- Email: [email protected]
Asymptotics for eigenvalues of the Laplacian with a Neumann boundary condition on a thin cut-out tube
C. R. Mecanique 331 (2003) 531–536
Asymptotics for eigenvalues of the Laplacian with a Neumann boundary condition on a thin cut-out tube Marina Yu. Planida The Bashkir State Pedagogical University, October Revolution st., 3a, 450000 Ufa, Russia Received 26 May 2003; accepted 6 June 2003 Presented by Évariste Sanchez-Palencia
Abstract We consider the eigenvalue problem for the Laplace operator in a bounded three-dimensional domain where a thin tube is cut out. Imposing a Neumann boundary condition on the boundary of this tube, we construct asymptotics for eigenvalues on the small parameter that is a diameter of the tube. To cite this article: M.Yu. Planida, C. R. Mecanique 331 (2003). 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Résumé Comportement asymptotique des valeurs propres du laplacien avec conditions de Neumann sur un tube extrait. On considère l’opérateur de Laplace dans un domaine tridimensionnel borné dont on a extrait un tube fin, avec la condition aux limites de Neumann. Nous construisons le développement asymptotique des valeurs propres pour des valeurs petites du diamètre du tube. Pour citer cet article : M.Yu. Planida, C. R. Mecanique 331 (2003). 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: Vibrations; Asymptotique; Valeur propre Mots-clés : Vibrations ; Asymptotics ; Eigenvalue
1. Introduction We consider the eigenvalue problem for the Laplace operator in a bounded domain and impose a Neumann boundary condition on a boundary of a thin cut-out tube. The asymptotics for an eigenvalue of this singular perturbed problem is constructed under the assumption the tube shrinks to a closed curve. Let x = (x1 , x2 , x3 ), Ω ⊂ R3 be a bounded simply-connected domain having an infinitely differentiable boundary Γ, γ ∈ C ∞ be a closed curve with no self-intersection lying in the plane x3 = 0, γ ⊂ Ω. In a vicinity of γ we introduce coordinates (y, s), y = (y1 , y2 ), where s is a natural parameter (the arc) of the curve γ , y1 = x3 , y2 is a distance to γ in the plane x3 = 0 measured along the inward normal to the curve γ considered as a boundary of two-dimensional domain in the plane x3 = 0. By ω we denote a simply-connected domain in R2 having E-mail address: [email protected] (M.Yu. Planida). 1631-0721/$ – see front matter 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved. doi:10.1016/S1631-0721(03)00125-6
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M.Yu. Planida / C. R. Mecanique 331 (2003) 531–536
Fig. 1.
Fig. 2.
smooth boundary and containing the origin. We set ωε = {y ∈ R2 : ε−1 y ∈ ω}, γε = {x ∈ R3 : s ∈ γ , y ∈ ωε }, Ωε = Ω \ γε , 0 < ε 1 (cf. Figs. 1, 2). In [1] it was proved that if λ0 is a simple eigenvalue of the boundary value problem ∂ψ0 −ψ0 = λ0 ψ0 as x ∈ Ω, σ1 (1) + σ2 ψ0 = 0 as x ∈ Γ ∂τ where τ denotes inward normal to ∂Ωε , (σ1 , σ2 ) = (1, 0) or (σ1 , σ2 ) = (0, 1), then there exists a unique eigenvalue λε of the problem ∂ψε ∂ψε = 0 on ∂γε , + σ2 ψε = 0 on Γ −ψε = λε ψε in Ωε , σ1 (2) ∂τ ∂τ converging to λ0 as ε → 0, and this perturbed eigenvalue is simple. In the present paper using the method of matched asymptotics expansions [2–4], the asymptotics for this λε are constructed as ε → 0. We notice that for particular case when the curve is a circle the asymptotics for the solution to the Poisson equation was constructed in [5].
