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International Journal of Differential Equations
Volume 2011 (2011), Article ID 619623, 22 pages
http://dx.doi.org/10.1155/2011/619623
Research Article

On Spectrum of the Laplacian in a Circle Perforated along the Boundary: Application to a Friedrichs-Type Inequality

1Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow Lomonosov State University, Moscow 119991, Russia
2Narvik University College, Postboks 385, 8505 Narvik, Norway
3Department of Engineering Science and Mathematics, Luleå University of Technology, 971 87 Luleå, Sweden

Received 24 May 2011; Accepted 30 August 2011

Academic Editor: Mayer Humi

Copyright © 2011 G. A. Chechkin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we construct and verify the asymptotic expansion for the spectrum of a boundary-value problem in a unit circle periodically perforated along the boundary. It is assumed that the size of perforation and the distance to the boundary of the circle are of the same smallness. As an application of the obtained results, the asymptotic behavior of the best constant in a Friedrichs-type inequality is investigated.

1. Introduction

We study a two-dimensional eigenvalue problem for the Laplace operator in a unit circle periodically perforated along the boundary. It is assumed that the size of perforation and the distance to the boundary of the circle are of the same smallness. The asymptotic behavior of the spectrum of the considered boundary-value problem is investigated in this paper. We construct and verify the asymptotic expansion for the eigenvalues with respect to the small parameter describing the microinhomogeneous structure of the domain. A similar problem was considered in [1] for the case of perforation located along the plane part of the boundary. The case studied in this paper is much more complicated since the eigenvalues of multiplicity more than one can appear. The technique for asymptotic analysis of such kind of problem can be found, for example, in [2, 3].

The obtained results are used for asymptotic expansion of the best constant in a Friedrichs-type inequality for functions from the space 𝐻1, vanishing on the boundary of the perforation and satisfying homogeneous Neuman condition on the boundary of the circle. Analogous questions concerning the asymptotic behavior of the best constant in Friedrichs-type inequality in domains having microinhomogeneous structure in a neighborhood of the boundary were studied in [1, 411]. In the remaining part of this introduction, we will give a short description of some of the most important results in these papers to put the results obtained in this paper into a more general frame.

In paper [4], the authors proved a Friedrichs-type inequality for functions, having zero trace on the small periodically alternating pieces of the boundary of a two-dimensional domain. The total measure of the set, where the function vanishes, tends to zero. It turns out that for this case the constant in the Friedrichs-type inequality is bounded. Moreover, the precise asymptotics of the constant in the derived Friedrichs-type inequality is described as the small parameter characterizing the microinhomogeneous structure of the boundary, tends to zero.

Paper [5] is devoted to the asymptotic analysis of functions depending on the small parameter, which characterizes the microinhomogeneous structure of the domain where the functions are defined. The authors considered a boundary-value problem in a two-dimensional domain perforated nonperiodically along the boundary in the case when the diameter of circles and the distance between them have the same order. In particular, it was proved that the Dirichlet problem is the limit for the original problem. Moreover, some numerical simulations were used to illustrate the results. As an application, a Friedrichs-type inequality was derived for functions vanishing on the boundary of the cavities. It was proved that the constant in the obtained inequality is close to the constant in the inequality for functions from 𝐻1. The three-dimensional case of the same problem is considered in [8].

In paper [9], the author considered a three-dimensional domain, which is aperiodically perforated along the boundary in the case when the diameter of the holes and the distance between them have the same order. A Friedrichs-type inequality was derived for functions from the space 𝐻1 vanishing on the boundaries of cavities. In particular, it was shown that the constant in the derived inequality tends to the constant of the classical inequality for functions from 𝐻1 when the small parameter describing the size of perforation tends to zero.

Paper [1] (see also [7]) deals with the construction of the asymptotic expansion for the first eigenvalue of a boundary-value problem for the Laplacian in a perforated domain. This asymptotics gives an asymptotic expansion for the best constant in a corresponding Friedrichs-type inequality.

