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ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2006, Vol. 46, No. 1, pp. 97–110. © MAIK “Nauka /Interperiodica” (Russia), 2006.

Asymptotics of Simple Eigenvalues and Eigenfunctions for the Laplace Operator in a Domain with an Oscillating Boundary Y. Amirat*, G. A. Chechkin**, and R. R. Gadyl’shin*** , ****1 * Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal 63177 Aubière cedex, France e-mail: [email protected] ** Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119992 Russia e-mail: [email protected] *** Institute of Mathematics (with Computing Center), Russian Academy of Sciences, Ufa, 450077 Russia **** Department of Mathematical Analysis, Faculty of Physics and Mathematics, Bashkir State Pedagogical University, Ufa, 450000 Russia e-mail: [email protected] Received June 7, 2005

Abstract—The asymptotic behavior of solutions to spectral problems for the Laplace operator in a domain with a rapidly oscillating boundary is analyzed. The leading terms of the asymptotic expansions for eigenelements are constructed, and the asymptotics are substantiated for simple eigenvalues. DOI: 10.1134/S0965542506010118 Keywords: oscillating boundary, spectrum of the Laplacian, asymptotics, matching of asymptotic expansions. 1

INTRODUCTION In recent years many mathematical works have been devoted to asymptotic analysis of problems in domains with rapidly oscillating boundaries. See, for instance, [1–23]. Such problems appear in many fields of physics and engineering sciences, such as the scattering of acoustic waves on small periodic obstacles, the free vibrations of strongly nonhomogeneous elastic bodies, the behavior of fluids over rough walls, and of coupled fluid–solid periodic structures. In [4], the authors considered problems with the Dirichlet type of boundary conditions on the oscillating part of the boundary. They proved a convergence theorem for the eigenvalues and eigenfunctions of a general 2m-order elliptic operator for a special type of the domains. Notice that convergence results were given in [24] as application of the method for the approximation of eigenvalues and eigenvectors of self-adjoint operators. In this paper, we consider spectral problems for the Laplace operator in a bounded domain, a part of which boundary, depending on a small parameter ε, is rapidly oscillating. Our aim is to construct accurate asymptotic approximations of the eigenvalues and corresponding eigenfunctions as ε 0. We consider the case where the frequency and the amplitude of the oscillations of the boundary are of the same order ε. Using the method of matching of asymptotic expansions (see [25]), we construct the leading terms of the asymptotic expansions for the eigenelements. Then, for a simple eigenvalue of the limit problem, we prove the asymptotic estimated of the difference between the solutions of the original problem and the approximate asymptotic expansions. The Neumann boundary-value problems were considered in [26] and also in [20, 21]. The outline of the paper is as follows: In Section 1, we introduce the notations, set the problem, and formulate the main result. In Section 2, we study the convergence of the solutions, and, in Section 3, we derive the formal asymptotics for the eigenelements and give a rigorous justification of the asymptotics. 1 The

text was submitted by the authors in English.

97

98

AMIRAT et al. Γ1

Ωε

x2

Γ2

Γ3

–1/2

1/2

x1

Γε Fig. 1. Membrane with oscillating boundary.

1. SETTING OF THE PROBLEM AND FORMULATION OF THE MAIN RESULT 2

We start with the construction of the domain. Denote by Ω a domain in
that lies in the upper half plane. We assume the boundary to be piecewise smooth, which consists of the parts ∂Ω = Γ0 ∪ Γ1 ∪ Γ2 ∪ Γ3, where Γ0 is the segment (–1/2, 1/2) on the abscissa axis, and Γ2 and Γ3 belong to the straight lines x1 = –1/2 and x1 = 1/2, respectively. It remains to equip the latter domain Ω with the oscillating boundary. Assume that ε = 1/(2
+ 1) is a small parameter, and
is a large positive number. Given a smooth negative 1-periodic in ξ1 even function F(ξ1) such that F '(ξ1) = 0 for ξ1 = ±1/2 and ξ1 = 0, we set x ⎧ ⎫ 2 Π ε = ⎨ x ∈
: ( x 1, 0 ) ∈ Γ 0, εF ⎛ -----1⎞ < x 2 ≤ 0 ⎬ ⎝ ε⎠ ⎩ ⎭ and, finally, define the desired domain as follows Ωε = Ω ∪ Πε (see Fig. 1). Thus, the boundary of Ωε consists of four parts: ∂Ωε = Γε ∪ Γ1 ∪ Γ2, ε ∪ Γ3, ε where x ⎫ ⎧ 2 Γ ε = ⎨ x ∈
: ( x 1, 0 ) ∈ Γ 0, x 2 = εF ⎛ -----1⎞ ⎬, ⎝ ε⎠ ⎩ ⎭ ⎧ ⎫ 2 1 1 Γ 2, ε = Γ 2 ∪ ⎨ x ∈
: x 1 = – ---, εF ⎛ – -----⎞ ≤ x 2 ≤ 0 ⎬, ⎝ 2ε⎠ 2 ⎩ ⎭ ⎧ ⎫ 1 1 2 Γ 3, ε = Γ 3 ∪ ⎨ x ∈
: x 1 = ---, εF ⎛ -----⎞ ≤ x 2 ≤ 0 ⎬. ⎝ 2ε⎠ 2 ⎩ ⎭ We study the following spectral problem ε

