（二）背景及目的
有些化學工廠的廢水經由水井排放到地底，地面的有毒廢棄物也會溶入雨水而進入地底，核
能電廠儲存在地底下的核廢棄物因為時間過長而容器腐蝕、或因為地層變動造成容器破裂等因
素，核廢棄物也因此進入地下水。這些都會造成日常飲用水的不安全。這些問題不只發生在臺灣，
世界其它國家也有同樣的情形。歐美有些國家甚至已設立專責機構負責這些污染源的清除工作。
此外身體內的血液在血管中的流動，地底下的碳氫化合物的抽取等。以上這些都屬於多孔介質中
的多相流問題[4,8,10,34]。因此了解多孔介質中的多相流的變化對解決很多實際問題是很有幫助
的。
一般討論多孔介質中多相流的變化的數學模式都是由 Darcy law 與 transport equation 所組
成[5,6,7,9,12,13,14,15]。前者為橢圓方程式而後者為抛物線方程式，在高度非均勻多孔介質中
以上兩個方程式都是屬於非均勻的情形。換句話說，一個為非均勻橢圓方程式另一個為 非均勻抛
物線方程式。理論分析的方法很難得到有實質義意的結果[1,2,3,24,28,30,31,35,37,38,42,43,
44,45]，因此我們計劃探討數值計算方法來了解非均勻橢圓方程式與 非均勻抛物線方程式的特
性。
這是一個二年期的計劃。我們希望探討計算非均勻橢圓方程式與 非均勻抛物線方程式的方法
及這計算方法的誤差估計問題。假設ε代表一個很小的數字，
是一個以原點為中心半徑小於 1/2
的球，
,
，
，
，
。 Denote absolute
permeability by
in
and
in
, phase pressure by
in
and
in
,
and external source by
in
and
in
.我們希望考慮的橢圓方程式為[11,16,
21,22]
mY
fD
(
Y
j
)
D
m
m K
nZ
j
f
F
Q ,
m n fR
D
D
k
m f
q ,
f
f
m m m
D
P
fp
m
P
Q
F
K
(
)
in
f,
(
k
p
q
)
f
in
m(
K
P
Q
)
n
k
(
p
q
)
n
on
m,
P
p
on
m顯然的，當ε很小時，計算
的代價會非常的高。如何使用很小的資源，穫得滿意的結果，
這是我們有興趣討論的。我們希望考慮的抛物線方程式為
p
K
P
Q
F
P
t
(
)
in
f
( T
0
,
)
,
tp
(
k
p
q
)
f
in
m
( T
0
,
)
(
K
P
Q
)
n
k
(
p
q
)
n
on
m
(
0
,
T
)
,
P
p
on
m
(
0
,
T
)
P
P
0in
f
p
p
0in
m同樣的，我們也要找出，當ε很小時，如何計算
p
的方法。
此計劃是之前計劃的沿續。在之前的計劃中，我們了解一些描述流體在破裂多孔介質中的微
觀模式與宏觀模式的關係，同時也了解一些微觀模式的解的均勻估計的問題，這對此次計劃會有
很大的幫助。
（三）研究方法、進行步驟及執行進度。
基本上，均勻橢圓方程式與均勻抛物線方程式的計算方法已經被研究了五、六十年，甚至
更久[18,19,20,23,25,26,27,29,32,34,36,39,40,41,46]。這部份的知識是垂手可得也很
完備。不過非均勻的情形仍十分欠缺，若只是直接把計算均勻問題的方法套用在非均勻的
問題上，可以想像出這是十分沒有效率，非常糟糕的想法。我們則是想將計算均勻問題的
方法與之前微觀模式的解的均勻估計做一結合，發展出一套有效率的計算方法。同時也討
論計算方法的誤差估計問題。
（四）成果與自評。
1.得到不均勻抛物線微分方程式的解的導數的
norm 的均勻估計結果[51]。此結果對我
們以後要發展出有效率的、可處理複雜情形的計算方法有很重要的理論依據。理論推導的
過程也是可做為進一步研究不均勻介質問題的有效工具。
pL
2. 找出一個計算非均勻橢圓方程式的解的數值方法，並且得到數值解與正確解之間的誤
差估計，此結果有很大的實用價值，它可以用來計算多重尺度的問題[52]。
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Fundamentals of Numerical Reservoir Simulation
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microstructure, J. Accoust. Soc. Am. 70(4) 1140-1146, 1981.
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variational principles, and rigorous bounds. J. Fluid Mech. 206, 25-46, 1989.
[44] A. Yu. Beliaev, S.M. Kozlov, Darcy equation for random porous media, Comm. Pure App.
Math. XLIX 1-34, 1996
[45] V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of differential operators and
integral functionals, Springer-Verlag, 1991.