2. Construction of asymptotics Since the function ψ0 does not meet the needed boundary condition on γε , we employ the method of matched asymptotics expansions in order to construct the asymptotics of ψε in a vicinity of γε . These asymptotics are constructed in terms of ‘inner’ variables ξ = yε−1 . For small y the Laplace operator rewritten to (y, s) becomes ∞ ∞ ∞ ∂ ∂ ∂2 (3) = y − t (s) + di (s)y2i + 1+ hi−1 (s)y2i−1 − pi (s)y2i ∂y2 ∂s 2 ∂s i=1
i=2
i=1
where y is the two-dimensional Laplace operator with respect to variables y, t, and bi , hq , pm ∈ C ∞ (γ ); |t| is the curvature of the curve γ , while eigenfunction ψ0 is expanded into a series (4) ψ0 x(y; s) = P0 (s) + P1 (y; s) + P2 (y; s) + O r 3 , r → 0 2 2 i+j c20 y1 + 2c11 y1 y2 + c02 y2 ∂ ψ0 , cij (s) = (5) P0 = c00 , P1 = c10 y1 + c01 y2 , P2 = j 2 ∂y1i ∂y2 y=0 where r = |y|. Moreover, by (1) and (3), c20 + c02 = tc01 − c00 − λ0 c00
(6)
In accordance with the method of matched asymptotics expansions [2], we rewrite (4) in the variables ξ and see that asymptotics for the eigenfunction ψε in a vicinity of γε should be sought as
M.Yu. Planida / C. R. Mecanique 331 (2003) 531–536
533
ψε = v0 + εv1 + ε2 v2 + · · ·
(7)
vi (ξ ; s) ∼ Pi (ξ ; s),
(8)
ρ →∞
where ρ = |ξ |. Substituting (3), (7) into (2), and substituting in the equality obtained for the variables ξ , we get the boundary value problem for v0 : ∂v0 ξ v0 = 0, ξ ∈ R2 \ ω, = 0, ξ ∈ ∂ω (9) ∂τ It is clear that the function v0 (ξ ; s) ≡ P0 (s)
(10)
satisfies (9) and (8). Substituting (3), (7) in (2) and writing out the problems for v1 , v2 , we see that ξ v1 = 0,
ξ ∈ R2 \ ω,
∂v1 = 0, ∂τ
ξ ∈ ∂ω
(11)
∂v1 ∂ 2 v0 ∂v2 = 0, ξ ∈ ∂ω (12) − − λ0 v0 , ξ ∈ R2 \ ω, 2 ∂ξ2 ∂τ ∂s Here we also bear in mind that v0 is independent of ξ . Now we are going to study the solvability of the problems (11), (12) and find the asymptotics of v1 , v2 as ρ → ∞. We start from the following obvious statement. ξ v2 = t
Lemma 2.1. Let Zn (ξ ; s) be an arbitrary harmonic polynomial of n-th degree with respect to the variables ξ whose coefficients depend on s. Then there exists a harmonic function V (ξ ; Zn (ξ ; s)) defined in R2 \ ω satisfying the homogeneous Neumann condition on ∂ω and the asymptotics (13) V ξ ; Zn (ξ ; s) = Zn (ξ ; s) + a(s) sin ϕ + b(s) cos ϕ ρ −1 + O ρ −2 as ρ → ∞. Here ϕ is polar angle. Corollary 2.2. (a) There exists, harmonic in R2 \ ω, functions V (ξ ; ξ1 ), V (ξ ; ξ2 ) satisfying the homogeneous Neumann condition on ∂ω and asymptotics V (ξ ; ξi ) = ξi +
2 ∂ ln ρ 1 mij + O ρ −2 , 2π ∂ξj
ρ →∞
(14)
j =1
(b) The matrix M(ω) = (mij )i,j =1,2 is positive defined, symmetric and determined by the domain ω. Item (a) is implied by Lemma 2.2 while the validity of item (b) can be established by analogy with [3,7]. Remark 1. We note that when ω is a unit disk, we have V (ξ ; ξ1 ) = (ρ + ρ −1 ) sin ϕ, V (ξ ; ξ2 ) = (ρ + ρ −1 ) cos ϕ and, therefore, the assertion (14) yields that M = 2πE, where E is a unit matrix. Due to Corollary 2.2 the function v1 (ξ ; s) = c10 (s)V (ξ ; ξ1 ) + c01 (s)V (ξ ; ξ2 ) is a solution of the boundary value problem (11) and has the asymptotics v1 (ξ ; s) = P1 (ξ ; s) + A(s) sin ϕ + B(s) cos ϕ ρ −1 + O ρ −2
(15)
(16)
as ρ → ∞. Thus, it satisfies (8), moreover, by (15), (14), A=
1 (c10 m12 + c01 m22 ), 2π
B=
1 (c10 m11 + c01 m21 ) 2π
(17)
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M.Yu. Planida / C. R. Mecanique 331 (2003) 531–536