– ∆u ε = λ ε u ε in Ω , u ε = 0 on Γ ε ,

(1)

∂u ε /∂ν = 0 on Γ 1 ∪ Γ 2, ε ∪ Γ 3, ε ,

where ν is an outward unit normal to the boundary of Ωε. 2

2

Denote by Γ = {ξ ∈
: –1/2 < ξ1 < 1/2, ξ2 = F(ξ1)} in ξ = x/ε variables, and let Π = {ξ ∈
: –1/2 < ξ1 < 1/2, ξ2 > F(ξ1)} be a semi-infinite strip (see Fig. 2). In Sect. 3 we prove the following statement. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

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99

ξ2

–1/2

ξ1

1/2

Γ Fig. 2. Cell of periodicity.

Proposition 1. Problem ∆ ξ X = 0 in Π, X = 0 on Γ,

∂X/∂ξ 1 = 0 as ξ 1 = ± 1/2,

X ∼ ξ 2 as ξ 2

(2)

+∞

has a solution with the asymptotics X ( ξ ) = ξ 2 + C ( F ) as ξ 2

(3)

+∞,

up to exponentially small terms, where C(F) > 0 is a constant depending on the function F. The following theorem is the main result of this paper. Theorem 1. Assume that λ0 is a simple eigenvalue of the problem – ∆u 0 = λ 0 u 0 in Ω, u 0 = 0 on Γ 0 ,

(4)

∂u 0 /∂ν = 0 on Γ 1 ∪ Γ 2 ∪ Γ 3 ,

and u0 is the respective normalized in L2(Ω) eignenfunction. Then, the following are true: a) there exists a simple eigenvalue λε of problem (1) converging to λ0 as ε 0; b) the asymptotics of λε have the form λ ε = λ 0 + ελ 1 + o ( ε ),

(5)

∂u 2 λ 1 = – C ( F ) ⎛ --------0⎞ ds. ⎝ ∂ν ⎠

(6)

∫

Γ0

2. CONVERGENCE OF SOLUTIONS In this section, we study the behavior of solutions to the following boundary-value problem: ε

– ∆U ε = λU ε + F ε in Ω , U ε = 0 on Γ ε ,

(7)

∂U ε /∂ν = 0 on Γ 1 ∪ Γ 2, ε ∪ Γ 3, ε ,

as ε 0. We define solutions to problem (7) in a weak form (see [27], [28]). Denote by H1(D, S) the completion of the set of functions of the form C∞( D ) vanishing in a neighborhood of S in the norm H1(D), where COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

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S is a part of the boundary of the domain D. Suppose that Fε ∈ L2(Ωε). A function Uε ∈ H1(Ωε, Γε) is a solution to problem (7) if it satisfies the integral identity ( ∇U ε, ∇V ) L ( Ωε ) = λ ( U ε, V ) L ( Ωε ) + ( F ε, V ) L ( Ωε ) 2

2

(8)

2

for any V ∈ H1(Ωε, Γε). In an analogous way, we define a solution to the boundary-value problem – ∆U 0 = λU 0 + F 0 in Ω, U 0 = 0 on Γ 0 ,

(9)

∂U 0 /∂ν = 0 on Γ 1 ∪ Γ 2 ∪ Γ 3 .

Theorem 2. Assume that Fε ∈ L2(Ωε), F0 ∈ L2(Ω), and Q are arbitrary compact in the complex plane that does not contain eigenvalues of problem (4). Then, for λ ∈ Q, the following are true: a) Problem (7) has a unique solution for sufficiently small ε, and the solution satisfies the following uniform estimate: Uε

ε

1

H (Ω )

≤ C1 Fε

(10)

ε

L2 ( Ω )

on ε and λ. b) If U0 is a solution to problem (9) and Fε – F0

+ Fε

L2 ( Ω )

0, then ||Uε – U0 || H 1 ( Ω ) + ||Uε || H 1 ( Ωε \Ω )

as ε

(11)

0

ε

L 2 ( Ω \Ω )

0 uniformly on λ.

Proof. Note that, from (8), we evidently deduce the a priori estimate Uε

ε

1

H (Ω )

≤ C2 ( U ε

ε

L2 ( Ω )

+ Fε

ε

L2 ( Ω )

)

(12)

uniform on sufficiently small ε and λ ∈ Q. To prove item (a) of the theorem, it is sufficient to deduce the estimate (10). Suppose that this estimate ε

0, λ = λk ∈ Q, Fε = F εk ∈ L2( Ω k ), such that, for U εk , the in-

is false. Hence, there exist sequences εk equality U εk

εk

1

H (Ω )

≥ k F εk

(13)

εk

L2 ( Ω )

holds true. Without loss of generality, we assume that functions Uε are normalized in L2(Ωε): Uε

ε

L2 ( Ω )

(14)

= 1.

