HOMOGENIZATION IN DOMAINS RANDOMLY

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS SERIES B Volume 12, Number 4, November 2009

doi:10.3934/dcdsb.2009.12.713 pp. 713–730

HOMOGENIZATION IN DOMAINS RANDOMLY PERFORATED ALONG THE BOUNDARY

Gregory A. Chechkin Department of Differential Equations Faculty of Mechanics and Mathematics Moscow Lomonosov State University Moscow 119991, Russia & Narvik University College Postboks 385, 8505 Narvik, Norway

Tatiana P. Chechkina Department of Higher Mathematics Moscow Engineering Physics Institute (State University) Kashirskoe sh., 31, Moscow 115409, Russia

Ciro D’Apice Dipartimento di Ingegneria dell’Informazione e Matematica Applicata Universit` a degli Studi di Salerno Via Ponte don Melillo, 1, Fisciano (SA) 84084, Italia

Umberto De Maio Dipartimento di Matematica e Applicazioni “R.Caccioppoli” Universit` a degli Studi di Napoli “Federico II” Complesso Monte S.Angelo Via Cintia, 80126 Napoli, Italia

(Communicated by Yalchin Efendiev) Dedicated to the Centenary Anniversary of Sergei L’vovich Sobolev Abstract. We study the asymptotic behavior of the solution of the Laplace equation in a domain perforated along the boundary. Assuming that the boundary microstructure is random, we construct the limit problem and prove the homogenization theorem. Moreover we apply those results to some spectral problems.

2000 Mathematics Subject Classification. Primary: 35B27, 35P05; Secondary: 76M50. Key words and phrases. Homogenization, partial differential equation, perforated domain, spectral problem. The work of the first and the second authors was partially supported by RFBR (project No 09-01-00353) and by the program “Leading Scientific Schools” (HIII-1698.2008.1).

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Introduction. Homogenization is one of the most important parts of PDE theory. Basic results and methods could be found in [35], [36], [11], [4], [34], [30], [22], [17]. The asymptotic analysis of the boundary value problem in perforated domain with small holes (not domains perforated along the boundary) has been widely studied by many authors. We could mention Marchenko and Khruslov [31], [32], Allaire and Murat [3], Cioranescu, Donato, Murat and Zuazua [20], Cioranescu and Saint Jean Paulin [22], Cioranescu and Murat [21], Dal Maso and Murat [26], Jikov, Kozlov and Oleinik [30], Efendiev and Durlofsky [28], Aarnes and Efendiev [1], Chechkin and Chechkina [15], [16] and Chechkin, Piatnitski and Shamaev [17]. It is well known the interesting effect of homogenization of the Poisson equation with (zero) Dirichlet conditions on the boundary of the holes, when a “strange term” appears in the limit equation (see [31], [21]). Another effect of homogenization of the same equation with a critical size of the holes, when nonhomogeneous Neumann conditions on the boundary of the holes are assumed, is studied by Conca and Donato in [23]. In this case, a constant, proportional to the limit of the total flux of the solution through the boundary of the holes, appears in the limit equation. Cardone, D’Apice and De Maio in [12] and Corbo Esposito, D’Apice and Gaudiello in [24] examined the same equation with mixed boundary conditions on the holes (see also a paper of Belyaev, Chechkin and Piatnitski [10]). In the paper [18] of Piatnitski and Chechkin, one can find some results obtained for domains with locally periodic structure and Fourier boundary conditions on the boundary of holes. As proved in [24] in the context of perforated domains with a rather simple geometry of the holes, an interference phenomenon in the homogenization of such boundary value problems is present. Interesting case of the Steklov, Neumann and Dirichlet type spectral problems is considered in [33]. The author proved the homogenization theorem and constructed the asymptotic expansion of eigenelements. For the asymptotic analysis of the boundary value problem in randomly perforated domains with small holes in nonlinear case, we could mention Balzano, Corbo Esposito and Paderni [5] (in particular, for integral functionals of the Calculus of Variations) and Balzano and Durante [6] (for p-Laplace operator). Also, randomly perforated domains were studied in [30] and [17]. The case of the Stokes equation with random potential was studied by Belyaev and Efendiev in [8]. Moreover, the asymptotic analysis of the boundary value problem in domains perforated along the boundary for the periodic case was studied in [9]. Also we can mention the paper [14] where the authors studied the asymptotic behavior of the constant in Friedrichs inequalities in domains perforated along the boundary. In many applications, one can meet geometrical structures, which are not purely periodic but similar to periodic. In such cases, it is natural to consider statistically homogeneous random models. This paper is devoted to the investigations of such problem. We consider a membrane randomly perforated along the boundary, which is fixed on holes and we study the asymptotic behavior of this membrane. As example of a membrane fixed on the holes along the boundary, we can consider a hand made drum. Taking a skin and fixing it on holes along the boundary on a wooden tube, one can use it as a musical instrument. The question is about the eigenvibrations of such a membrane or how it must be fixed theoretically to get a good sound (see Figure 1). In this paper we study the asymptotic behavior, for ε → 0, of solutions to a boundary value problem for the Laplace equation in a domain randomly perforated along the