Let us proceed to the problem (12). We seek v2 in the form: t ρ 2 v2 = ξ2 v1 − (c00 + λ0 c00 ) + v˜2 2 4 By (18), (12), (11), (10), (8), (5) and (6), the boundary value problem for v˜2 can be written as follows: t ∂ξ2 1 ∂ρ 2 ∂ v˜2 = − v1 + (c + λ0 c00 ), ξ ∈ R2 \ ω, ∂τ 2 ∂τ 4 ∂τ 00 2 v˜2 (ξ ; s) = Z2 (ξ ; s) + o ρ , ρ → ∞
ξ v˜2 = 0,
In turn, v˜2 is constructed as v˜2 (ξ ; s) = V ξ ; Z2 (ξ ; s) + vˆ2 (ξ ; s),
vˆ2 (ξ ; s) = o(ρ),
ξ ∈ ∂ω
ρ→∞
(18)
(19)
(20)
It follows from (20) and (19) that the boundary value problem for vˆ2 has the form t ∂ξ2 1 ∂ρ 2 ∂ vˆ2 = − v1 + (c + λ0 c00 ), ξ ∈ ∂ω ∂τ 2 ∂τ 4 ∂τ 00 It is known that there exists the solution to the problem (21) having the asymptotics ∂ vˆ2 1 dlξ vˆ2 (ξ ; s) = G(s) ln ρ + O(1), G(s) = 2π ∂τ ξ vˆ2 = 0,
ξ ∈ R2 \ ω,
(21)
(22)
∂ω
as ρ → ∞. Calculating G(s) by the boundary condition in (21), (15) and employing (14), we derive that t (s) 1 (c10 m12 + c01 m22 ) + |ω|(c00 + λ0 c00 ) (23) 4π 2π It follows from (18), (20), (22) that the boundary value problem (12) has a solution v2 satisfying the asymptotics G(s) = −
v2 (ξ ; s) = P2 (ξ ; s) + G(s) ln ρ + O(1),
ρ→∞
(24)
and, therefore, this solution satisfies (8). Remark 2. Observe that rewriting asymptotics of the function ε2 v2 (as ρ → ∞) in the variables x is the origin of the term (−ε2 ln εG(s)). To eliminate this term, in the inner expansion we introduce an additional term ε2 ln εv2,1 obeying the asymptotics: v2,1 (ξ ; s) = G(s) + o(1)
(25)
Thus, the leading terms of the asymptotics ψε should be constructed as ψε ≈ v0 + εv1 + ε2 ln εv2,1 + ε2 v2
(26)
Now we substitute (26), (3) into (2), pass to the variables ξ in the equality obtained and equate the coefficient of ε2 ln ε. This procedure gives the following boundary value problem: ∂v2,1 = 0, ξ ∈ ∂ω (27) ∂τ It is obvious that v2,1 (ξ ; s) ≡ G(s) is a solution of the boundary value problem (27) and has the asymptotics (25). Replacing vi by their asymptotics at infinity (10), (16), (24), (25) in (26) and rewriting these asymptotics in the variable y, in accordance with method of matched asymptotics expansions, we see that the asymptotics of the eigenfunction outside a neighbourhood of γε and the asymptotics of the eigenvalue should be constructed as follows: ξ v2,1 = 0,
ξ ∈ R2 \ ω,
M.Yu. Planida / C. R. Mecanique 331 (2003) 531–536
ψε (x) = ψ0 (x) + ε2 ψ2 (x) + · · · ψ2 (x) = r −1 A(s) sin ϕ + B(s) cos ϕ + G(s) ln r + O(1),
535
(28) r →0
(29)
λε = λ0 + ε λ2 + · · · 2
(30)
Substituting (30) and (28) into (2), we arrive at the boundary value problem for ψ2 : ∂ψ2 + σ2 ψ2 = 0, ∂τ By analogy with [4] one can prove the following statement: −ψ2 = λ0 ψ2 + λ2 ψ0 ,
x ∈ Ω \ γ,
σ1
x ∈Γ
(31)
Lemma 2.3. Let a, b, g ∈ C ∞ (γ ), ψ0 L2 (Ω) = 1. Then there exist functions Ψ2 (x; g), Ψ2 (x; a, b) ∈ C ∞ (Ω \ γ ), having the asymptotics (0) (2) Ψ2 x; g(s) (x) ∼ g(s) ln r, Ψ2 x; a(s), b(s) ∼ r −1 a(s) sin ϕ + b(s) cos ϕ (0)
(2)
(2) as r → 0 and being the solutions of the boundary value problem (31) for λ2 = Λ(0) 2 and λ2 = Λ2 , respectively, where (0) (2) Λ2 = 2π c00 g ds, Λ2 = −2π (c10 b + c01 a) ds γ
γ
It follows from Lemma 2.3 that the function ψ2 = Ψ2(2) (x; A, B) + Ψ2(0) (x; G), where the quantities A, B, G are defined by the equalities (17), (23), has the asymptotics (29) and is a solution to the boundary value problem (31) as ∂ψ0 2 1 λ2 = |ω| λ0 ψ02 − ds − ∇y ψ0 M(ω) ∇y ψ0 + t (s)ψ0 e2 ds (32) ∂s 2 γ
γ
where e2 = (0, 1). Thus, the eigenvalue of the problem (2) has the asymptotics (30), (32). Rigorous justification of this asymptotics can be carried by analogy with [4].