Substituting (14) and (13) in (12), we obtain uniform on ε and λ the estimate U εk

εk

1

H (Ω )

≤ C3 .

(15)

On the other hand, from (13) and (15), it follows that F εk

εk

L2 ( Ω )

k→∞

(16)

0.

˜ = Ω ∪ ((–1/2, 1/2) × [0, – ω )), Γ˜ 0 = {x : –1/2 < x < 1/2, x = – ω } Let δ > 0 be a fixed number. Denote Ω 1 2 ˜ 1 ε (see Fig. 3). It is clear that functions from H (Ω, Γ ) and from H (Ω , Γ ) extended by zero in Ω \Ω and in 0

1

ε

ε

˜ \Ω , respectively, belong to H1( Ω ˜ , Γ˜ 0 ). We keep the same notation for the continued functions. Under Ω this notation, the estimate (10) takes the form Uε

1

˜) H (Ω

≤ C1 Fε

ε

L2 ( Ω )

(17)

.

Hence, as a result of inequality (17), the compactness of Q, the weak compactness of a bounded set in a ˜ ) inL ( Ω ˜ ), we conclude that there exists a Hilbert space, and the compactness of the embedding of H1( Ω 2 COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

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ASYMPTOTICS OF SIMPLE EIGENVALUES Γ1

~ Ω

101

x2

Γ2

Γ3

–1/2

1/2

x1

~ Γ0 Fig. 3. Extended membrane.

subsequence εk' such that U εk'

k' → ∞

1 ˜ 1 ˜ ˜ ), U* ∈ H (Ω , Γ 0 ) weakly in H ( Ω ) and strongly in L 2 ( Ω

(18)

lim λ k' = λ * ∈ Q.

(19)

U∗ ≠ 0,

(20)

k' → ∞

Moreover, due to (14) and (18), we have

˜ \Ωε and the convergence (18) we get and by virtue of Uε = 0 outside Ω U * ∈ H ( Ω, Γ 0 ), 1

˜ \Ω. U * = 0 in Ω

(21)

Let V be an arbitrary function from H1(Ω, Γ0). Let us continue it by zero outside Ω. Obviously, this function belongs to H1(Ωε, Γε). Passing to the limit in (8) as ε = εk' 0 and λ εk' , keeping in mind (16)–(19), we obtain ( ∇U , ∇V ) L2 ( Ω ) = λ * ( U , V ) L2 ( Ω ) . * * Then, due to (20), (21) and the integral identity of problem (9), we conclude that U∗ is an eigenfunction and λ∗ is an eigenvalue of the limit problem that contradicts (19). Hence, we proved the estimate (10). Bearing in mind the estimate (10), we obtain the proof of item (b) by means of the same scheme. Rewrite the integral identity (8) in the form ( ∇U ε, ∇V ) L ( Ω˜ ) = λ ( U ε, V ) L ( Ω˜ ) + ( F ε, V ) L ( Ωε ) . 2

2

(22)

2

Let εk

0 be an arbitrary sequence. Due to the inequality (17), the weak compactness of a bounded set ˜ ) in L ( Ω ˜ ), we deduce that there exists in a Hilbert space, and the compactness of the embedding of H1( Ω 2 a subsequence εk' such that the convergence (18) and (21) holds. Passing to the limit as εk' 0 in (22) for V ∈ H1(Ω, Γ0) by means of (18), (21), and (11), we obtain that U∗ = U0 is a solution of the boundary value problem (9). Due to the uniqueness of the solution to the boundary value problem (9) and the arbitrariness of the sequence εk 0, we deduce that Uε

ε→0

1 ˜ ˜ ). U 0 weakly in H ( Ω ) and strongly in L 2 ( Ω

On the other hand, passing to the limit in (22) as ε

(23)

0 for V = Uε and keeping in mind (23) and (11), we

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conclude that ∇U ε

2 ˜) ε→0 L2 ( Ω

∇U 0

2 ˜ ). L2 ( Ω

(24)

From (23) and (24), it follows that Uε – U0

2

0.

1 ˜ H (Ω ) ε→0

Finally, we proved item (b) of the theorem. The theorem is proved. Lemma 1. Assume that the multiplicity of the eigenvalue λ0 of problem (4) is equal to p. Then, (i) there are p eigenvalues of problem (1) (with multiplicities taken into account) converging to λ0 as ε 0; (ii) for any λ close to λ0, the solution Uε to problem (7) admits the estimate Uε

H

1

F ε L ( Ωε ) 2 -------------------------,  ≤ p (Ω )

(25)

ε

∏

j λε

–λ

j=1

where

1 λε ,

p λε

…,

are the eigenvalues of problem (1) that converge to λ0;

(iii) if a solution Uε to problem (7) is orthogonal in L2(Ωε) to the eigenfunction u ε of problem (1) correk

sponding to λ ε , then it satisfies the estimate k

Uε

H

1

F ε L ( Ωε ) 2 ----------------------------------.  ≤ p (Ω )

(26)

ε

∏

j λε

–λ

j = 1; j ≠ k

Due to Theorem 2, the proof repeats the proofs of the analogous propositions in [29], [30] (see also [31] and [32]) for various types of singular perturbations. Remark 1. Using the methods of [6], [7], and [24], it is possible to prove item (i) of Theorem 2, i.e., to obtain the structure of the limit (homogenized) problem. However, for the justification of the asymptotics, we also need to prove items (ii) and (iii), which cannot be retrieved by the mentioned methods. 3. CONSTRUCTION OF THE ASYMPTOTICS In this section, we use the method of matching of asymptotic expansions (see [25], [33–35], and also [36–38]). We construct the asymptotics inside the domain (external expansion) in the form uε(x) = u0(x) + … and also the series for the eigenvalues as follows: λ ε = λ 0 + ….