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[47] L.M. Yeh. L.M. Yeh. Elliptic equations in highly heterogeneous porous media, Mathematical
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differential equation,250,2011,1828-1849.
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gradient estimate for non-uniform elliptic equations with
discontinuous coefficients. preprint.
p
L
[52] L.M. Yeh. A numerical method for elliptic equations in highly heterogeneous media.
preprint.
May 29, 2011
Lp gradient estimate for non-uniform elliptic equations with discontinuous coefficients
Li-Ming Yeh
Department of Applied Mathematics
National Chiao Tung University, Hsinchu, 30050, Taiwan, R.O.C. liming@math.nctu.edu.tw
Many elliptic equations defined in heterogeneous media are non-uniform elliptic equa-tions with discontinuous coefficients. The heterogeneous media considered are periodic and consist of a connected high permeability sub-region and a disconnected matrix block sub-region with low permeability. Let ǫ denote the size ratio of matrix blocks to the whole domain and assume the permeability ratio of the matrix block sub-region to the con-nected high permeability sub-region is of the order ǫ2. Elliptic equations with diffusion
depending on the permeability of the media have fast diffusion in high permeability sub-region and slow diffusion in low permeability sub-region, and they are non-uniform elliptic equations. It is proved that the Lpnorm of the gradient of the elliptic solutions
in the high permeability sub-region are bounded uniformly in ǫ. One example also shows that the Lp norm of the second order derivatives of the elliptic solutions in the high
permeability sub-region in general are not bounded uniformly in ǫ.
Keywords: permeability, perforated domain, fractured media, heterogeneous media AMS Subject Classification: 35J15, 35J25, 35J67
1. Introduction
A priori Lp estimate for the gradient of the solutions of non-uniform elliptic equa-tions with discontinuous coefficients is presented. Let Y (≡ [0, 1]n, n ≥ 3) be a unit cube, B1/24(~12) denote a ball centered at ~12 ≡ (12, · · · ,12) with radius 1/24, Ym ⊂ B1/24(~12) be a smooth domain, Yf ≡ Y \ Ym, ǫ ∈ (0, 1), Ωǫm ≡ {x|x ∈ ǫ(Ym− j) for j ∈ Zn} be a disconnected subset of Rn, Ωǫf (≡ Rn\ Ωǫm) denote a connected subset of Rn, and ∂Ωǫ
m represent the boundary of Ωǫm. The equations that we considered are
−∇ · (Kǫ∇Ψǫ+ Gǫ) = Vǫ in Ωǫf, −ǫ∇ · (ǫKǫ∇ψǫ+ gǫ) = vǫ in Ωǫm, (Kǫ∇Ψǫ+ Gǫ) · ~nǫ= ǫ(ǫKǫ∇ψǫ+ gǫ) · ~nǫ on ∂Ωǫm, Ψǫ= ψǫ on ∂Ωǫm, (1.1)
where ~nǫ is the unit normal vector on ∂Ωǫ
m and Kǫ, Gǫ, gǫ, Vǫ, vǫ are given func-tions. System (1.1) has applications in flows in highly heterogeneous media, the stress in composite materials, and so on (see [3, 10, 15] and references therein).
May 29, 2011
2 Lpgradient estimate
Since ǫ ∈ (0, 1), they are non-uniform elliptic equations with discontinuous coeffi-cients. If Kǫ is positive and smooth as well as if Gǫ, gǫ, Vǫ, vǫ are smooth and have compact supports, the regular solution of (1.1) in Hilbert space Hk (k ≥ 1) exists uniquely for each ǫ [14]. By energy method, it is easy to see that the H1 norm of the solutions of (1.1) in the connected region Ωǫ
f are bounded uniformly in ǫ if Gǫ, gǫ, Vǫ, vǫ are bounded uniformly in ǫ in L2(Rn). There are some literatures related to this problem. Lipschitz estimate and W2,p estimate for uniform elliptic equations with discontinuous coefficients could be found in [15, 18]. Uniform H¨older, Lp, and Lipschitz estimates in ǫ for uniform elliptic equations in periodic domains were proved in [4, 5]. Uniform Lp estimate in ǫ for Laplace equation in perforated domains was considered in [17] and the same problem in Lipschitz estimate was considered in [23]. For non-uniform elliptic equations with smooth coefficients, exis-tence of C2,α solution was studied in [11]. Uniform H¨older and Lipschitz estimates in ǫ for non-uniform elliptic equations with discontinuous coefficients were shown in [26]. Here we shall consider the non-uniform elliptic equations with discontinuous coefficients in Lp space case. It is proved that Lp estimate for the gradient of the solutions of (1.1) in the connected sub-region Ωǫ
f are bounded uniformly in ǫ but the Lp gradient estimate in the disconnected sub-region Ωǫ
m may not be. This is different from uniform elliptic equation case, in which uniform bound holds in the whole domain [4, 5, 11, 15, 18]. We also note that the Lp estimate of the second derivatives of the solutions of (1.1) may not be bounded uniformly in ǫ [26]. In [26], Lipschitz estimate for the solutions of (1.1) was derived under some regularity requirements on Gǫ, gǫ, Vǫ, vǫ and under smallness of vǫ. In this work, no regularity requirements and no smallness of vǫ are needed. The assumption that the diameter of Ym is less than 1/12 is only for convenience of presentation. Indeed the results still hold if the diameter of Ymis less than 1.