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Figure 1. Drum boundary. We set nonhomogeneous Neumann conditions on the outer boundary of the domain and homogeneous Dirichlet conditions on the boundary of holes. Also, it should be noted that we apply the approach developed in the paper [7]. The paper is organized as follows. In Section 2, we establish a convergence result for the solutions uε to the problem (1) (see Theorem 2.2) for perforated domains with selfsimilar structure. In Section 3, we introduce the notion of randomness, give rigorous definitions and formulate necessary propositions. In Section 4, we prove a Homogenization Theorem (see Theorem 4.1) for random structure which satisfies the condition of selfsimilarity. In Section 5, we apply those results to some spectral problems. 1. Setting of the problem. Let D be a bounded smooth domain in R2 . Denote by σε the curve which lies along the boundary ∂D at the distance ε from the boundary. The existence of this curve is a consequence of the smoothness of the boundary. Denote also by Dε the domain D \ Bε , where Bε is a union of circles of radius ε,that are along the boundary ∂D at distance ε from it (see Figure 2). Let us describe the set Bε in detail. Assume that B is a set of circles on a curve ̟ε . Consider a contraction of this set in 1ε times, such that ̟ε maps to the curve σε , then Bε is the obtained set of circles. The boundary ∂Dε consists of two parts: ∂Dε = ∂D ∪ Γε , where Γε = ∂Bε . We study the asymptotic behavior, for ε → 0 of the solutions uε to the boundary value problem:  in Dε ,   ∆uε = 0 uε = 0 on Γε , (1)   ε ∂uε = g(x) on ∂D. ∂ν Here ν is the outward normal to ∂D.

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Figure 2. Domain Perforated Along the Boundary We denote by H01 (Dε , Γε ) the completion by the Sobolev norm k · kW21 (Dε ) of the set of functions C ∞ (D\Bε ) which vanish in the neighborhood of Γε . Remark 1. Further we identify functions from the space H01 (Ωε , Γε ) with functions from H 1 (Ω), equal to zero in Bε , and functions from the space L2 (Ωε ) with functions from L2 (Ω), equal to zero in Bε . On the other hand for the restrictions of functions from L2 (Ω) on Ωε we keep their notations. The dependence of Γε on the small parameter plays the principal role for the asymptotic behaviour of the solution uε as ε → 0, and the multiplier ε in the boundary conditions of (1) could be excluded by changing of variables. We introduced this multiplier in order to obtain the order 1 leading term of the asymptotic behaviour of uε as ε → 0. 2. Compactness Theorem. Following [7], let us formulate the basic property of the considered structures without using of the geometrical parameters. Definition 2.1. A family of closed sets Γε ⊂ ∂Dε is called selfsimilar, if there exist constants C1 > 0 and s, 1 < s ≤ 2, independent of ε, such that for any ε, 0 < ε ≤ ε0 and for any smooth function ϕ ∈ C ∞ (Dε ) with supp ϕ ∩ Γε = ∅, the following inequality holds:  Z  1s  Z  12 |ϕ|s dx ≤ C1 ε ϕ2 + (∇ϕ)2 dx . (2) Dε

∂Dε

Remark 2. Note that, if there exists a “spot” of fixed size on σε , nonintersecting with Γε as ε → 0, the inequality (2) does not hold. If the family {Γε } is selfsimilar, then solutions uε to the problem (1) satisfy the following asymptotic properties as ε → 0. ′

Theorem 2.2. Assume that g ∈ Ls (∂D), where s′ is a number mutual to s from the Definition 2.1., i.e. 1s + s1′ = 1. Then (i) the sequence uε is bounded in Ls (∂D) as ε → 0;

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(ii) there exists a measurable function K : ∂D → [0, +∞) and a subsequence ′ εk → 0, independent of the function g ∈ Ls (∂D), such that uεk weakly converges to K(x) g(x) in Ls (∂D) as εk → 0; (iii) the sequence uε is compact in Lp (Dε ), and the subsequence uεk strongly converges in Lp (Dε ) to a harmonic function u⋆ equal to K(x) g(x) on the boundary ∂D of the domain D, where p < 2s. To prove the theorem, we use the following lemma from [7]. Lemma 2.3. Suppose that Q is a smooth domain in Rn and a sequence of harmonic functions vε defined in Q, is weakly compact in Ls (∂Q), s > 1. Then, this sequence ns is strongly compact in the space Lp (Q), p < n−1 . Remark 3. Classical results concerning the embedding theorems were proved by S. L. Sobolev (see, for instance, [37]). Proof of Theorem 2.2. To prove the statement (i), we use the integral identity of the problem (1). The function uε ∈ H01 (Dε , Γε ) satisfies Z Z ε ∇uε ∇vdx = gvdx, (3) Dε

∂D

H01 (Dε , Γε ).

for any functions v ∈ In further analysis we use the Poincar´e inequality in the following form: !  Z 2 Z Z 2 2 ϕ dx ≤ C2 |ϕ|dx + (∇ϕ) dx , Dε

∂Dε

(4)