(27)

The function u0 is not defined in the neighborhood of Γε. Then, we introduce the internal expansion in the neighborhood of Γε, and we shall use cut-off functions for both expansions to construct the expansion in the whole domain Ωε. Consider the Taylor series for u0 with respect to x2 as x2 0. By virtue of problem (1), we conclude that u 0 ( x ) = α 0 ( x 1 )x 2 + O ( x 2 ),

(28)

α '0 ( ± 1/2 ) = 0.

(29)

3

where α0(x1) = ∂u0/∂ x 2

x2 = 0

and

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Changing the variables ξ2 = x2/ε, we find from (28) that u 0 ( x 1, εξ 2 ) = εα 0 ( x 1 )ξ 2 + O ( ε ξ 2 ). 3 3

(30)

The leading term of the internal expansion satisfies the boundary conditions of problem (1) on Γε and has the asymptotics at the infinity (as ξ2 +∞) as in (30) (this is the main idea of the method of matching asymptotic expansions, see [25]). Consequently, the internal expansion has the structure u ε ( x ) = εv 1 ( ξ; x 1 ) + …,

(31)

where ξ = x/ε, v 1 ( ξ; x 1 ) ∼ α 0 ( x 1 )ξ 2 as ξ 2

(32)

+∞,

and x1 plays the role of the “slow variable”. In the variables (ξ, x1), the Laplacian and the normal derivative operator have the form ∂ ∂ –2 –1 ∆ = ε ∆ ξ + 2ε ----------------- + --------2 , ∂x 1 ∂ξ 1 ∂x 1 2

∂ ∂ –1 ∂ ------ = ε -------- + -------- on Γ 3 , ∂ξ 1 ∂x 1 ∂ν

2

(33)

∂ ∂ –1 ∂ ------ = – ε -------- – -------- on Γ 2 . ∂ν ∂ξ 1 ∂x 1

(34)

Remark 2. Further, we construct the coefficients of the internal expansion (31) in the form of 1-periodic in ξ1 functions (depending on the fixed “slow” parameter x1). Substituting (27), (31), (33), and (34) in (1) and collecting terms with minimal orders of ε (with ε–1 for the equation and with ε0 for the boundary conditions), keeping in mind the last remark, we obtain the following boundary value problem: ∆ ξ v 1 = 0 in Π, v 1 = 0 on Γ,

∂v 1 /∂ξ 1 = 0 as ξ 1 = ± 1/2,

v 1 ∼ α 0 ( x 1 )ξ 2 as ξ 2

(35)

+∞.

Proof of Proposition 1. Consider the problem ∆ ξ Y = 0 in Π,

Y = – ξ 2 on Γ,

∂Y 1 -------- = 0 as ξ 1 = ± --- . ∂ξ 1 2

(36)

Arguing as in [2], where periodic boundary conditions are imposed on the lateral sides of Π (see also [39]), one can show that problem (36) admits a unique solution with ∇Y ∈ (L2(Π))2. Moreover, the mean of Y over a horizontal cross section (for ξ2 > 0) is constant, thus, denoting 1/2

C( F) =

∫ Y ( ξ , ξ ) dξ , 1

2

1

ξ 2 > 0.

– 1/2

There exists a positive constant κ > 0 such that ∀δ > 0, ∀α ∈ 2 α

∂ ( Y ( ξ 1, ξ 2 ) – C ( F ) ) ≤ C δ, α e

– κξ 2

for any (ξ1, ξ2) ∈ Π with ξ2 > δ, where Cδ, α is a constant depending only on δ and α. Setting X(ξ) = Y(ξ) + ξ2, we obtain the statement for some C(F). Let us prove that the constant C(F) is positive. We first note that the function Y defined by (36) is positive on Γ. Let us prove that C(F) is nonnegative by contradiction. Assume the opposite, then there exists a curve F1 ˜ with F as connecting the lateral sides of the cell Π, where Y = 0. From the uniqueness of the solution in Π 1 ˜ a bottom side, it follows that Y ≡ 0 in Π and we obtain the contradiction. 2

Let us prove that C(F) ≠ 0. Assume that C(F) = 0 and denote Πµ = {ξ ∈
: –1/2 < ξ1 < 1/2, ξ2 > F + µ, µ > 0} with sufficiently small µ, such that F + µ is also negative. We can easily verify that, in this semiinfinite strip, the analogue of the function Y has negative asymptotics as ξ2 +∞, which is equal to –µ. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