2. Notation and main result
Lp (resp. Hs, Ws,p) denotes a Sobolev space with norm k · k
Lp (resp. k · kHs, k ·
kWs,p), Ck,α denotes a H¨older space with norm k · kCk,α, [ζ]C0,α is the H¨older
semi-norm of ζ, Wlocs,p(D) ≡ {ζ|ζ ∈ Ws,p(D) for any compact subset D in D}, and H1
loc(D) ≡ W 1,2
loc(D), where s ≥ −1, p ∈ [1, ∞], k ≥ 0, α ∈ (0, 1). Hper1 (Y ) contains functions in H1(Y ) satisfying periodic boundary conditions on ∂Y , C(Rn) is a space of continuous functions in Rn, and C∞
0 (D) is a space of infinitely differentiable functions with compact support in D. Define kζ1, · · · , ζkkB≡ kζ1kB+ · · · + kζkkB for any Banach space B, B(x) is a ball centered at x, and Br(x) is a ball centered at x with radius r > 0. For any set D, |D| is the volume of D, D is the closure of D, XD is the characteristic function on D, dist(x, D) is the distance from x to D, D/r ≡ {x|rx ∈ D} for r > 0, and − Z D ζ(y)dy ≡ 1 |D| Z D ζ(y)dy if ζ ∈ L1(D).
May 29, 2011 Lpgradient estimate 3 If ζ ∈ L1(Rn), (ζ) x,r ≡ − Z Br(x)
ζ(y)dy. Define Ωm≡ {x|x ∈ Ym− j for j ∈ Zn} and Ωf ≡ Rn\ Ωm. So Ωm= Ωǫm/ǫ and Ωf = Ωǫf/ǫ. For any ǫ, ω > 0 and j ∈ Zn,
Eǫ≡ ( 1 in Ωf, ǫ2 in Ω m, e Eǫ,j≡ ( 1 in Rn\ (Y m− j), ǫ2 in Y m− j, Eǫω(x) ≡ Eǫ(x ω), Ee ǫ,j ω (x) ≡ eEǫ,j( x ω). Define |||ζ|||C0,α(D∩Ωω f)≡ kηkC0,α(D/ω∩Ωωf/ω)and |||ζ|||C0,α(D∩Ωωm)≡ kηkC0,α(D/ω∩Ωωm/ω)
where η(x) = ζ(ωx), ω > 0, and α ∈ (0, 1). Define the left and the right limits of ζ (denoted by ζ,−and ζ,+) on ∂Ymas
ζ,−(x) ≡ lim x′→0 x+x′∈Ym ζ(x + x′), ζ,+(x) ≡ lim x′→0 x+x′∈Yf ζ(x + x′) for x ∈ ∂Ym.
Denote by xi(i = 1, · · · , n) the i-th component of x ∈ Rn, Rn+ ≡ {x|xn > 0}, and ∂Rn+ ≡ {x|xn = 0}. Let B ≡ [0, M]n for some M ∈ N, Bǫm≡ {x|x ∈ ǫ(Ym+ j) ⊂ B for j ∈ Zn}, and Bǫ
f ≡ B \ Bǫm. For any function ζ on B, ΠeB(ζ) and ΠoB(ζ) defined in Rn are extensions of ζ, are periodic with period [0, 2M]n, and satisfy
( Πe
B(ζ)(x1, ··, xi−1, xi, xi+1, ··, xn) = ΠeB(ζ)(x1, ··, xi−1, −xi, xi+1, ··, xn), Πo
B(ζ)(x1, ··, xi−1, xi, xi+1, ··, xn) = −ΠoB(ζ)(x1, ··, xi−1, −xi, xi+1, ··, xn), for all i = 1, · · · , n. Πe
B(ζ) (resp. ΠoB(ζ)) is symmetric (resp. antisymmetric) with respect to all coordinate planes (that is, xi = 0, i = 1, · · · , n) and is called even (resp. odd) extension of ζ in Rn. For any function ζ on B, bΠe,i
B (ζ) (resp. bΠ o,i B (ζ)) in Rn is extensions of ζ, is periodic with period [0, 2M]n, is symmetric (resp. antisym-metric) in xi= 0 plane, and is antisymmetric (resp. symmetric) in other coordinate planes, that is,
( b Πe,iB (ζ)(x1, ··, xj−1, xj, xj+1, ··, xn) = bδei,jΠb e,i B (ζ)(x1, ··, xj−1, −xj, xj+1, ··, xn), b Πo,iB (ζ)(x1, ··, xj−1, xj, xj+1, ··, xn) = bδ o i,jΠb o,i B (ζ)(x1, ··, xj−1, −xj, xj+1, ··, xn), where bδe i,j = ( 1 if i = j, −1 if i 6= j, bδ o i,j = ( −1 if i = j,
1 if i 6= j, and i, j = 1, · · · , n. For any vector function ζ = (ζ1, · · · , ζn), we define bΠeBζ ≡ (bΠ
e,1 B ζ1, · · · , bΠ e,n B ζn) and define b Πo Bζ ≡ (bΠ o,1 B ζ1, · · · , bΠ o,n B ζn).