3. Conclusing remarks We note that if we impose Dirichlet boundary conditions on the tube’s boundary, the variational properties of the eigenvalues say that λε − λ0 > 0. At the same time, for the problem considered in this paper, the quantity λε − λ0 has no definite sign. Indeed, assume that SR1 is a disk of radius R1 > 1 with center at the origin, Ω = SR1 × (−π/2, π/2), γ ⊂ Ω ∩ {x3 = 0} is a unit circle with center at the origin, (R, Θ, x3 ) are cylindrical coordinates associated with x. It is known that a simple eigenvalue of the problem (1) for (σ1 , σ2 ) = (0, 1) (i.e., of the Dirichlet problem) is given by λ0 = 8 2 + (νj /R1 )2 , and the associated eigenfunction normalized in L2 (Ω) is determined by the equality ψ0 (x) =
2 I0 (νj R/R1 ) X(x3 ) πR1 I0 (νj )
(33)
where X(x3 ) = sin(8x3 ) for even 8; X(x3 ) = cos(8x3 ) for odd 8; νj are zeroes for Bessel function I0 (ν) of zero order. Since γ is a circle, it follows from (33) that ∂ψ0 /∂s = 0 on γ (and, therefore, the second term in the first integral in (32) disappears). Assume, in addition, that ω is a unit circle. Therefore, M = 2πE (see Remark 1).
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M.Yu. Planida / C. R. Mecanique 331 (2003) 531–536
Let us consider the case 8 = 2, and νj /R1 is not a zero of the Bessel function I0 (ν). Clearly, there exists such an R1 . Then ψ0 = ∂ψ0 /∂s = 0 on γ and it follows from the formula (32) that: λ2 = −2π∇y ψ0 2L2 (γ ) < 0
(34)
Now assume that 8 = 1 and νs /R1 is a zero of first derivative of the Bessel function I0 (ν) (clearly, for s large enough such an R1 does exist). Then ∇y ψ0 = 0 on γ and it follows from (32) that: λ2 = 2πλ0 ψ0 2L2 (γ ) > 0
(35)
Therefore, the assertions (30), (34) and (35) imply that in the first case λε − λ0 < 0, while in the second λε − λ0 > 0. We note that similar phenomena appears in two- and three-dimensional boundary value problems for the Laplace operator in domains with a small cavity, when the Neumann condition is imposed on the boundary of the cavity while the cavity shrinks to a point [6,7]. In conclusion we also note that in the case of the Robin condition on Γ , the convergence of the eigenvalues and the estimates of the inverse operator needed for a rigorous justification of the asymptotics can be proved exactly by analogy with [1] (in this paper they considered Dirichlet and Neumann boundary conditions on Γ ). The construction of the asymptotics λε does not differ from that above.
Acknowledgements The author thanks R.R. Gadyl’shin and D.I. Borisov for attention to this work. The author is supported by RFBR (02-01-00693), by the program ‘Leading Scientific School’ (NSh-1446.2003.1) and by the program ‘Universities of Russia’ (UR.04.01.010).
References [1] M.Yu. Planida, On the convergence of solutions of singularly perturbed boundary-value problems for the Laplace operator, Math. Notes 71 (2002) 867–877. [2] A.M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, American Mathematical Society, Providence, RI, 1992. [3] R.R. Gadyl’shin, Ramification of a multiple eigenvalue of the Dirichlet problem for the Laplacian under singular perturbation of the boundary condition, Math. Notes 52 (1992) 1020–1029. [4] M.Yu. Planida, Asymptotics of eigenvalues for a cylinder insulated on a narrow strip, Comput. Math. Math. Phys. 43 (2003) 403–413. [5] S.A. Nazarov, M.V. Paukshto, Discrete models and homogenization in problems of elastisity theory, Lenigrad University, Leningrad, 1984 (in Russian). [6] S. Ozawa, Spectra of domains spherical Neumann boundary, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983) 259–277. [7] V.G. Maz’ya, S.A. Nazarov, B.A. Plamenevskij, Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes, Math. USSR-Izv. 24 (1985) 321–345.