Paper [11], is devoted to the Friedrichs-type inequality, where the domain is periodically and rarely perforated along the boundary. It is assumed that the functions satisfy homogeneous Neumann boundary conditions on the outer boundary and that they vanish on the perforation. In particular, it is proved that the best constant in the inequality converges to the best constant in a Friedrichs-type inequality as the size of the perforation goes to zero much faster than the period of perforation. The limit Friedrichs-type inequality is valid for functions in the Sobolev space 𝐻1.

Some generalizations of Friedrichs-type inequalities are Hardy-type inequalities. There exist several books devoted to this topic, see [1216]. The first attempts to generalize the classical results concerning Hardy-type inequalities in fixed domains to domains with microinhomogeneous structure one can find in [6, 10].

Paper [6] deals with a three-dimensional weighted Hardy-type inequality in the case when the domain Ω is bounded and has nontrivial microstructure. It is assumed that the small holes are distributed periodically along the boundary. The main result is the validity of a weighted Hardy-type inequality for the class of functions from the Sobolev space 𝐻1 having zero trace on the small holes under the assumption that a weight function decreases to zero in a neighborhood of the microinhomogenity on the boundary.

In paper [10], the author derived a new two-dimensional weighted Hardy-type inequality in a rectangle for the class of functions from the Sobolev space 𝐻1 vanishing on small alternating pieces of the boundary. The dependence of the best constant in the derived inequality on the small parameter describing the size of microinhomogenity was established.

This paper is organized as follows: in Section 2 we give all necessary definitions and state the spectral problem. Section 3 is devoted to the construction of the leading terms of asymptotic expansion, while the complete expansions for the simple and multiple eigenvalues are constructed in Sections 4 and 5, respectively. The verification of the constructed asymptotics is given in Section 6. Finally, in Section 7, the obtained results are applied to describe the asymptotic behavior for the best constant in a Friederichs-type inequality considered in a perforated domain.

2. Preliminaries

Consider a unit circle Ω centered at the origin. We introduce the polar system of coordinates (𝜃,𝑟) in Ω. Introduce a small parameter 𝜀=2/𝑁,𝑁1, and consider the open set 𝐵𝜀 which is the union of small sets periodically distributed along the boundary. Each of these small sets can be obtained from the neighboring one by rotation about the origin through the angle 𝜀𝜋. Finally, we define Ω𝜀=Ω𝐵𝜀 and 𝜕𝐵𝜀=Γ𝜀, see Figure 1. Let us describe the geometry of 𝐵𝜀 in details. Consider the semi-strip:𝜋Π=𝜉2<𝜉1<𝜋2,𝜉2𝜋>0,Γ=𝜉2<𝜉1<𝜋2,𝜉2.=0(2.1) Let 𝐵 be an arbitrary two-dimensional open domain with a smooth boundary that is symmetric the vertical axis and lies in a disk of a fixed radius 𝑎<1 centered at the point (0,1), see Figure 2. Let 𝐵𝑎 be the union of the 𝜋-integer translations of 𝐵 along the axis 𝜉1. Then we define 𝐵𝜀 as the image of 𝐵𝑎 under the mapping 𝜃=𝜀𝜉1,𝑟=1𝜀𝜉2.

619623.fig.001
Figure 1: Perforated circle.
619623.fig.002
Figure 2: Cell of periodicity.

Consider the following spectral problem:Δ𝑢𝜀=𝜆𝜀𝑢𝜀inΩ𝜀,𝑢𝜀=0onΓ𝜀,𝜕𝑢𝜀𝜕𝑟=0on𝜕Ω.(2.2)

The problem,Δ𝑢0=𝜆0𝑢0𝑢inΩ,0=0on𝜕Ω,(2.3) is the limit one for (2.2). This fact can be established analogously as in [17, 18], by using the same technique.

Remark 2.1. In particular, it can be proved that the number of eigenvalues (bearing in mind the multiplicities) of the original problem converging to the eigenvalue of the limit (homogenized) problem is equal to the multiplicity of the mentioned eigenvalue of the limit problem (for the method of proof see, e.g., [19]).