The following statements are assumed throughout this work:
A1. Kǫ ∈ [d1, d2] for some d1, d2 > 0 and kKǫkC0,α(Rn) for some α ∈ (0, 1) is
bounded independent of ǫ(< 1),
A2. Ym⊂ B1/24(~12) is a smooth domain, where~12 ≡ ( 1 2, · · · ,
1 2). Our main results are:
May 29, 2011
4 Lpgradient estimate
Theorem 2.1. Any solution of (1.1) satisfies k∇ΨǫkLp(Ωǫ f)+ ǫk∇ψǫkLp(Ωǫm)≤ c(kΨǫXΩǫf + ǫψǫXΩǫm, GǫXΩǫf + gǫXΩǫmkLp(Rn) +kVǫXΩǫ f + vǫXΩǫmkW−1,p(Rn)+ ǫ −1kv ǫkW−1,p(Ωǫ m)), (2.1)
where p ∈ (1, ∞) and c is a constant independent of ǫ(< 1). Theorem 2.2. (1) In addition to
A3. Ym− ~1/2 is symmetric with respect to all coordinate planes xi = 0, i = 1, · · · , n, the solution of −∇ · (Kǫ∇Ψǫ+ Gǫ) = Vǫ in Bǫf, −ǫ∇ · (ǫKǫ∇ψǫ+ gǫ) = vǫ in Bǫm, (Kǫ∇Ψǫ+ Gǫ) · ~nǫ= ǫ(ǫKǫ∇ψǫ+ gǫ) · ~nǫ on ∂Bǫm, Ψǫ= ψǫ on ∂Bǫm, Ψǫ= 0 on ∂B, (2.2) satisfies k∇ΨǫkLp(Bǫ f)+ ǫk∇ψǫkLp(Bǫm)≤ c(kGǫXBǫf+ gǫXBǫmkLp(B) +kΠoB(VǫXBǫ f + vǫXBǫm)kW−1,p([−M,2M]n)+ ǫ −1kv ǫkW−1,p(Bǫ m)), (2.3)
where p ∈ (1, ∞), c is independent of ǫ, and ~nǫ is the unit normal vector on ∂Bǫ f. (2) In addition to A3 and A4. RVǫXBǫ f + vǫXBǫmdx = 0, the solution of −∇ · (Kǫ∇Ψǫ+ Gǫ) = Vǫ in Bǫf, −ǫ∇ · (ǫKǫ∇ψǫ+ gǫ) = vǫ in Bǫm, (Kǫ∇Ψǫ+ Gǫ) · ~nǫ= ǫ(ǫKǫ∇ψǫ+ gǫ) · ~nǫ on ∂Bǫm, Ψǫ= ψǫ on ∂Bǫm, (Kǫ∇Ψǫ+ Gǫ) · ~nǫ= 0 on ∂B, R ΨǫXBǫ f + ψǫXBǫmdx = 0, (2.4) satisfies k∇ΨǫkLp(Bǫ f)+ ǫk∇ψǫkLp(Bǫm)≤ c(kGǫXBǫf+ gǫXBǫmkLp(B) +kΠeB(VǫXBǫ f + vǫXBǫm)kW−1,p([−M,2M]n)+ ǫ −1kv ǫkW−1,p(Bǫ m)), (2.5)
where p ∈ (1, ∞), c is independent of ǫ, and ~nǫ is the unit normal vector on ∂Bǫ f. Clearly if the right hand side of (2.1) (resp. (2.3) and (2.5)) is bounded, Lp norm for the gradient of the elliptic solutions of (1.1) (resp. (2.2) and (2.4)) in the connected sub-region is bounded uniformly in ǫ. But this is not the case for the elliptic solutions in the disconnected sub-region.