Asymptotics for eigenvalues of the Laplacian with a Neumann boundary condition on a thin cut-out tube Marina Yu. Planida The Bashkir State Pedagogical University, October Revolution st., 3a, 450000 Ufa, Russia Received 26 May 2003; accepted 6 June 2003 Presented by Évariste Sanchez-Palencia
Abstract We consider the eigenvalue problem for the Laplace operator in a bounded three-dimensional domain where a thin tube is cut out. Imposing a Neumann boundary condition on the boundary of this tube, we construct asymptotics for eigenvalues on the small parameter that is a diameter of the tube. To cite this article: M.Yu. Planida, C. R. Mecanique 331 (2003). 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Résumé Comportement asymptotique des valeurs propres du laplacien avec conditions de Neumann sur un tube extrait. On considère l’opérateur de Laplace dans un domaine tridimensionnel borné dont on a extrait un tube fin, avec la condition aux limites de Neumann. Nous construisons le développement asymptotique des valeurs propres pour des valeurs petites du diamètre du tube. Pour citer cet article : M.Yu. Planida, C. R. Mecanique 331 (2003). 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: Vibrations; Asymptotique; Valeur propre Mots-clés : Vibrations ; Asymptotics ; Eigenvalue
1. Introduction We consider the eigenvalue problem for the Laplace operator in a bounded domain and impose a Neumann boundary condition on a boundary of a thin cut-out tube. The asymptotics for an eigenvalue of this singular perturbed problem is constructed under the assumption the tube shrinks to a closed curve. Let x = (x1 , x2 , x3 ), Ω ⊂ R3 be a bounded simply-connected domain having an infinitely differentiable boundary Γ, γ ∈ C ∞ be a closed curve with no self-intersection lying in the plane x3 = 0, γ ⊂ Ω. In a vicinity of γ we introduce coordinates (y, s), y = (y1 , y2 ), where s is a natural parameter (the arc) of the curve γ , y1 = x3 , y2 is a distance to γ in the plane x3 = 0 measured along the inward normal to the curve γ considered as a boundary of two-dimensional domain in the plane x3 = 0. By ω we denote a simply-connected domain in R2 having E-mail address: [email protected] (M.Yu. Planida). 1631-0721/$ – see front matter 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved. doi:10.1016/S1631-0721(03)00125-6
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M.Yu. Planida / C. R. Mecanique 331 (2003) 531–536
Fig. 1.
Fig. 2.
smooth boundary and containing the origin. We set ωε = {y ∈ R2 : ε−1 y ∈ ω}, γε = {x ∈ R3 : s ∈ γ , y ∈ ωε }, Ωε = Ω \ γε , 0 < ε 1 (cf. Figs. 1, 2). In [1] it was proved that if λ0 is a simple eigenvalue of the boundary value problem ∂ψ0 −ψ0 = λ0 ψ0 as x ∈ Ω, σ1 (1) + σ2 ψ0 = 0 as x ∈ Γ ∂τ where τ denotes inward normal to ∂Ωε , (σ1 , σ2 ) = (1, 0) or (σ1 , σ2 ) = (0, 1), then there exists a unique eigenvalue λε of the problem ∂ψε ∂ψε = 0 on ∂γε , + σ2 ψε = 0 on Γ −ψε = λε ψε in Ωε , σ1 (2) ∂τ ∂τ converging to λ0 as ε → 0, and this perturbed eigenvalue is simple. In the present paper using the method of matched asymptotics expansions [2–4], the asymptotics for this λε are constructed as ε → 0. We notice that for particular case when the curve is a circle the asymptotics for the solution to the Poisson equation was constructed in [5].