Remark 2.2. The limit spectral problem (2.3) is studied very well. In particular, if the eigenvalue 𝜆0 is simple, then the corresponding eigenfrequency 𝑘0=𝜆0 of (2.3) is the zero-point of the Bessel-function 𝒥0, and the corresponding eigenfunction has the form 𝒥0(𝑘0𝑟). One can find the definition of Bessel-functions, for example, in [20, Section 4.7].

The goal of this paper is to construct and verify the asymptotic expansion for the eigenvalues of (2.2). The obtained asymptotics is used for studying the behavior of the best constant in a Friedrichs-type inequality for functions belonging to the Sobolev class 𝐻1(Ω𝜀,Γ𝜀) (see the definition of 𝐻1(Ω𝜀,Γ𝜀) in Section 7). One of the main results of this paper is the following asymptotics for 𝜆𝜀 converging to 𝜆0:𝜆𝜀=𝜆0+𝑖=1𝜀𝑖𝜆𝑖,(2.4) where 𝜆𝑖 are some fixed constants which can be calculated according to (4.23) and (4.15) in the case of simple 𝜆𝜀 and according to (5.10) and (4.15) when 𝜆𝜀 is of multiplicity two. In particular, 𝜆1<0 which implies that 𝜆𝜀<𝜆0.

3. Construction of the Leading Terms of the Asymptotic Expansion

Suppose that 𝜆0 is the simple eigenvalue for (2.3) and the corresponding eigenfunction 𝑢0 is normalized in 𝐿2(Ω). Our aim is to construct the leading terms of the asymptotic expansions for 𝜆𝜀 converging to 𝜆0 as well as 𝑢𝜀 converging to 𝑢0. We use the method of boundary-layer functions (see [21]) for this purpose. We are looking for eigenvalues and eigenfunctions in the following form:𝜆𝜀=𝜆0+𝜀𝜆1𝑢+,𝜀(𝑥)=𝑢0(𝑥)+𝜀𝑢1(𝑥)+𝜀𝛼0(𝜃)𝑣(𝜉)+,(3.1) where 𝜉=(𝜉1,𝜉2),𝜉1=𝜃/𝜀,𝜉2=(1𝑟)/𝜀, and𝑢0(𝑥)=𝛼0(𝜃)(1𝑟)+𝑂(1𝑟)2as𝑟1,𝛼0(𝜃)=𝜕𝑢0||||𝜕𝑟𝑟=1,𝑢1(𝑥)=𝑢1𝑟=1+𝛼1(𝜃)(1𝑟)+𝑂(1𝑟)2as𝑟1,𝛼1(𝜃)=𝜕𝑢1||||𝜕𝑟𝑟=1.(3.2)

Substituting the first expansion from (3.1) and the sum 𝑢0+𝜀𝑢1 from the second expansion in (2.2) and equating terms at the same power of 𝜀, we get the equation for 𝑢1: Δ𝑥𝑢1=𝜆0𝑢1+𝜆1𝑢0inΩ.(3.3) The existence of the solution for (3.3) is given in the following proposition.

Proposition 3.1. For any 𝜆1, there exists the smooth solution of (3.3) satisfying the boundary condition 𝑢1=𝜆1𝛼0(𝜃)02𝜋𝛼20𝑑𝜃1on𝜕Ω.(3.4)

Proof. The existence of the smooth solution follows from the classical results on regular solutions of elliptic equations (see e.g., [22]). In order to get 𝑢1 as the unique solution, one can add the condition of mutual orthogonality: Ω𝑢0𝑢1𝑑𝑥=0.(3.5)
By multiplying (3.3) by 𝑢0, integrating (3.3) over Ω, and twice integrating by parts the obtained equation, we find that Ω𝑢1Δ𝑢0𝑑𝑥𝜕Ω𝑢1𝜕𝑢0𝜕𝑟𝑑𝜃+𝜕Ω𝑢0𝜕𝑢1𝜕𝑟𝑑𝜃=𝜆1Ω𝑢20𝑑𝑥+𝜆0Ω𝑢1𝑢0𝑑𝑥.(3.6) Taking into account the fact that 𝑢0 is the normalized (in 𝐿2(Ω)) solution of (2.3) and since 𝑢1 satisfies (3.5), we can deduce that 𝜆1=𝜕Ω𝜕𝑢0𝑢𝜕𝑟1𝑑𝜃=𝜕Ω𝛼0(𝜃)𝑢1𝑑𝜃.(3.7) Then (3.7) leads to (3.4) and the proof is complete.