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This function is evidently equal to Z(ξ) = Y(ξ1, ξ2 – µ) – µ. In fact, Z(ξ1, F(ξ1) + µ) = Y(ξ1, F(ξ1) + µ – µ) – µ = Y(ξ1, F(ξ1)) – µ = –F(ξ1) – µ; and this is equal to –ξ2 on the bottom side F + µ. We obtained the contradiction. The proposition is proved. Note that, due to the evenness of the function F, the solution X is even in ξ1 and can be extended to a 1-periodic function: ξ1. Later on, we use the same notation X for the extension. Remark 3. The existence of periodic in (n – 1) variables solutions and their behavior at infinity in n-dimensional semi-space are studied, for instance, in [40]. See also [42]. Thus, v 1 ( ξ; x 1 ) = α 0 ( x 1 ) X ( ξ )

(37)

where X is a solution of (2) and, due to (3), has the asymptotics v 1 ( ξ; x 1 ) = α 0 ( x 1 ) ( ξ 2 + C ( F ) ) as ξ 2

(38)

+∞,

up to exponentially small terms. Note that due to (29), (37) ∂v 1 1 --------- ( ξ; x 1 ) = 0 as x 1 = ± --- , 2 ∂x 1

(39)

and, hence, by (34), (39) and the boundary condition from (2), we have ∂v x ---------1 ⎛ --; x 1⎞ = 0 on Γ 2, ε ∪ Γ 3, ε . ∂ν ⎝ ε ⎠

(40)

Constructing the internal expansion, we obtain a new discrepancy in the asymptotics at infinity: εC(F)α0(x1) (see (31), (38)). Introducing a new term in the external expansion of order O(ε), we eliminate this discrepancy. Rewriting the asymptotics εv1 as ξ2 +∞ in the external variables x by (38) we obtain that the external expansion does have the form u ε ( x ) = u 0 ( x ) + εu 1 ( x ) + …,

(41)

where, u 1 ( x ) ∼ C ( F )α 0 ( x 1 ) as x 2

(42)

0,

because of the smoothness of u1, it is equivalent to the following boundary conditions: u 1 ( x 1, 0 ) = C ( F )α 0 ( x 1 ).

(43)

Thus, we obtain the boundary conditions for u1 on Γ0. Since the corrected external expansion is constructed in the form of (41), acting in an analogous way as for problems with a rapidly changing type of boundary conditions [36–38], we consider the corrected asymptotics of the eigenvalue in a similar form λ ε = λ 0 + ελ 1 + ….

(44)

Substituting the series (41), (44) in problem (1), collecting the terms with ε1, and bearing in mind (43), we deduce the boundary value problem for u1: – ∆u 1 = λ 0 u 1 + λ 1 u 0 in Ω, u 1 = C ( F )α 0 on Γ 0 ,

(45)

∂u 1 /∂ν = 0 on Γ 1 ∪ Γ 2 ∪ Γ 3 .

The constant λ1 can be defined from the solvability condition of problem (45). Multiplying the equation in (45) by u0 and keeping in mind its normalization in L2(Ω), we deduce that

∫

∫

– ∆u 1 u 0 d x = λ 0 u 1 u 0 d x + λ 1 . Ω

(46)

Ω

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Applying the Green’s formula two times to the left-hand side of this equation, we obtain

∫

– ∆u 1 u 0 d x = Ω

∫

= – u 1 ∆u 0 d x – Ω

∫

∂Ω

∂u 1

- u ds ∫ ( ∇u , ∇u ) dx – ∫ ------∂ν 1

0

0

Ω

∂u --------1 u 0 ds + ∂ν

∫

∂Ω

∂Ω

∫

Ω

(47)

1/2

∂u 0 -------- u 1 ds = λ 0 u 1 u 0 d x – C ( F ) ∂ν

∫

2 α0 ( x 1 ) dx 1 .

– 1/2

We used here that on Γ0 ∂u ∂u --------0 = – --------0 = – α 0 ( x 1 ). ∂ν ∂x 2 Substituting (47) in (46), we obtain (6) or 1/2

λ1 = –C ( F )

∫ α ( x ) dx . 2 0

1

(48)

1

– 1/2

To uniquely determine the solution of problem (45), we assume that

∫ u ( x )u ( x ) dx = 0. 0

1

Ω

Thus, using the method of matching asymptotic expansions, we obtained the first term of the perturbation theory for the eigenvalue of the spectral problem (1). However, to correctly verify the constructed term, we need to continue the construction of the formal asymptotics of the eigenfunction. 0. Since u1(x) is sufficiently Let us continue the process. Consider the Taylor series of u1(x) as x2 smooth, then ∂u u 1 ( x ) = u 1 ( x 1, 0 ) + --------1 ( x 1, 0 )x 2 + … ∂x 2 or, taking into account (43), u 1 ( x ) = C ( F )α 0 ( x 1 ) + α 1 ( x 1 )x 2 + …,

(49)