2. Construction of asymptotics Since the function ψ0 does not meet the needed boundary condition on γε , we employ the method of matched asymptotics expansions in order to construct the asymptotics of ψε in a vicinity of γε . These asymptotics are constructed in terms of ‘inner’ variables ξ = yε−1 . For small y the Laplace operator rewritten to (y, s) becomes ∞ ∞ ∞ ∂ ∂ ∂2 (3) = y − t (s) + di (s)y2i + 1+ hi−1 (s)y2i−1 − pi (s)y2i ∂y2 ∂s 2 ∂s i=1
i=2
i=1
where y is the two-dimensional Laplace operator with respect to variables y, t, and bi , hq , pm ∈ C ∞ (γ ); |t| is the curvature of the curve γ , while eigenfunction ψ0 is expanded into a series (4) ψ0 x(y; s) = P0 (s) + P1 (y; s) + P2 (y; s) + O r 3 , r → 0 2 2 i+j c20 y1 + 2c11 y1 y2 + c02 y2 ∂ ψ0 , cij (s) = (5) P0 = c00 , P1 = c10 y1 + c01 y2 , P2 = j 2 ∂y1i ∂y2 y=0 where r = |y|. Moreover, by (1) and (3), c20 + c02 = tc01 − c00 − λ0 c00
(6)
In accordance with the method of matched asymptotics expansions [2], we rewrite (4) in the variables ξ and see that asymptotics for the eigenfunction ψε in a vicinity of γε should be sought as
M.Yu. Planida / C. R. Mecanique 331 (2003) 531–536
533
ψε = v0 + εv1 + ε2 v2 + · · ·
(7)
vi (ξ ; s) ∼ Pi (ξ ; s),
(8)
ρ →∞
where ρ = |ξ |. Substituting (3), (7) into (2), and substituting in the equality obtained for the variables ξ , we get the boundary value problem for v0 : ∂v0 ξ v0 = 0, ξ ∈ R2 \ ω, = 0, ξ ∈ ∂ω (9) ∂τ It is clear that the function v0 (ξ ; s) ≡ P0 (s)
(10)
satisfies (9) and (8). Substituting (3), (7) in (2) and writing out the problems for v1 , v2 , we see that ξ v1 = 0,
ξ ∈ R2 \ ω,
∂v1 = 0, ∂τ
ξ ∈ ∂ω
(11)
∂v1 ∂ 2 v0 ∂v2 = 0, ξ ∈ ∂ω (12) − − λ0 v0 , ξ ∈ R2 \ ω, 2 ∂ξ2 ∂τ ∂s Here we also bear in mind that v0 is independent of ξ . Now we are going to study the solvability of the problems (11), (12) and find the asymptotics of v1 , v2 as ρ → ∞. We start from the following obvious statement. ξ v2 = t
Lemma 2.1. Let Zn (ξ ; s) be an arbitrary harmonic polynomial of n-th degree with respect to the variables ξ whose coefficients depend on s. Then there exists a harmonic function V (ξ ; Zn (ξ ; s)) defined in R2 \ ω satisfying the homogeneous Neumann condition on ∂ω and the asymptotics (13) V ξ ; Zn (ξ ; s) = Zn (ξ ; s) + a(s) sin ϕ + b(s) cos ϕ ρ −1 + O ρ −2 as ρ → ∞. Here ϕ is polar angle. Corollary 2.2. (a) There exists, harmonic in R2 \ ω, functions V (ξ ; ξ1 ), V (ξ ; ξ2 ) satisfying the homogeneous Neumann condition on ∂ω and asymptotics V (ξ ; ξi ) = ξi +
2 ∂ ln ρ 1 mij + O ρ −2 , 2π ∂ξj
ρ →∞
(14)
j =1
(b) The matrix M(ω) = (mij )i,j =1,2 is positive defined, symmetric and determined by the domain ω. Item (a) is implied by Lemma 2.2 while the validity of item (b) can be established by analogy with [3,7]. Remark 1. We note that when ω is a unit disk, we have V (ξ ; ξ1 ) = (ρ + ρ −1 ) sin ϕ, V (ξ ; ξ2 ) = (ρ + ρ −1 ) cos ϕ and, therefore, the assertion (14) yields that M = 2πE, where E is a unit matrix. Due to Corollary 2.2 the function v1 (ξ ; s) = c10 (s)V (ξ ; ξ1 ) + c01 (s)V (ξ ; ξ2 ) is a solution of the boundary value problem (11) and has the asymptotics v1 (ξ ; s) = P1 (ξ ; s) + A(s) sin ϕ + B(s) cos ϕ ρ −1 + O ρ −2
(15)
(16)
as ρ → ∞. Thus, it satisfies (8), moreover, by (15), (14), A=
1 (c10 m12 + c01 m22 ), 2π
B=
1 (c10 m11 + c01 m21 ) 2π
(17)
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M.Yu. Planida / C. R. Mecanique 331 (2003) 531–536