May 29, 2011
Lpgradient estimate 5
References
1. E. Acerbi, V. Chiado Piat, G. Dal Maso, and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Analysis 18 (1992) 481–496.
2. R. A. Adams, Sobolev Spaces, Academic Press, 2003.
3. Gregoire Allaire, Homogenization and two-scale convergence, SIAM I. Math. Anal. 23 (1992) 1482–1518.
4. Marco Avellaneda, Fang-Hua Lin, Compactness methods in the theory of homogeniza-tion, Communications on Pure and Applied Mathematics Vol. XI (1987) 803–847. 5. Marco Avellaneda, Fang-Hua Lin, Lp bounds on singular integrals in homogenization,
Communications on Pure and Applied Mathematics Vol. XLIV (1991) 897–910. 6. G. Chen and J. Zhou, Boundary Element Methods (Academic Press, 1992). 7. L. Escauriaza, Marius Mitrea, Transmission problems and spectral theory for singular
integral operators on Lipschitz domains, Journal of Functional Analysis 216 (2004) 141–171.
8. M. Giaquinta, Multiple integrals in the calculus of variations, Study 105, Annals of Math. Studies, Princeton Univ. Press., 1983.
9. D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, second edition, 1983.
10. Jianguo Huang, Jun Zou, Some new a priori estimates for second-order elliptic and parabolic interface problems, Journal of Differential Equations 184 (2002) 570–586. 11. A. V. Ivanov, Quasilinear degenerate and nonuniformly elliptic and parabolic
equations of second order, American Mathematical Society, Providence, RI, 1984. 12. V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential
Opera-tors and Integral Functions, Springer-Verlag, 1994.
13. Carlos E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems. CBMS Regional Conference Series in Mathematics 83, 1994.
14. O. A. Ladyzhenskaya, Nina N. Ural’tseva, Elliptic and Quasilinear Elliptic Equa-tions. Academic Press, 1968.
15. Yan Yan Li, Michael Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal. 153 (2000) 91–151.
16. W. Littman, G. Stampacchia, and H.F. Weinberger, Regular points for elliptic equa-tions with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963) 43–77.
17. Nader Masmoudi, Some uniform elliptic estimates in porous media, C. R. Acad. Sci. Paris Ser. I 339 (2004) 849–854.
18. A. Maugeri, Dian Palagachev, Lubomira G. Softova, Elliptic and parabolic equa-tions with discontinuous coefficients, Wiley-VCH, Berlin ; New York, 2000. 19. N. G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic
divergence equations, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963) 189–206. 20. H. L. Royden, Real Analysis (Macmillan, 1970).
21. Thomas Runst, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations(Berlin ; New York : Walter de Gruyter, 1996).
22. M. Schechter, Principles of Functional Analysis, American Mathematical Society, 2002.
23. Ben Schweizer, Uniform estimates in two periodic homogenization problems, Commu-nications on Pure and Applied Mathematics Vol.LIII (2000) 1153–1176.
May 29, 2011
6 Lpgradient estimate
24. Elias M. Stein, Singular Integrals and Differentiability Properties of Func-tions, Princeton University Press, 1970.
25. Hans Triebel, Theory of function spaces (Geest & Portig, Leipzig, 1983 and Birkh¨auser, Basel, 1983).
26. Li-Ming Yeh, A priori estimates for non-uniform elliptic equations, Journal of Differ-ential Equations 250 (2011) 1828–1849.
October 31, 2012
A numerical method for elliptic equations in highly heterogeneous media
Li-Ming Yeh
Department of Applied Mathematics
National Chiao Tung University, Hsinchu, 30050, Taiwan, R.O.C. liming@math.nctu.edu.tw
The approximation of the solutions of elliptic equations in highly heterogeneous media is concerned. The media consist of a connected fractured subregion with high permeability and a disconnected matrix block subset with low permeability. Let ω denote the size ratio of the matrix blocks to the whole domain and let the permeability ratio of the matrix block subset to the fractured subregion be of the order ω2λ for some λ > 0.
The solutions of elliptic equations in highly heterogeneous media change smoothly in the fractured subregion but change rapidly in the disconnected matrix block subset. Indeed, it is shown that in the fractured subregion, the elliptic solutions can be bounded uniformly in ω in Lipschitz norm; but in the matrix block subset, the elliptic solutions may be unbounded in ω in general. Besides, a numerical method is proposed to find the approximation of the solutions of elliptic equations in highly heterogeneous media. The convergence rate of the numerical method in L∞norm is also derived.