∂u where α1(x1) = --------1 (x1, 0). Note that, due to the smoothness of the function u1 and the homogeneity of the ∂x 2 Neumann boundary conditions on Γ2 and Γ3 for u1, we have 1 α '1 ⎛ ± ---⎞ = 0. ⎝ 2⎠

(50)

Constructing the external expansion, we obtain a new discrepancy in the asymptotics at zero: ε2α1(x1) (see (41), (49)). Introducing a new term in the internal expansion of order O(ε2), we eliminate this discrepancy. 0) with the substitution x2 = εξ2, by virtue of (41), Namely, rewriting the asymptotics u0 + εu1 (as x2 (28), and (49), we deduce that the specified internal expansion has the form u ε ( x ) = εv 1 ( ξ; x 1 ) + ε v 2 ( ξ; x 1 ) + …, 2

(51)

where v 2 ( ξ; x 1 ) ∼ α 1 ( x 1 )ξ 2 as ξ 2

(52)

+∞.

Substituting (51), (44), (33), and (34) in (1) and collecting the terms with ε0 for the equation and with ε for the boundary conditions, keeping in mind Remark 2 and formula (39), we obtain the following boundary COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

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value problem: ∂ v1 - in Π, – ∆ ξ v 2 = 2 ---------------∂x 1 ∂ξ 1 2

v 2 = 0 on Γ,

∂v 1 ---------2 = 0 as ξ 1 = ± --- . 2 ∂ξ 1

(53)

Let us consider an auxiliary problem ∂X ∆ ξ X˜ = -------- in Π, ∂ξ 1

X˜ = 0 on ∂Π.

(54)

Proposition 2. Problem (54) has a solution with the asymptotics X˜ ( ξ ) = 0 as ξ 2

(55)

+∞,

up to exponentially small terms. The proof hereafter is inspired from the method used in [41, pp. 49–58] for a problem in a semi-infinite strip with a flat bottom with periodic boundary conditions on the lateral part of the domain. Proof of Proposition 2. According to the proof of Proposition 1, there exists a positive constant κ > 0 such that ∀δ > 0, ∀α ∈ 2 – κξ α ∂X ∂ ⎛ --------⎞ ( ξ 1, ξ 2 ) ≤ C δ, α e 2 ⎝ ∂ξ 1⎠

(56)

for any (ξ1, ξ2) ∈ Π with ξ2 ≥ δ, where Cδ, α is a constant depending only on δ and α. Let us introduce the Hilbert space ⎧ ⎫ κξ κξ ∂v V κ = ⎨ v : e 2 v ∈ L 2 ( Π ), e 2 -------- ∈ L 2 ( Π ) for j = 1, 2, v = 0 on ∂Π ⎬, ∂ξ j ⎩ ⎭ with the scalar product ( v , w )κ =

∫e

2κξ 2

∇v ⋅ ∇wdξ.

Π

We consider the bilinear form aκ on Vκ × Vκ aκ ( v , w ) =

∫ ∇v ⋅ ∇ ( e

2κξ 2

w )dξ for v , w ∈ V κ ,

Π

and the linear form Lκ on Vκ ∂X 2κξ L κ ( v ) = – -------- e 2 v dξ for v ∈ V κ . ∂ξ 1

∫

Π

Note here that, due to (56), ∂X/∂ξ1 ∈ Vκ and then Lκ is well defined. We extend any v ∈ Vκ by 0 outside Π , and we use the same notation v for the extension. For v ∈ Vκ and ξ2 ∈
, we have the Poincaré inequality 1/2

∫

1/2

1 v ( ξ 1, ξ 2 ) dξ 1 ≤ --2 2

– 1/2

∫ – 1/2

2 ∂v -------- ( ξ 1, ξ 2 ) dξ 1 . ∂ξ 1

(57)

Now, for any v ∈ Vκ, aκ ( v , v ) =

∫e

Π

2κξ 2

∫

∇v dξ + 2κ e 2

Π

2κξ 2 ∂v

-------- v dξ, ∂ξ 2

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and since

∫

2 e Π

2κξ 2

2κξ ∂v 2 2κξ ∂v 2 -------- v dξ ≤ e 2 ------dξ + e 2 v dξ, ∂ξ 2 ∂ξ 2

∫

∫

Π

Π

using (57), it follows that

∫

aκ ( v , v ) ≥ ( 1 – κ ) e

2κξ 2

∇v dξ. 2

Π

Thus, for κ < 1 (which we may suppose), the bilinear form aκ is coercive on Vκ. Using (57), one can also show that aκ is continuous on Vκ × Vκ and Lκ is continuous on Vκ. Then, by the Lax–Milgram theorem, we obtain the existence of a unique solution X˜ ∈ Vκ of the variational equation a κ ( X˜ , v ) = L κ ( v ) ∀v ∈ V κ , from which it follows that X˜ is a weak solution of problem (54). Since X˜ ∈ Vκ, we deduce that X˜ decays exponentially fast in the Dirichlet integral; i.e., for any δ > 0, there is a constant Cδ such that