Let us proceed to the problem (12). We seek v2 in the form: t ρ 2 v2 = ξ2 v1 − (c00 + λ0 c00 ) + v˜2 2 4 By (18), (12), (11), (10), (8), (5) and (6), the boundary value problem for v˜2 can be written as follows: t ∂ξ2 1 ∂ρ 2 ∂ v˜2 = − v1 + (c + λ0 c00 ), ξ ∈ R2 \ ω, ∂τ 2 ∂τ 4 ∂τ 00 2 v˜2 (ξ ; s) = Z2 (ξ ; s) + o ρ , ρ → ∞
ξ v˜2 = 0,
In turn, v˜2 is constructed as v˜2 (ξ ; s) = V ξ ; Z2 (ξ ; s) + vˆ2 (ξ ; s),
vˆ2 (ξ ; s) = o(ρ),
ξ ∈ ∂ω
ρ→∞
(18)
(19)
(20)
It follows from (20) and (19) that the boundary value problem for vˆ2 has the form t ∂ξ2 1 ∂ρ 2 ∂ vˆ2 = − v1 + (c + λ0 c00 ), ξ ∈ ∂ω ∂τ 2 ∂τ 4 ∂τ 00 It is known that there exists the solution to the problem (21) having the asymptotics ∂ vˆ2 1 dlξ vˆ2 (ξ ; s) = G(s) ln ρ + O(1), G(s) = 2π ∂τ ξ vˆ2 = 0,
ξ ∈ R2 \ ω,
(21)
(22)
∂ω
as ρ → ∞. Calculating G(s) by the boundary condition in (21), (15) and employing (14), we derive that t (s) 1 (c10 m12 + c01 m22 ) + |ω|(c00 + λ0 c00 ) (23) 4π 2π It follows from (18), (20), (22) that the boundary value problem (12) has a solution v2 satisfying the asymptotics G(s) = −
v2 (ξ ; s) = P2 (ξ ; s) + G(s) ln ρ + O(1),
ρ→∞
(24)
and, therefore, this solution satisfies (8). Remark 2. Observe that rewriting asymptotics of the function ε2 v2 (as ρ → ∞) in the variables x is the origin of the term (−ε2 ln εG(s)). To eliminate this term, in the inner expansion we introduce an additional term ε2 ln εv2,1 obeying the asymptotics: v2,1 (ξ ; s) = G(s) + o(1)
(25)
Thus, the leading terms of the asymptotics ψε should be constructed as ψε ≈ v0 + εv1 + ε2 ln εv2,1 + ε2 v2
(26)
Now we substitute (26), (3) into (2), pass to the variables ξ in the equality obtained and equate the coefficient of ε2 ln ε. This procedure gives the following boundary value problem: ∂v2,1 = 0, ξ ∈ ∂ω (27) ∂τ It is obvious that v2,1 (ξ ; s) ≡ G(s) is a solution of the boundary value problem (27) and has the asymptotics (25). Replacing vi by their asymptotics at infinity (10), (16), (24), (25) in (26) and rewriting these asymptotics in the variable y, in accordance with method of matched asymptotics expansions, we see that the asymptotics of the eigenfunction outside a neighbourhood of γε and the asymptotics of the eigenvalue should be constructed as follows: ξ v2,1 = 0,
ξ ∈ R2 \ ω,
M.Yu. Planida / C. R. Mecanique 331 (2003) 531–536
ψε (x) = ψ0 (x) + ε2 ψ2 (x) + · · · ψ2 (x) = r −1 A(s) sin ϕ + B(s) cos ϕ + G(s) ln r + O(1),
535
(28) r →0
(29)
λε = λ0 + ε λ2 + · · · 2
(30)
Substituting (30) and (28) into (2), we arrive at the boundary value problem for ψ2 : ∂ψ2 + σ2 ψ2 = 0, ∂τ By analogy with [4] one can prove the following statement: −ψ2 = λ0 ψ2 + λ2 ψ0 ,
x ∈ Ω \ γ,
σ1
x ∈Γ
(31)
Lemma 2.3. Let a, b, g ∈ C ∞ (γ ), ψ0 L2 (Ω) = 1. Then there exist functions Ψ2 (x; g), Ψ2 (x; a, b) ∈ C ∞ (Ω \ γ ), having the asymptotics (0) (2) Ψ2 x; g(s) (x) ∼ g(s) ln r, Ψ2 x; a(s), b(s) ∼ r −1 a(s) sin ϕ + b(s) cos ϕ (0)
(2)
(2) as r → 0 and being the solutions of the boundary value problem (31) for λ2 = Λ(0) 2 and λ2 = Λ2 , respectively, where (0) (2) Λ2 = 2π c00 g ds, Λ2 = −2π (c10 b + c01 a) ds γ
γ
It follows from Lemma 2.3 that the function ψ2 = Ψ2(2) (x; A, B) + Ψ2(0) (x; G), where the quantities A, B, G are defined by the equalities (17), (23), has the asymptotics (29) and is a solution to the boundary value problem (31) as ∂ψ0 2 1 λ2 = |ω| λ0 ψ02 − ds − ∇y ψ0 M(ω) ∇y ψ0 + t (s)ψ0 e2 ds (32) ∂s 2 γ
γ
where e2 = (0, 1). Thus, the eigenvalue of the problem (2) has the asymptotics (30), (32). Rigorous justification of this asymptotics can be carried by analogy with [4].