Dε

where the constant C2 does depend only on the domain. From (2) and (4), we derive that the functionals in the lefthand side and the righthand side of (3) satisfy the conditions of the Lax–Milgram lemma (see, for instance [38], [17]). Hence, the unique solution uε ∈ H01 (Dε , Γε ) does exist. Moreover the inequalities (2) and (4) hold true for functions from H01 (Dε , Γε ). In addition, there exists a continuous trace operator from H01 (Dε , Γε ) to Ls (∂D), hence the values of functions from H01 (Dε , Γε ) on the boundary of the domain Dε can be treated as the values of their traces. Substituting v = uε in (3) and using the H¨older inequality, we obtain that: Z  1′  Z  1s Z s ′ ε |∇uε |2 dx ≤ |g|s dl |uε |s dl . Dε

∂D

∂D

Estimating the righthand side of such inequality by (2), we derive that: v Z Z u u 2 ε |∇uε | dx ≤ C3 tε (u2ε + |∇uε |2 ) dx. Dε

Dε

Hence, by using the Young inequality with small enough δ > 0, we derive that:  Z 2  Z 2  Z 2 2 2 2 2 2 ε |∇uε | dx ε (uε + |∇uε | ) dx ≤ 2δ ε uε dx + δ Dε

Dε

Dε

 Z 2 Z 2 ≤ 2δ ε (u2ε + |∇uε |2 ) dx + C32 ε (u2ε + |∇uε |2 ) dx δ Dε

Dε

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or ε

Z

(u2ε + |∇uε |2 )dx ≤ C32

Dε

2 δ

1 − 2δ

Hence from (5) we derive s v Z Z u u ε |∇uε |2 dx ≤ C3 tε (u2ε + |∇uε |2 ) dx ≤ C32 Dε

Dε

(5)

.

2 δ

1 − 2δ

= C4

(6)

and, using (2), we have

Z

∂D



|uε |s dl ≤ C1 C3

s

2 δ

1 − 2δ

s

 = C5 ,

(7)

where C4 , C5 do not depend on ε. The statement (i) is completely proved. To prove the statement (ii) we consider an auxiliary problem  = 0 in Dε ,   ∆wε wε = 0 on Γε , (8) ∂w  ε  ε = 1 on ∂D ∂ν which coincides with the given problem (1), if g = 1. The corresponding integral identity in H01 (Dε , Γε ) assumes the form Z Z ε ∇wε ∇vdx = vdx. (9) Dε

The solution wε ∈

∂D

H01 (Dε , Γε )

ε

Z

Dε

satisfies some inequalities analogues to (6), (7). Z 2 |∇wε | dx ≤ C6 , |wε |s dx ≤ C7 , (10) ∂D

where C6 , C7 do not depend on ε. By (10), we can choose a subsequence ε = εk , such that wεk weakly converges in Ls (∂D) as εk → 0. We denote by K(x) the limit function on ∂D. Obviously, K ∈ Ls (∂D). Using the maximum principal for harmonic functions, we conclude that K(x) ≥ 0. Suppose that ψ ∈ C ∞ (D) is an arbitrary function. Substituting v = ψwε and v = ψuε in identities (3) and (9), respectively, and subtracting such identities, we obtain that: Z Z ε (wε ∇uε − uε ∇wε )∇ψdx = (g(x)wε − uε )ψdx. (11) Dε

∂D

Let us show that the lefthand side of (11) converges to zero as ε → 0. In fact, from (6), (7), (10) and the Poincar´e inequality, we have that functions √ the estimates √ εuε and εwε are bounded in W21 (Dε ). By the Rellich theorem (see, for instance, [17]), the sequences of these functions are strongly compact in L2 (Dε ), and they converge to zero by the norm of the space Lp (Dε ), p < 2s. Hence, they converge to zero in L2 (Dε ). Thus, in the expressions of the lefthand side of (11), one multiplier is bounded in L2 (Dε ) as ε → 0, and the second one converges to zero. Let us pass to the limit at the righthand side of (11) as εk → 0. The family of functions wεk weakly converges to K(x) in Ls (∂D). Functions uεk are bounded in Ls (∂D). If we consider a subsequence ε′k from the sequence εk , such that uε′k

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weakly converges in Ls (∂D) as ε′k → 0 to a limit function u⋆ on ∂D, and pass to the limit, we get that: Z   g(x)K(x) − u⋆ ψdx = 0. ∂D

Due to the arbitrariness in the choice of ψ ∈ C ∞ (D) on ∂D, the function u = g K, i.e. the limit function u⋆ does not depend on the choice of the second subsequence. Hence, the whole sequence uεk has unique limit. The statement (iii) of the theorem follows from (i), (ii) and Lemma 2.3. Theorem 2.2 is proved. ⋆

3. Random structures. In this section we consider a domain perforated randomly along the boundary which satisfies the conditions of the previous section. We introduce the notion of randomness, give rigorous definitions and formulate necessary propositions. Assume that (Ω, A, µ) is a probability space, where A is the σ-algebra of subsets of Ω and T (s) : Ω → Ω, s ∈ R, is an ergodic dynamical system (for definitions see [30], [17]). Definition 3.1. A measurable function ϕ(s, ω) on R × Ω is called random statistically homogeneous, if ϕ = ϕ(T (s) ω). Definition 3.2. Random set in R2 situated along the line y2 = const, is called homogeneous, if the characteristic function χ of the set is statistically homogeneous with respect to y1 , i.e. χ(y1 , y2 , ω) = χ(T (y1 ) ω, y2 ). In further analysis, we shall use homogeneous sets to construct the family {Γε }. Let B(ω) be a random homogeneous set, that generates the family Bε . To simplify the analysis, we consider the case of a unit square, i.e. D = (0, 1)×(0, 1), and assume that the domain Dε = D\Bε is perforated along the lower side Q = {(x1 , x2 ) : 0 < x1 < 1, x2 = 0} of the square (see Figure 3). Also we suppose that Γε = ∂Dε \Q.