∫ ∇ X˜ Π

2

dξ ≤ C δ e

– 2κδ

,

δ

where Πδ = (–1/2, 1/2) × (δ, +∞). Then, using the local regularizing properties of the Laplace operator and the Sobolev embedding theorem (see, for instance, [27], [28]), we deduce that ∀δ > 0, ∀α ∈ 2, – κξ α ∂ X˜ ( ξ 1, ξ 2 ) ≤ C δ, α e 2

for any (ξ1, ξ2) ∈ Π with ξ2 ≥ δ, where Cδ, α is another constant depending only on δ and α. The proposition is proved. Note that, due to the evenness of the functions F, the solution X˜ of problem (54) is odd in ξ1, and thus has a 1-periodic in ξ extension for which we keep the same notation X˜ . 1

Then, it is easy to see that, due to (2), (3), (29), (50), (54), and (55), the function v 2 ( ξ; x 1 ) = α 1 ( x 1 ) X ( ξ ) – 2α 0' ( x 1 ) X˜ ( ξ )

(58)

is the 1-periodic solution to problem (53) that has the asymptotics v 2 ( ξ; x 1 ) = α 1 ( x 1 )ξ 2 + C ( F )α 1 ( x 1 ) as ξ 2

(59)

+∞

up to exponentially small terms and also satisfies condition (52). It is easy to verify that, due to (50), (58) and the boundary condition from (54), ∂v 2 x 1 --------- ( ξ; x 1 ) = 0 as ξ = -- and x 1 = ± --- . ε 2 ∂x 1

(60)

Hence, by (34), (60) and the boundary condition from (53), we have ∂v x ---------2 ⎛ --; x 1⎞ = 0 on Γ 2, ε ∪ Γ 3, ε . ∂ν ⎝ ε ⎠

(61)

Denote by χ(s) a smooth cut-off function that is equal to zero when s < 1 and equal to one when s > 2. x Define χ˜ t (x2) = χ ⎛ -----2⎞ , which equals zero when x2 < t and equals one when x2 > 2t, where t > 0 is a suffi⎝ t⎠ ciently large fixed number such that (2t)–1 < min{|Γ2|, |Γ3|}. Denote λ˜ ε = λ 0 + ελ 1 , COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

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AMIRAT et al.

and x x x 2 2 u˜ ε ( x ) = [ u 0 ( x ) + εu 1 ( x ) + ε u 2 ( x ) ]χ ⎛ ----β-2⎞ + εv 1 ⎛ --; x 1⎞ + ε v 2 ⎛ --; x 1⎞ ⎝ε ⎠ ⎝ε ⎠ ⎝ε ⎠

x 1 – χ ⎛ ----β-2⎞ , ⎝ε ⎠

(63)

where β is a fixed number (0 < β < 1), and u 2 ( x ) = C ( F )α 1 ( x 1 ) [ 1 – χ˜ t ( x 2 ) ].

(64)

u 2 ( x ) = C ( F )α 1 ( x 1 ) as x 2

(65)

It is clear that 0.

Note that, due to (64) and the boundary conditions from (4), (45) for u0, u1 and the boundary conditions (40), (61) for v1, v2, we obtain that the function u˜ ε , which is defined by (63), satisfies the boundary conditions of problem (1): ∂u˜ ε -------- = 0 on Γ 1 ∪ Γ 2, ε ∪ Γ 3, ε . ∂ν

u˜ ε = 0 on Γ ε ,

(66)

On the other hand, due to the definition of u˜ ε , formula (33), and equations (4), (45), (35), and (53) for u0, u1, v1, and v2, we have ε – ∆u˜ ε = λ˜ ε u˜ ε + f ε in Ω ,

where x x x ⎧ x ⎫ –β – 2β f ε ( x ) = – ⎨ χ ⎛ ----β-2⎞ I 1 ( x; ε ) + 1 – χ ⎛ ----β-2⎞ I 2 ( x; ε ) + 2ε χ' ⎛ ----β-2⎞ I 3 ( x; ε ) + ε χ'' ⎛ ----β-2⎞ I 4 ( x; ε ) ⎬, ⎝ε ⎠ ⎝ε ⎠ ⎝ε ⎠ ⎝ε ⎠ ⎩ ⎭ I 1 = ε ( λ 1 u 1 + λ 0 u 2 + ελ 1 u 2 + ∆u 2 ), 2

2 2 2 ⎛ ∂ v2 ∂ v ⎞ ∂ v 2 - + ε ----------2-2⎟ , I 2 = ε ⎜ λ 0 v 1 + ελ 0 v 2 + ελ 1 v 1 + ε λ 1 v 2 + ----------2-1 + 2 ---------------∂x 1 ∂ξ 1 ∂x 1 ⎠ ∂x 1 ⎝

∂u ∂v ∂v ∂u 2 ∂u I 3 = --------0 + ε --------1 + ε --------2 – ⎛ ---------1 + ε ---------2⎞ , ∂x 2 ∂x 2 ⎝ ∂ξ 2 ∂ξ 2 ⎠ ∂x 2 I 4 = u 0 + εu 1 + ε u 2 – ( εv 1 + ε v 2 ). 2