However, the approximation 𝑢0+𝜀𝑢1 does not satisfy the condition on Γ𝜀. This forces us to introduce an additional term 𝛼0𝑣 in second expansion of (3.1) to satisfy the appropriate boundary condition. We assume that the function 𝑣 has exponential decay as 𝜉2 and is 𝜋-periodical with respect to 𝜉1. Under this assumption, 𝛼0𝑣 “almost” does not destroy (2.2) in the sense that the norm of additional contribution is small. The rigorous explanation is given in Section 6. Proceeding, we have thatΔ𝑥𝑢0+𝜀𝑢1+𝜀𝛼0=𝜆𝑣+0+𝜀𝜆1𝑢+0+𝜀𝑢1+𝜀𝛼0𝑣+.(3.8) Taking into account (2.3) and (3.3), we see that 𝑣 has to satisfy the equationΔ𝑥𝛼0𝑣=𝜆0𝛼0𝑣.(3.9)

Rewrite Δ𝑥 in polar coordinates and pass to the 𝜉-variables in the argument of 𝑣: Δ𝑥𝛼0𝑣=1𝑟𝜕𝑟𝜕𝜕𝑟𝛼𝜕𝑟0𝑣+1𝑟2𝜕2𝛼0𝑣𝜕𝜃2=𝛼0𝜕2𝑣𝜕𝑟2+𝛼0𝑟𝜕𝑣+1𝜕𝑟𝑟2𝑣𝜕2𝛼0𝜕𝜃2+2𝜕𝛼0𝜕𝜃𝜕𝑣𝜕𝜃+𝛼0𝜕2𝑣𝜕𝜃2=𝛼0𝜀2𝜕2𝑣𝜕𝜉22𝛼0𝜀𝜀2𝜉2𝜕𝑣𝜕𝜉2+11𝜀𝜉22𝑣𝜕2𝛼0𝜕𝜃2+2𝜀𝜕𝛼0𝜕𝜃𝜕𝑣𝜕𝜉1+𝛼0𝜀2𝜕2𝑣𝜕𝜉21.(3.10) Finally, replacing formulas 1/(𝜀𝜀2𝜉2) and 1/(1𝜀𝜉2)2 with Taylor series with respect to 𝜀, substituting the obtained formula for Δ𝑥(𝛼0𝑣) in (3.9), and equating terms at 𝜀2, we deduce thatΔ𝜉𝑣=0.(3.11)

Now we derive the boundary conditions for function 𝑣. Substituting the second series from (3.1) in boundary conditions from (2.2) and using (3.2), we have 0=𝑢𝜀=𝑢0+𝜀𝑢1+𝜀𝛼0𝛼𝑣+=𝜀0𝜉2+𝑢1𝑟=1+𝛼0𝑣𝜀+𝑂2,0=𝜕𝑢𝜀=𝜕𝑟𝜕𝑢0𝜕𝑟+𝜀𝜕𝑢1𝜕𝑟+𝜀𝛼0𝜕𝑣𝜕𝑟+=𝛼0𝜀𝛼1𝛼0𝜕𝑣𝜕𝜉2+,(3.12) which implies that 𝛼0𝜉2+𝑢1𝑟=1+𝛼0𝑣=0,𝛼0𝛼0𝜕𝑣𝜕𝜉2=0.(3.13)

Taking into account (3.4), we derive the boundary conditions for 𝑣 on 𝜕𝐵 and on Γ:𝑣=𝜉2+𝜆102𝜋𝛼20𝑑𝜃1on𝜕𝐵,𝜕𝑣𝜕𝜉2=1onΓ.(3.14)