Keywords: heterogeneous media, fractured region. AMS Subject Classification: 35J25, 35J67, 35M20
1. Introduction
A numerical method is proposed to find the approximation of the solutions of elliptic equations in highly heterogeneous media. The media Ω ⊂ Rn(n = 2, 3) contain two subsets, a connected subregion with high permeability and a disconnected matrix block subset with low permeability. Let Y ≡ (0, 1)n be a cell consisting of a sub-domain Ym completely surrounded by another connected sub-domain Yf (≡ Y \ Ym), XYm be the characteristic function of Ym and be extended Y -periodically to
Rn, and ω ∈ (0, 1) be the size ratio of the matrix blocks to the whole domain. If Ω(2ω) ≡ {x ∈ Ω|dist(x, ∂Ω) > 2ω}, the disconnected matrix block subset is Ωω
m ≡ {x|x ∈ ω(Ym+ j) ⊂ Ω(2ω) for some j ∈ Zn}, the connected subregion is Ωω
f ≡ Ω \ Ω ω
m, and the boundary of Ω (resp. Ωωm) is ∂Ω (resp. ∂Ωωm). The problem
October 31, 2012 2 Elliptic equations that we considered is −∇ · (Kω∇Ψǫ,ω) + γΨǫ,ω = G in Ωωf, −∇ · (ǫ2λk ω∇ψǫ,ω) + ǫτγψǫ,ω= ǫτgǫ,ω in Ωωm, Kω∇Ψǫ,ω· ~nω= ǫ2λkω∇ψǫ,ω· ~nω on ∂Ωωm, Ψǫ,ω= ψǫ,ω on ∂Ωωm, Ψǫ,ω= 0 on ∂Ω, (1.1)
where ǫ, ω ∈ (0, 1), λ, τ > 0, and γ ∈ [0, M] are constants, Kω(x) ≡ K(xω), kω(x) ≡ k(x
ω), K(1 − XYm) + kXYm is a periodic positive function in R
n with period Y , and ~nω is a unit normal vector on ∂Ωωm. It is known that if K, k, G, gǫ,ω are smooth, a piecewise smooth solution of (1.1) exists uniquely and, by energy method, the H1 norm of the solution in the high permeability subregion Ωω
f is bounded uniformly in ǫ, ω, but not in the low permeability subregion Ωω
m. However, the higher order norm of the solution of (1.1) may not be bounded uniformly in ǫ, ω. Indeed, the higher order norm may grow fast even in the high permeability subregion Ωω
f when ǫ, ω become small. Therefore, if standard finite element or finite difference method is used to compute the approximation of the solution of (1.1), then the mesh size should be very small in order to obtain good approximation of the solution of (1.1) when ǫ, ω are small. It is expensive to obtain the good approximation by classical methods. On the other hand, homogenization theory tells us that when ǫ, ω become small, the solution of (1.1) approaches to some function which satisfies a simple elliptic differential equation. So one may expect that an approximation of the simple differential equation is a good approximation of the solution of (1.1) when ǫ, ω become small.
There are some literatures related to this work. Lipschitz estimate and W2,p esti-mate for uniform elliptic equations with discontinuous coefficients had been proved in [22, 26]. Uniform H¨older, Lp, and Lipschitz estimates in ǫ for uniform elliptic equations with smooth oscillatory coefficients were proved in [5, 4]. Uniform Lips-chitz estimate in ǫ for the Laplace equation in perforated domains was considered in [30]. Uniform H¨older and Lipschitz estimates in ǫ for non-uniform elliptic equations with discontinuous oscillatory coefficients were shown in [34]. By homogenization theory, the solutions of elliptic equations in perforated domains in general converge to a solution of some homogenized elliptic equation with convergence rate ǫ in L2 norm and with convergence rate ǫ1/2in H1norm as ǫ closes to 0 (see [6, 20, 28] and references therein). Higher order asymptotic expansion for the solutions of elliptic equations in perforated domains could be found in [7, 23]. Rigorous proof of higher order convergence rate for the solution of (2.3) in Hilbert spaces was considered in [6, 10, 28]. In this work, we shall derive uniform H¨older and Lipschitz estimates in ω for the solution of (1.1) and derive some convergence results of the approximation for the solution of (1.1) in L∞
October 31, 2012
Elliptic equations 3
2. Notation and main result
For any set D, D denotes the closure of D, |D| is the volume of D, and XD is the characteristic function on D. Let Ck,αdenote the H¨older space with norm k · k
Ck,α
and Lp (resp. Hs, Ws,p, Hs
0) denote the Sobolev space with norm k · kLp (resp.
k · kHs, k · kWs,p, k · kHs) for k ≥ 0, α ∈ (0, 1], s ≥ −1, p ∈ (1, ∞) [18]. Br(x) is
a ball centered at x with radius r > 0. If D is a bounded set in Rn, we define D(x, r) ≡ D ∩ Br(x). For any ϕ ∈ L1(D) and r > 0,
(ϕ)x,r≡ − Z D(x,r) ϕ(y)dy ≡ 1 |D(x, r)| Z D(x,r) ϕ(y)dy. Remark 2.1. Next we recall an extension result [1].
