(d−n+2)−
1
. ,
(d n+1)
y e − −
| − |∗
where we used Gλ to denote the Green’s function of λ+ . Thus, by the same
∗
∇ ∇
application of the spectral gap estimate (2.16), we have
Ψλ .1
k kkq
whend>2n 1. ByLemmaA.1,thereexistsastationaryrandomfieldΨ (y)such
k
−
that Ψ (0) Lq(Ω) and Ψλ(0) Ψ (0) weakly in Lq(Ω). Let Ψ = (Ψ ,...,Ψ ),
k ∈ k → k 1 d
and we claim
Ψ=F.
∗
∇
From (2.25), we have
(λ+∇∗∇)∇∗kΨλk =∇∗k∇kF,