Figure 3. Square Perforated Along the Boundary

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Definition 3.3. A random homogeneous closed set B(ω) ⊂ R2 situated along the straight line parallel to the abscissa axis, is called non-degenerate, if there exists a positive statistically homogeneous function h = h(y1 , ω), such that for almost all ω and for any functions ϕ ∈ C0∞ (R2 \B(ω)) with compact support, supp ϕ ∩ B(ω) = ∅, the following inequality holds: Z

2

h (T (y1 ) ω) ϕ (y1 , 0) dx1 ≤

R

Z∞ Z 0

| ▽ ϕ|2 dy1 dy2

(12)

R

and the mathematical expectation satisfies

−1−δ  < +∞ h

(13)

for some δ, 0 < δ ≤ +∞.

The following definitions and theorems are well known and could be found, for instance, in [30]. Definition 3.4. The number M {f } is called the spatial average value of a given function f , if Z   x lim f dx = |K| M {f } (14) ε→0 ε K

for any bounded measurable set K ⊂ R2 . Under additional conditions for f (x), the definition of average value can be represented in terms of weak convergence. For instance, suppose that a family f ( xε ) is 2 bounded in Lα loc (R ) for α ≥ 1. Since the linear capsule of characteristic functions ′ 1 1 2 of the sets K is dense in Lα loc (R ), where α + α′ = 1, one can substitute (14) with: x 2 f ⇀ M {f } weakly in Lα loc (R ). ε In the next section, we shall use the Birkhoff theorem (see, for instance, [30], [27]) in the following form. Theorem 3.5. (Birkhoff ergodic theorem) Assume that f ∈ Lα (Ω) α ≥ 1. Then, for almost all ω ∈ Ω, the realization f (T (s) ω) has the average value in the sense (14). Moreover, the average value M {f (T (s) ω)} is invariant as a function of ω ∈ Ω and Z Z hf i ≡ f (ω)dµ = M {f (T (s) ω)}dµ Ω

Ω

In particular, if the system T (s) is ergodic, then for almost all ω ∈ Ω, we have that: hf i = M {f (T (s) ω)}. Now, we give an example of a function h, which could be applied for a wide class of random sets. Assume that the domain B(ω) is a union of disks centered in isolated points Y i = (y1i , 32 ), with diameter equal to one. Denote by r = rω (y1 ) the distance between the point Y = (y1 , 32 ) and the nearest center Y i . If the random domain B(ω) is homogeneous, then the function r is also statistically homogeneous.

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Lemma 3.6. The inequality (12) holds true if h = h(y1 , ω) = q r2 +

1 9 4

log(r2 + 94 )

.

Proof. Let us represent the line {(y1 , y2 ) : y1 ∈ R, y2 = 23 } as a union of segments Vi , consisting of points for which Y i represents the nearest center. In each semistrip (∞) Vi ≡ {(y1 , y2 ) : (y1 , 32 ) ∈ Vi , y2 ∈ [0, +∞)}, we define the polar system of (∞) coordinates (ρ, θ), where ρ = |y − Y i |, y ∈ Vi . It is easy to check that the (2) boundary of the rectangle Vi ≡ {(y1 , y2 ) : (y1 , 32 ) ∈ Vi , y2 ∈ [0, 2]} is defined in a unique way by the function ρ = R(θ). Also, it should be noted that, in polar

Figure 4. The sets Vi × [0, +∞) 1 coordinates, h = h(R(θ)) = 2R(θ) log R(θ) . Consider an arbitrary function ϕ ∈ C0∞ (R2 \B(ω)). In each semistrip, ϕ(ρ, θ) ≡ 0 if ρ ≤ 1. (2) For any points (R(θ), θ) from the segment vi ≡ {(y1 , y2 ) ∈ Vi : y2 = 0}, we represent the function ϕ(R(θ), θ) in the form R(θ) Z

ϕ(R(θ), θ) =

∂ϕ dρ. ∂ρ

1

Using the Cauchy—Schwarz—Bunyakovsky inequality and the inequality  2  2  2 ∂ϕ ∂ϕ ∂ϕ ≤2 +2 = 2|∇ϕ|2 , ∂ρ ∂y1 ∂y2 we derive that: 2

ϕ (R(θ), θ) ≤ 2

R(θ) Z 1

2

|∇ϕ| ρ dρ

R(θ) Z 1

dt . t

(15)