Summing up (3.11) and (3.14), we get the following boundary-value problem for 𝑣Δ𝜉𝑣=0Π𝐵,𝑣=𝜉2+𝜆102𝜋𝛼20𝑑𝜃1on𝜕𝐵,𝜕𝑣𝜕𝜉2=1onΓ.(3.15) Define the function 𝑌 as the solution of the following boundary-value problem in the cell of periodicity:Δ𝑌=0inΠ𝐵,𝑌=0on𝜕𝐵,𝜕𝑌𝜕𝜉1=0on𝜕ΠΓ,𝜕𝑌𝜕𝜉2=0onΓ,𝜕𝑌𝜕𝜉2=1as𝜉2.(3.16)

It was proved in [7] that there exists the solution of (3.16), which is even with respect to 𝜉1 and has the asymptotics: 𝑌(𝜉)=𝜉2𝑒+𝐶(𝐵)+𝑂𝛼𝜉2as𝜉2,(3.17) where𝐶(𝐵)=Π𝐵||𝑌𝜉2||2||𝐵||𝑑𝜉+>0,(3.18) and |𝐵| is the area of the domain 𝐵.

The following lemma gives the conditions to obtain 𝑣 as an exponentially decaying function as 𝜉2.

Lemma 3.2. Assume that 𝐹 is 𝜋-periodic with respect to 𝜉1 function with exponential decay as 𝜉2, and let 𝑣 be a 𝜋-periodic solution of the boundary-value problem: Δ𝑣=𝐹,𝜉2>0;𝑣=𝐴1,𝜉𝜕𝐵;𝜕𝑣𝜕𝜉2=𝐴2,𝜉Γ;(3.19) with finite Dirichlet integral in Π. Then there exists the unique weak solution, which has asymptotics 𝑣=𝐶+𝑂(𝑒𝛼𝜉2),𝛼>0. To obtain 𝑣 as a function with exponential decay as 𝜉2, it is necessary and sufficient to have Π𝐵𝑌𝐹𝑑𝜉+𝜕𝐵𝐴1𝜕𝑌𝜕𝜈𝐵𝑑𝑆𝐵+Γ𝐴2𝑌𝑑𝜉1=0.(3.20)

Proof. The existence of the solution with asymptotics 𝑣=𝐶+𝑂(𝑒𝛼𝜉2) follows from the classical results on elliptic boundary-value problems in cylindric domains (see, e.g., [23] and [24, Chapters 2, 5]). Let us verify (3.20). Define Π𝑅=Π{0<𝜉2<𝑅} and Γ𝑅={𝜉𝜋/2<𝜉1<𝜋/2,𝜉2=𝑅}. By multiplying the equation from (3.15) by 𝑌, integrating it over Π𝑅𝐵, and using the property of 𝑌, we get that Π𝑅𝐵𝐹𝑌𝑑𝜉=Π𝑅𝐵𝑣𝑌𝑑𝜉+Γ𝑅𝜕𝑣𝜕𝜉2𝑌𝑑𝜉1Γ𝜕𝑣𝜕𝜉2𝑌𝑑𝜉1=Π𝑅𝐵𝑣Δ𝑌𝑑𝜉Γ𝑅𝑣𝜕𝑌𝜕𝜉2𝑑𝜉1+Γ𝑣𝜕𝑌𝜕𝜉2𝑑𝜉1𝜕𝐵𝑣𝜕𝑌𝜕𝜈𝑑𝑆𝐵+Γ𝑅𝜕𝑣𝜕𝜉2𝑌𝑑𝜉1Γ𝜕𝑣𝜕𝜉2𝑌𝑑𝜉1=Γ𝑅𝑣𝜕𝑌𝜕𝜉2𝑑𝜉1𝜕𝐵𝐴1𝜕𝑌𝜕𝜈𝑑𝑆𝐵+Γ𝑅𝜕𝑣𝜕𝜉2𝑌𝑑𝜉1Γ𝐴2𝑌𝑑𝜉1.(3.21) Passing to the limit as 𝑅, we obtain that Π𝐵𝐹𝑌𝑑𝜉=𝜋𝐶𝜕𝐵𝐴1𝜕𝑌𝜕𝜈𝑑𝑆𝐵Γ𝐴2𝑌𝑑𝜉1.(3.22) This can be rewritten as 1𝐶=𝜋Γ𝐴2𝑌𝑑𝜉1𝜕𝐵𝐴1𝜕𝑌𝜕𝜈𝑑𝑆𝐵Π𝐵.𝐹𝑌𝑑𝜉(3.23) Then 𝑣 has exponential decay as 𝜉2 if and only if 𝐶=0 which is equivalent to (3.20). The proof is complete.