For 1 ≤ p < ∞, there is a constant γ1(Yf, p) and a linear continuous extension operator Πω: W1,p(Ωωf) → W1,p(Ω) such that
1) If ϕ ∈ W1,p(Ωω f), then Πωϕ = ϕ in Ωωf almost everywhere, kΠωϕkW1,p(Ω)≤ γ1kϕkW1,p(Ωω f), γ2≤ Πωϕ ≤ γ3 if ϕ ∈ L∞(Ωωf) and γ2≤ ϕ ≤ γ3, Πωϕ = ζ in Ω if ϕ = ζ|Ωω
f for some linear function ζ in Ω.
2) For any constant r > 0, Πω/rζ(x) = (Πωϕ)(rx) where ζ(x) ≡ ϕ(rx).
It is well-known that if τ ≥ λ, K, k ∈ C0,1(Rn) are positive functions, and kGkL2(Ωω
f)+kgǫ,ωkL2(Ωωm)are bounded independent of ǫ, ω, the solution of (1.1) exists
uniquely and satisfy kΨǫ,ωkH1(Ωω f)+ kǫ τ /2ψ ǫ,ω, ǫλ∇ψǫ,ωkL2(Ωω m)≤ c (independent of ǫ, ω). By compactness principle [2, 18, 20], ΠωΨǫ,ω→ Ψ in L2(Ω) strongly Kω∇Ψǫ,ωXΩω f → K∇Ψ in L 2(Ω) weakly GXΩω f + ǫ τg ǫ,ωXΩω m → |Yf|G in L 2(Ω) weakly as ǫ, ω → 0, (2.1)
where K is a constant symmetric positive definite matrix. Moreover, the function Ψ in (2.1) satisfies
(
−∇ · (K∇Ψ) + |Yf|γΨ = |Yf|G in Ω,
Ψ = 0 on ∂Ω, (2.2)
where |Yf| is the volume of Yf. For any δ > 0, define µ ≡ δ
n+δ and take α ∈ (µ, 1). We assume A1. Ω ⊂ Rn for n = 2, 3 is C3,α, Y
mis a smooth simply connected domain, A2. K, k ∈ C0,1(Rn) are positive periodic functions in Rn with period Y and
k∇KkL∞(Y )+ k∇kkL∞(Y )is small compared with minx∈Y{K, k},
October 31, 2012
4 Elliptic equations
A4. kGkLn+δ(Ωω
f)+ kgǫ,ωkLn+δ(Ωωm) for some δ ∈ (0, 3) is bounded independent
of ǫ, ω.
We have the following convergence result.
Theorem 2.1. Under A1-4, the solutions of (1.1) and (2.2) satisfy kΨǫ,ω− ΨkL∞(Ωω f)+ ǫ 2λkψ ǫ,ω− ΨkL∞(Ωω m)≤ c max{ω, ǫ τ}, where constant c is independent of ǫ, ω.
Define B ≡ (0, 1)n, Bω
m≡ {x|x ∈ ω(Ym+ j) ⊂ B for some j ∈ Zn}, and Bωf ≡ B \ Bωm. Suppose γ > 0 and −∇ · (Kω∇Ψǫ,ω) + γΨǫ,ω = G in Bfω, −∇ · (ǫ2λk ω∇ψǫ,ω) + ǫτγψǫ,ω= ǫτgǫ,ω in Bmω, Kω∇Ψǫ,ω· ~nω= ǫ2λkω∇ψǫ,ω· ~nω on ∂Bmω, Ψǫ,ω= ψǫ,ω on ∂Bmω, Ψǫ,ωXΩω
f + ψǫ,ωXΩωm satisfies periodic boundary condition.
(2.3)
Under A1–3 and kGkL2(Bω
f)+ kgǫ,ωkL2(Bmω) is bounded independent of ǫ, ω, the
solution of (2.3) exists uniquely and satisfies kΨǫ,ωkH1(Bω
f)+ kǫ
τ /2ψ
ǫ,ω, ǫλ∇ψǫ,ωkL2(Bω m)≤ c,
where c is independent of ǫ, ω. By compactness principle [2, 18, 20], ΠωΨǫ,ω→ Ψ in L2(B) strongly Kω∇Ψǫ,ωXBω f → K∇Ψ in L 2(B) weakly GXBω f + ǫ τg ǫ,ωXBω m → |Yf|G in L 2(B) weakly as ǫ, ω → 0,
where K is a constant symmetric positive definite matrix. Moreover, the function Ψ satisfies
(
−∇ · (K∇Ψ) + |Yf|γΨ = |Yf|G in B,
Ψ satisfies periodic boundary condition. (2.4) Next we consider Lipschitz estimate.