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Let I be defined as: I = I(θ) = ϕ2 (R(θ), θ) h(R(θ)) R(θ). Integrating with respect to θ from θ0 to θ1 , and making a sum with respect to i, we obtain the lefthand side of the inequality (12). Here, [R(θ0 ) cos θ0 , R(θ1 ) cos θ1 ] ≡ vi . On the other hand, from (15), we deduce that: I≤2

R(θ) Z

2

|∇ϕ| ρdρ

1

 R(θ) Z 1

R(θ)  Z dt 1 R(θ) = |∇ϕ|2 ρdρ. t 2R(θ) log R(θ) 1

Note that: +∞Z Zθ1 R(θ) Z Z2π R(θ) Z Z 2 2 |∇ϕ| ρdρ dθ ≤ |∇ϕ| ρdρ dθ ≤ |∇ϕ|2 dy1 dy2 .

θ0

1

0

1

0

(16)

vi

Finally, integrating with respect to θ over [θ0 , θ1 ], considering the (16) and making a sum with respect to i, we obtain the (12). Lemma is proved. 4. Homogenization Theorem. If the family of fine-scale sets Γε ⊂ ∂Dε is obtained from the nondegenerate homogeneous set B(ω), the asymptotic behaviour of solutions uε to the problem (1) can be described more explicitly, than in Theorem 2.2. Theorem 4.1. If B(ω) is a nondegenerate closed set with index δ > 0 from the δ definition 3.3, then a.s. {Γε } is a selfsimilar family with s = 1 + (2+δ) and, for the sequence of solutions uε to problem (1), the statement of Theorem 2.2 holds true. Moreover, the limit function u⋆ is unique and non-random. The boundary function K(x) is independent of the choice of subsequences, vanishes on ∂D\Q and is equal to a positive constant on Q. Proof. First, let us prove that the family {Γε } is selfsimilar. Let ϕ be a smooth function in Dε with support in Dε \Γε . Assume that:  δ 2 2 1 1 s=1+ , α= , β= + =1 . (2 + δ) s 2−s α β By the H¨older inequality, we have that: Z      α1  Z  β1 Z  x   β x1 1 s 2 −α |ϕ| dx1 ≤ h T ω |ϕ| dx1 h T ω dx1 , ε ε Q

Q

(17)

Q

where h(y1 , ω) is the function as defined by 3.3.

x2  . The first cofacε ε tor in the righthand side of (17) is estimated by the integral over the cube D: Z     Z x1 h T ω ϕ2 (x1 , 0)dx1 ≤ ε |∇ϕ|2 dx1 dx2 . ε In the inequality (12), we change variables (y1 , y2 ) →

Q

x

1

,

D

Because of the Birkhoff theorem, the second cofactor a.s. has a finite limit, which

1 β is equal to h−(1+δ) β , since α = 1 + δ. Thus, the inequality (2) follows from (17).

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It means that the family {Γε } is selfsimilar with probability one, and the statement of Theorem 2.2 holds true for the solutions uε . The boundary function K(x) is the limit function for the sequence of solutions wε to the auxiliary problem (8). Note that wε = 0 on ∂D\Q, K(x) ≡ 0 on ∂D\Q independently of the choice of subsequences εk → 0. Now, we want to show that the boundary function K(x) is uniquely determined and is equal to a positive nonrandom constant on Q. Together with (8), let us consider the following auxiliary problem in R2+ = R × [0, +∞):  2 ∂ W ∂2W   in R2+ ,  ∂y 2 + ∂y 2 = 0   1 2  (18) W = 0 on B(ω),     ∂W   = −1 on {y2 = 0}. ∂y2

The equations (18)correspond to the auxiliary problem (8), obtained by them x1 x2  changing (y1 , y2 ) → , . But the unboundedness of the boundary of the ε ε 2 domain R+ is a more essential distinction. Let us describe the function space, in which we define solutions to (18). By S, we denote the linear space of functions W : Ω × [0, +∞) → R,

the realizations W (T (y1 ) ω, y2 ) of which are smooth functions in R2+ uniformly on ω ∈ Ω, bounded together with their derivatives, and which have a support in R2+ \B(ω), which is bounded in the direction x2 . The operator of the problem (18) is defined on S. We define a solution to the problem (18) as a function from the closure of S in an appropriate norm. For non-degenerate domains, B(ω) functions from S satisfy the following properties. Lemma 4.2. The inequalities

2  W (ω, 0)h(ω) ≤

Z∞ 0



 |∇W (T (y1 ) ω, y2 )|2 dy2 ,


1


1 |W (ω, 0)|t t ≤ C(t) W 2 (ω, 0)h(ω) 2

(19)

(20)

holds for any function W ∈ S. Here h(ω) is the weighting function from the defδ inition 3.3, t is a constant, 1 ≤ t ≤ s = 1 + 2+δ , and C(t) is a positive constant,

−1  12 C(1) = h . The proof of Lemma 4.2 is similar to the proof of Lemma 1 from [7]. Note that the probability measure µ is invariant with respect to mappings T (y1 ) and hence the righthand side of (19) is independent of y1 . The square root of this expression can be considered as a norm. Denote by S the completion of the space S by this norm. The inequality (20) shows that for functions W ∈ S, one can define a trace W (ω, 0) and the trace operator is a continuous operator from S to Ls (R).