3. Conclusing remarks We note that if we impose Dirichlet boundary conditions on the tube’s boundary, the variational properties of the eigenvalues say that λε − λ0 > 0. At the same time, for the problem considered in this paper, the quantity λε − λ0 has no definite sign. Indeed, assume that SR1 is a disk of radius R1 > 1 with center at the origin, Ω = SR1 × (−π/2, π/2), γ ⊂ Ω ∩ {x3 = 0} is a unit circle with center at the origin, (R, Θ, x3 ) are cylindrical coordinates associated with x. It is known that a simple eigenvalue of the problem (1) for (σ1 , σ2 ) = (0, 1) (i.e., of the Dirichlet problem) is given by λ0 = 8 2 + (νj /R1 )2 , and the associated eigenfunction normalized in L2 (Ω) is determined by the equality ψ0 (x) =
2 I0 (νj R/R1 ) X(x3 ) πR1 I0 (νj )
(33)
where X(x3 ) = sin(8x3 ) for even 8; X(x3 ) = cos(8x3 ) for odd 8; νj are zeroes for Bessel function I0 (ν) of zero order. Since γ is a circle, it follows from (33) that ∂ψ0 /∂s = 0 on γ (and, therefore, the second term in the first integral in (32) disappears). Assume, in addition, that ω is a unit circle. Therefore, M = 2πE (see Remark 1).
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M.Yu. Planida / C. R. Mecanique 331 (2003) 531–536
Let us consider the case 8 = 2, and νj /R1 is not a zero of the Bessel function I0 (ν). Clearly, there exists such an R1 . Then ψ0 = ∂ψ0 /∂s = 0 on γ and it follows from the formula (32) that: λ2 = −2π∇y ψ0 2L2 (γ ) < 0
(34)
Now assume that 8 = 1 and νs /R1 is a zero of first derivative of the Bessel function I0 (ν) (clearly, for s large enough such an R1 does exist). Then ∇y ψ0 = 0 on γ and it follows from (32) that: λ2 = 2πλ0 ψ0 2L2 (γ ) > 0
(35)
Therefore, the assertions (30), (34) and (35) imply that in the first case λε − λ0 < 0, while in the second λε − λ0 > 0. We note that similar phenomena appears in two- and three-dimensional boundary value problems for the Laplace operator in domains with a small cavity, when the Neumann condition is imposed on the boundary of the cavity while the cavity shrinks to a point [6,7]. In conclusion we also note that in the case of the Robin condition on Γ , the convergence of the eigenvalues and the estimates of the inverse operator needed for a rigorous justification of the asymptotics can be proved exactly by analogy with [1] (in this paper they considered Dirichlet and Neumann boundary conditions on Γ ). The construction of the asymptotics λε does not differ from that above.
Acknowledgements The author thanks R.R. Gadyl’shin and D.I. Borisov for attention to this work. The author is supported by RFBR (02-01-00693), by the program ‘Leading Scientific School’ (NSh-1446.2003.1) and by the program ‘Universities of Russia’ (UR.04.01.010).
References [1] M.Yu. Planida, On the convergence of solutions of singularly perturbed boundary-value problems for the Laplace operator, Math. Notes 71 (2002) 867–877. [2] A.M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, American Mathematical Society, Providence, RI, 1992. [3] R.R. Gadyl’shin, Ramification of a multiple eigenvalue of the Dirichlet problem for the Laplacian under singular perturbation of the boundary condition, Math. Notes 52 (1992) 1020–1029. [4] M.Yu. Planida, Asymptotics of eigenvalues for a cylinder insulated on a narrow strip, Comput. Math. Math. Phys. 43 (2003) 403–413. [5] S.A. Nazarov, M.V. Paukshto, Discrete models and homogenization in problems of elastisity theory, Lenigrad University, Leningrad, 1984 (in Russian). [6] S. Ozawa, Spectra of domains spherical Neumann boundary, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983) 259–277. [7] V.G. Maz’ya, S.A. Nazarov, B.A. Plamenevskij, Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes, Math. USSR-Izv. 24 (1985) 321–345.