Theorem 2.2. Under A1-3, γ > 0, and kGkLn+δ(Bω
f)+ kgǫ,ωkLn+δ(Bωm) is bounded
independent of ǫ, ω, the solutions of (2.3) and (2.4) satisfy sup x∈Bω f ∇Ψǫ,ω(x) − (I − ∇X(x ω))∇Ψ(x) +ǫ2λ sup x∈Bω m ∇ψǫ,ω(x) − (I − ∇X( x ω))∇Ψ(x) ≤ c max{ωµ/2, ǫτ},
October 31, 2012
Elliptic equations 5
Next we consider the following −∇ · (Kω∇Ψǫ,ω) = G in Bωf, −∇ · (ǫ2λk ω∇ψǫ,ω) = ǫτgǫ,ω in Bωm, Kω∇Ψǫ,ω· ~nω= ǫ2λkω∇ψǫ,ω· ~nω on ∂Bωm, Ψǫ,ω = ψǫ,ω on ∂Bωm,
Ψǫ,ω satisfies periodic boundary condition, Z GXBω f + ǫ τg ǫ,ωXBω mdx = Z Ψǫ,ωXBω f + ψǫ,ωXBωmdx = 0. (2.5)
Under A1–3 and kGkL2(Bω
f)+ kgǫ,ωkL2(Bωm) is bounded independent of ǫ, ω, the
solution of (2.5) exists uniquely and satisfies kΨǫ,ωkH1(Bω f)+ kǫ
λ∇ψ
ǫ,ωkL2(Bω m)≤ c
(independent of ǫ, ω). By compactness principle [2, 18, 20], ΠωΨǫ,ω → Ψ in L2(B) strongly Kω∇Ψǫ,ωXBω f → K∇Ψ in L 2(B) weakly GXBω f + ǫ τg ǫ,ωXBω m → |Yf|G in L 2(B) weakly as ǫ, ω → 0,
where K is a constant symmetric positive definite matrix. Moreover, the function Ψ satisfies −∇ · (K∇Ψ) = |Yf|G in B, Z B Ψdx = Z B Gdx = 0,
Ψ satisfies periodic boundary condition.
(2.6)
Theorem 2.3. Under A1-3 and kGkLn+δ(Bω
f)+ kgǫ,ωkLn+δ(Bωm)is bounded
indepen-dent of ǫ, ω, the solutions of (2.5) and (2.6) satisfy sup x∈Bω f ∇Ψǫ,ω(x) − (I − ∇X(x ω))∇Ψ(x) +ǫ2λ sup x∈Bω m ∇ψǫ,ω(x) − (I − ∇X( x ω))∇Ψ(x) ≤ c max{ωµ/2, ǫτ},
where constant c is independent of ǫ, ω. References
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3. T. Arbogast, J. Douglas, and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal. 21 (1990) 823–836. 4. Marco Avellaneda, Fang-Hua Lin, Homogenization of elliptic problems with Lp
October 31, 2012
6 Elliptic equations
5. Marco Avellaneda, Fang-Hua Lin, Compactness methods in the theory of homogeniza-tion, Communications on Pure and Applied Mathematics Vol. XI (1987) 803–847. 6. N. Bakhvalov and G. Panasenko, Homogenisation : averaging processes in
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Systems(AMS, 1998).
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Annals of Math. Studies, Princeton Univ. Press., 1983).
18. D. Gilbarg, N. S. Trudinger Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin, second edition, 1983.
19. Jianguo Huang, Jun Zou, Some new a priori estimates for second-order elliptic and parabolic interface problems, Journal of Differential Equations 184 (2002) 570–586. 20. V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential
Opera-tors and Integral Functions, (Springer-Verlag, 1994).
21. O. A. Ladyzhenskaya, Nina N. Ural’tseva Elliptic and Quasilinear Elliptic Equations. Academic Press, 1968.
22. Yan Yan Li, Michael Vogelius, Gradient estimates for solutions to divergence form el-liptic equations with discontinuous coefficients, Arch. Rational Mech. Anal. 153 (2000) 91–151.
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26. A. Maugeri, Dian Palagachev, Lubomira G. Softova, Elliptic and parabolic equa-tions with discontinuous coefficients, Wiley-VCH, Berlin ; New York, 2000. 27. N. G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic
October 31, 2012
Elliptic equations 7
28. O.A. Oleinik, A.S. Shamaev, G.A. Tosifan, Mathematical Problems in Elasticity and Homogenization(North-Holland, Amsterdam, 1992).
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