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For the realization W (T (y1 ) ω, y2 ) of the function W ∈ S, we call the solution of the auxiliary problem (18), if it satisfies the following integral identity: Z∞ 0

h∇W (T (y1 ) ω, y2 )∇W ′ (T (y1 ) ω, y2 )i dy2 = hW ′ (T (y1 ) ω, 0)i

(21)

for any functions W ′ ∈ S. From Lemma 4.2 it follows that the bilinear form and linear functional in (21) satisfy the conditions of the Lax–Milgram lemma. Thus, the solution W ∈ S to the problem (18) exists and is unique. Moreover, if W = W ′ , then from (21) one can derive the estimate

 hW (ω, 0)i ≤ h−1 . (22) The realization W (T (y1 ) ω, y2 ) of the solution satisfies the equations (18) in the 1,2 sense (21) as well as locally as a function from the Sobolev space Wloc (R2+ ).

Lemma 4.3. For almost all ω ∈ Ω the realization W (T (y1 ) ω, y2 ) of the solution 1,2 to the problem (18) belongs to Wloc (R2+ ) and satisfies the integral identity Z Z ∇W (T (y1 ) ω, y2 )∇Ψ(y1 , y2 ) dy1 dy2 = Ψ(y1 , 0) dy1 (23) R2+

R

for any smooth function Ψ(y1 , y2 ) with compact support in R2+ \B(ω). The proof of Lemma 4.3 is similar to the proof of a more general proposition from [30] and we omit it. Let us rewrite the identity (23) in the following form: Z Z  x  x  1 2 ε ∇W T ω, ∇v(x1 , x2 )dx1 dx2 = v(x1 , 0)dx1 . (24) ε ε R2+

R

Here, v(x1 , x2 ) is a smooth function with support in R2+ \εB(ω). Note that we derive the identity (24) from the identity (23) by changing of variables. Suppose that ψ ∈ C ∞ (D) is a smooth function in D with support   in D ∪ Q. Substituting v = ψ(x1 , x2 )wε (x1 , x2 ) in (24) and v = ψ(x1 , x2 )W T xε ω, xε2 in

(9), where wε is a solution to the problem (8), and subtracting such identities from each other, we get that: Z h  x  x  1 2 ω, ε wε (x1 , x2 )∇W T ε ε D  x  x  i 1 2 −W T ω, ∇wε (x1 , x2 ) ψ(x1 , x2 )dx1 dx2 (25) ε ε Z h   x  x i 1 2 = wε (x1 , 0) − W T ω, ψ(x1 , 0)dx1 . ε ε Q

Pass to the limit in (25) as ε → 0. The lefthand side converges to zero in the same way as in
(11)
in the proof of Theorem 2.2. In the righthand side the functions W T xε1 ω, 0 converges weakly to hW (ω, 0)i in Ls (Q) because of the ergodicity, and functions wε (x1 , 0) converge to K(x) by a subsequence εk → 0 due to Theorem 2.2. Due to the arbitrariness of the choice of the function ψ(x1 , x2 ), we have that

RANDOMLY PERFORATED DOMAIN

725

K(x) ≡ hW (·, 0)i on Q, independently on the choice of the subsequence. Hence, the whole sequence wε (x1 , 0) converges to this nonrandom limit on Q. Assuming that W ′ = W in the identity (21), we conclude that hW (·, 0)i ≥ 0. The upper bound (22) holds also for the function K(x). Theorem 4.1 is proved. 5. Application to spectral problems. In this section, we apply the results developed in [34] to the spectral problem associated with boundary–value problem (1). To simplify the analysis we consider a domain Dε defined in Section 1 and assume that ∂D = Γ1 ∪ Γ2 . The simplification is connected with the application of the Friedrichs inequality. We suppose that functions vanish on Γ1 — a fixed part of the boundary ∂D. Denote by H01 (Dε , Γε ∪ Γ1 ) the completion by the Sobolev norm k · kW21 (Dε ) of the set of functions C ∞ (D\Bε ) which vanish in the neighborhood of Γε ∪ Γ1 . We consider the following problems:   ∆(ukε ) + λkε ukε = 0 in Dε ,   on Γε ∪ Γ 1 , ukε = 0 (26)  ∂ukε   = 0 on Γ2 , k = 1, 2... ∂ν and



∆(uk ) + λk0 uk uk

= 0 = 0

on

∂D,

in D, k = 1, 2...