2

Since the functions ui are smooth, it is obvious that x χ ⎛ ----β-2⎞ I 1 ⎝ε ⎠

2

= O ( ε ). 4

ε

L2 ( Ω )

(67)

Estimating the second term of fε, due to (37), (38), (52), and (58), we obtain that x 1 – χ ⎛ ----β-2⎞ I 2 ⎝ε ⎠

2

= O ( ε ). 3β

ε

L2 ( Ω )

(68)

Bearing in mind the matching conditions (28), (38), (49), (59), (65) and that the derivatives of χ(x2/εβ) are not equal to zero only in the strip εβ < x2 < 2εβ, it is easy to see that x –β ε χ' ⎛ ----β-2⎞ I 3 ⎝ε ⎠

2 ε

L2 ( Ω )

+ ε

– 2β

x χ'' ⎛ ----β-2⎞ I 4 ⎝ε ⎠

2

= O ( ε ). 3β

ε

L2 ( Ω )

(69)

From (67), (68), and (69) it follows that, for 2/3 < β < 1, we have fε

ε

L2 ( Ω )

= o ( ε ).

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Thus, we obtained the formal asymptotics of the simple eigenvalue of the spectral problem (1) with a discrepancy of larger order than the first constructed term of the perturbation theory for λε. Finally, note that from (63) and (64), we derive u˜ ε

ε

L2 ( Ω )

= 1 + o ( 1 ) as ε

(71)

0.

Applying Lemma 1 for λ = λ˜ ε , Fε = fε, and Uε = u˜ ε , by virtue of (25), we obtain f ε L ( Ωε ) 2 -. λ ε – λ˜ ε ≤  -------------------u˜ ε L ( Ωε )

(72)

2

From (70), (71), and (72), it follows that λ ε = λ˜ ε + o ( ε ). Thus, the formal asymptotics of the eigenvalues constructed in the previous section are verified. 4. CONCLUDING REMARKS Continuing the algorithm introduced in Section 3, one can construct the complete asymptotic expansion of the eigenvalues λε and the respective eigenfunctions uε. It should be noted that an analogous problem was considered in [42]. ACKNOWLEDGMENTS The idea to write this paper was inspired by the stay of G.A. Chechkin at Blaise Pascal University (Clermont-Ferrand, France) in the spring of 2003. He wants to express his many thanks for the hospitality he received and for the wonderful working conditions and the support he received. The final version of the paper was completed at Bashkir State Pedagogical University (Ufa, Russia), whose support is also acknowledged, in May 2005. The work of G.A. Chechkin was partially supported by the program “Leading Scientific Schools” (project no. IO-1464.2003.1) and by the program “Universities of Russia” (project no. UR. 04.01.010). The work of R.R. Gadyl’shin was partially supported by the Russian Foundation for Basic Research (project no. 05-01-97912_p_agidel) and the program “Leading Scientific Schools” (project no. IO-1446.2003.1). REFERENCES 1. Y. Achdou, O. Pironneau, and F. Valentin, “Effective Boundary Conditions for Laminar Flows over Rough Boundaries,” J. Comput. Phys. 147, 187–218 (1998). 2. Y. Amirat and J. Simon, “Riblets and Drag Minimization,” Optimization Methods in PDE’s: Contemporary Mathematics (Am. Math. Soc., 1997), pp. 9–17. 3. Y. Amirat, D. Bresch, J. Lemoine, et. al, “Effect of Rugosity on a Flow Governed by Navier–Stokes Equations,” Q. Appl. Math. 59, 769–785 (2001). 4. I. Babu¡ka and R. Vyborny, “Continuous Dependence of Eigenvalues on the Domains,” Czech. Math. J. 15, 169– 178 (1965). 5. A. G. Belyaev, “Average of the Third Boundary-Value Problem for the Poisson Equation in a Domain with Rapidly Oscillating Boundary,” Vestn. Mosk. Gos. Univ., Ser. 1: Math. Mech. 6, 63–66 (1988). 6. A. G. Belyaev, Dissertation in Mathematics and Physics (Moscow State Univ, Moscow, 1990). 7. A. G. Belyaev, A. L. Piatnitski, and G. A. Chechkin, “Asymptotic Behavior of a Solution to a Boundary-Value Problem in a Perforated Domain with Oscillating Boundary,” Sib. Math. J. 39, 621–644 (1998). 8. G. Bouchitte, A. Lidouh, and P. Suquet, “Homogénéisation de frontière pour la modélisation du contact entre un corps déformable non linéaire un corps rigide,” C. R. Acad. Sci., Ser. I 313, 967–972 (1991). 9. R. Brizzi and J. P. Chalot, “Homogénéisation de frontière,” Ric. Mat. 46, 341–387 (1997). 10. G. A. Chechkin and T. P. Chechkina, “On Homogenization Problems in Domains of the “Infusorium” Type,” J. Math. Sci. 120, 1470–1482 (2004). 11. G. A. Chechkin and T. P. Chechkina, “Homogenization Theorem for Problems in Domains of the “Infusorian” Type with Uncoordinated Structure,” J. Math. Sci. 123, 4363–4380 (2004). COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

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