(27)

Here, ukε ∈ H01 (Dε , Γε ∪ Γ1 ), uk ∈ H01 (D), k = 1, 2, ... are orthogonal basis in L (Dε ) and L2 (D), respectively. The sets {λkε }, {λk0 }, k = 1, 2, ... are the corresponding eigenvalues such that 2

0 < λ1ε ≤ λ2ε ≤ ... ≤ λkε ≤ ...,

0 ≤ λ10 ≤ λ20 ≤ ... ≤ λk0 ≤ ...

and they repeat with respect to their multiplicities. For the sake of completeness, we state here the results on spectral convergence for positive, selfadjoint and compact operators on Hilbert spaces (see Section III.1 in [34] for the proof). Theorem 5.1. Let Hε and H0 be two separable Hilbert spaces with the scalar products (·, ·)ε and (·, ·)0 , respectively. Let Aε ∈ L(Hε ) and A0 ∈ L(H0 ). Let V be a linear subspace of H0 such that {v : v = A0 u, u ∈ H0 } ⊂ V . We assume that the following properties are satisfied: ε→0

• C1 There exists Rε ∈ L(H0 , Hε ) such that (Rε F, Rε F )Hε −−−−−−→γ0 (F, F )H0 , for all F ∈ V and certain positive constant γ0 . • C2 The operators Aε and A0 are positive, compact and selfadjoint. Moreover, kAε kL(Hε ) are bounded by a constant, independent of ε. ε→0

• C3 kAε Rε F − Rε A0 F kHε −−−−−−→0 for all F ∈ V. • C4 The family of operators Aε is uniformly compact, i.e., for any sequence F ε in Hε such that supε kF ε kHε is bounded by a constant independent of ε, ′ we can extract a subsequence F ε , that verifies the following: ′

kAε′ F ε − Rε′ v 0 kHε′ → 0, as ε′ → 0, for certain v 0 ∈ H0 .

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GREGORY A. CHECHKIN ET AL.

0 ∞ Let {µεi }∞ i=1 and {µi }i=1 be the sequences of the eigenvalues of Aε and A0 , respectively, with the classical convention of repeated eigenvalues. Let {wiε }∞ i=1 and ({wi0 }∞ , respectively) be the corresponding eigenfunctions in H , which are asε i=1 sumed to be orthonormal (H0 , respectively). Then, for each k, there exists a constant C8k , independent of ε, such that

|µεk − µ0k | ≤ C8k

sup u∈N (µk 0 ,A0 ), kukH =1 0

kAε Rε u − Rε A0 ukHε ,

where N (µk0 , A0 ) = {u ∈ H0 , A0 u = µk0 u}. Moreover, if µ0k has multiplicity s (µ0k = µ0k+1 = · · · = µ0k+s−1 ), then for any w eigenfunction associated with µ0k , with kwkH0 = 1, there exists a linear combination wε of eigenfunctions of Aε , {wjε }j=k+s−1 associated with {µεj }j=k+s−1 such that j=k j=k kwε − Rε wkHε ≤ C9k kAε Rε w − Rε A0 wkHε ,

where the constant C9k is independent on ε.

We denote by Hε the space L2 (Dε ) with the scalar product Z ε ε (f , g )Hε ≡ f ε (x) g ε (x) dx . Dε

We denote by H0 the space L2 (D), where the scalar product is Z (f 0 , g 0 )H0 ≡ f 0 g 0 dx. D

We define the operator Aε : L2 (Dε ) → H01 (Dε , Γε ∪ Γ1 ), Aε f = uε , where uε is the solution of problem    −∆uε = f uε = 0   ∂uε = 0 ∂ν We consider the operator:

in Dε , on Γε ∪ Γ1 ,

(28)

on Γ2 .

A0 : L2 (D) → H01 (D), A0 f = u, where u is the solution of problem:  −∆u0 = u0 =

f 0

in D, on ∂D.

(29)

In fact, Aε and A0 are operators associated to the eigenvalue problems (26) and (27), respectively. Now, considering the operators Aε : Hε → Hε and A0 : H0 → H0 , it is easy to establish the positiveness, self–adjointness and compactness of the operators Aε and A0 , respectively. In particular, the compactness of both operators follows from the compactness of the imbedding of H 1 (D) into the space L2 (D). Let V be V = H01 (D), which satisfies ImA0 ⊆ V ⊂ H0 , and let Rε be Rε : L2 (D) → L2 (Dε )

the restriction operator.

(30)

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Similarly to the proof of Theorem 2.2, we can show that the following statement is true. Theorem 5.2. Assume that f ∈ L2 (D). Then, the sequence uε is compact in L2 (Dε ), and the subsequence uεk strongly converges in L2 (Dε ) to a function u0 , that satisfies the problem (29). Let us verify the conditions C1 − C4 of Theorem 5.1 (Theorem 1.4 from Section III.1 in [34]). C1. The operator Rε : H0 → Hε is defined in (30). Obviously, Z Z (Rε F, Rε F )Hε = F 2 dx → F 2 dx = (F, F )H0 Dε

D

as ε → 0. Hence, we conclude that this condition is fulfilled with γ0 = 1. Let us prove that norms kAε kL(Hε ) are uniformly bounded with respect to ε. Keeping in mind the equivalence of norms, we obtain that: Z kuε k2H 1 (Dε ) ≤ C10 |∇uε |2 dx ≤ C11 kf kL2(Dε ) kuε kH 1 (Dε ) Dε

or kuε kH 1 (Dε ) ≤ C12 kf kL2(Dε ) .

Hence, kAε f k2Hε =

Z

uε (x)2 dx ≤ kuε k2H 1 (Dε ) ≤ C13 kf k2Hε .

Dε

Thus, kAε f kHε ≤ C14 kf kHε holds true and condition C2 is fulfilled. By Theorem 5.2, condition C3 takes place. Let us consider this condition in more detail. Using the definitions of the operators Aε , A0 for any f ∈ V , we obtain that: kAε Rε f − Rε A0 f k2Hε = kuε − uk2Hε −→ 0

as ε → 0.

Thus, condition C3 is valid. Let us prove the last condition, C4. If a sequence {fε } is bounded in Hε then by standard arguments we deduce that the solutions {uε = Aε fε }ε to the problem (28) are uniformly bounded with respect to ε in H 1 (Dε ). Therefore, there exists w ∈ H 1 (Dε ) and a subsequence ε′ → 0 such that uε′ → w in L2 (Dε ) and weakly in H 1 (Dε ). Thus, Z kAε fε − Rε wk2Hε =

(uε (x) − w(x))2 dx

Dε

and, then, we obtain that: kAε′ fε′ − Rε′ wkHε′ → 0 as ε′ → 0 ,

and the condition C4 is fulfilled. Now, we consider the spectral problems: Aε ukε = µkε ukε ,

ukε ∈ Hε ,

µ1ε ≥ µ2ε ≥ ... ≥ µkε ≥ · · · > 0, (ulε , ukε )Hε = δlk

k = 1, 2, ...,

and µ10

≥

µ20

A0 uk0 = µk0 uk0 , uk0 ∈ H0 , k ≥ ... ≥ µ0 ≥ · · · > 0, k = 1, 2, ...,

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GREGORY A. CHECHKIN ET AL.

(ul0 , uk0 )H0 = δlk . According to our definitions µkε = λ1k , and µk0 = λ1k , where λkε and λk0 are the ε 0 eigenvalues of problems (26) and (27), respectively. Finally, applying Theorem 5.1 (Theorems 1.4 and 1.7 in Section III.1 of [34]), we prove the following statements. Theorem 5.3. For the eigenvalues λkε , λk0 of problems (26) and (27), respectively, the convergence λkε → λk0 is valid as ε → 0. Theorem 5.4. Let us consider the same hypothesis as in Theorem 5.3. Suppose that k, l are integers, k ≥ 0, l ≥ 1, and λk0 < λk+1 = ... = λk+l < λk+l+1 . Then, 0 0 0 , there exists a for any w eigenfunction of (27), associated with the eigenvalue λk+1 0 linear combination uε of eigenfunctions uk+1 , ..., uk+l of problem (26) such that: ε ε uε → Rε w

as

ε → 0.

Acknowledgments. This paper was mostly written during the stay of Gregory A. Chechkin and Tatiana P. Chechkina at the Universit`a degli Studi di Salerno in July — August 2007. They want to thank the colleagues for the hospitality, for wonderful conditions to work and for the support. The authors express deep thanks to Professor E. S´anchez-Palencia for valuable remarks, comments and recommendation which allowed to improve the presentation of the results. Also the authors thank the reviewers for the helpful comments and suggestions. REFERENCES [1] J. Aarnes and Y. Efendiev, An adaptive multiscale method for simulation of fluid flow in heterogeneous porous media, Multiscale Model. Simul., 5 (2006), 918–939. [2] R. Adams, “Sobolev Spaces,” Academic Press, New York – London, 1975. [3] G. Allaire and F. Murat, Homogenization of the Neumann problem with non isolated holes, Asymptotic Analysis, 7 (1993), 81–95. [4] A. Bakhvalov and G. P. Panasenko, “Homogenization: Averaging Processes in Periodic Media,” Math. Appl. (Soviet Ser.), 36 Kluwer Academic Publishers, Dordrecht, 1989. [5] M. Balzano, A. Corbo Esposito and G. Paderni, Nonlinear Dirichlet problems in randomly perforated domains, Rendiconti di Matematica e delle sue Appl. (7), 17 (1997), 163–186. [6] M. Balzano and T. Durante, The p-Laplacian in domains with small random holes, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 6 (2003), 435–458. [7] A. Yu. Beliaev and G. A. Chechkin, Averaging operators with boundary conditions of fine scaled structure, Mathematical Notes, 65 (1999), 418–429 (Translated from Math. Zametki, 65 (1999), 496–510). [8] A. Yu. Beliaev and Y. Efendiev, Homogenization of the Stokes equations with random potential, Mathematical Notes, 59 (1996), 361–372 (Translated from Math. Zametki, 59 (1996), 504–520, 638). [9] A. G. Belyaev, “On Singular Perturbations of Boundary-value Problems” (Russian), Ph.D Thesis. Moscow State University, 1990. [10] A. G. Belyaev, G. A. Chechkin and A. L. Piatnitski, Averaging in a perforated domain with an oscillating third boundary condition, Russian Academy of Sciences. Sbornik. Mathematics, 192 (2001), 933–949 (Translated from Matematicheskii Sbornik, 192 (2001), 3–20). [11] A. Bensoussan, J. L. Lions and G. Papanicolaou, “Asymptotic Analysis for Periodic Structures,” North Holland, Amsterdam, 1978.

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