In order to obtain 𝑣 as function with exponential decay as 𝜉2, one must have0=𝜕𝐵𝜉2+𝐾𝜕𝑌𝜕𝜈𝐵𝑑𝑆𝐵+Γ𝑌𝑑𝜉1,(3.24) where we denote 𝐾=𝜆1(02𝜋𝛼20𝑑𝜃)1. However, (3.24) implies that𝐾=𝜕𝐵𝜉2𝜕𝑌𝜕𝜈𝐵𝑑𝑆𝐵+Γ𝑌𝑑𝜉1𝜕𝐵𝜕𝑌𝜕𝜈𝐵𝑑𝑆𝐵1.(3.25) Integrate the identities 0=Π𝑅𝐵Δ𝑌𝑑𝜉,0=Π𝑅𝐵𝜉2Δ𝑌𝑑𝜉: 0=Π𝑅𝐵Δ𝑌𝑑𝜉=𝜕Π𝑅𝐵𝜕𝑌𝜕𝑛𝑑𝑆=𝜕𝐵𝜕𝑌𝜕𝜈𝐵𝑑𝑆𝐵+Γ𝑅𝜕𝑌𝜕𝜉2𝑑𝜉1,0=Π𝑅𝐵𝜉2Δ𝑌𝑌Δ𝜉2𝑑𝜉=𝜕Π𝑅𝐵𝜉2𝜕𝑌𝜕𝑛𝑌𝜕𝜉2=𝜕𝑛𝑑𝑆𝜕𝐵𝜉2𝜕𝑌𝜕𝜈𝐵𝑑𝑆𝐵+Γ𝑌𝑑𝜉1+Γ𝑅𝜉2𝜕𝑌𝜕𝜉2𝑌𝑑𝜉1.(3.26)

Passing to the limit as 𝑅, we find that0=𝜕𝐵𝜕𝑌𝜕𝜈𝐵𝑑𝑆𝐵+𝜋,0=𝜕𝐵𝜉2𝜕𝑌𝜕𝜈𝐵𝑑𝑆𝐵+Γ𝑌𝑑𝜉1𝜋𝐶(𝐵).(3.27) Then (3.25) and (3.27) together with Remark 2.2 imply that𝜆1=𝐶(𝐵)02𝜋𝛼20𝑑𝜃=2𝜋𝐶(𝐵)𝑘20𝒥02𝑘0<0.(3.28)

4. Complete Expansion in the Case of the Simple Eigenvalue 𝜆0

Assume that 𝜆0 is the simple eigenvalue of the limit problem. Now we construct the complete expansion in the following form:𝑢𝜀(𝑥)=𝑢ex𝜀(𝑥)+𝜒(1𝑟)𝑢in𝜀1𝑟𝜀,𝜃𝜀,(4.1) where 𝜒 is a smooth cutoff function, which equals to one when 1/2<𝑟<1 and zero when 𝑟<1/4:𝑢ex𝜀(𝑥)=𝒥0(𝑘(𝜀)𝑟),(4.2)𝑢in𝜀(𝜉)=𝑖=1𝜀𝑖𝑣𝑖(𝜉).(4.3) Here 𝑘(𝜀)=𝜆𝜀,𝑣𝑖(𝜉) are 𝜋-periodic in 𝜉1 functions with exponential decay as 𝜉2. One can easily show that (4.2) solves the equation: Δ𝑥𝑢ex𝜀(𝑥)=𝜆𝜀𝑢ex𝜀(𝑥)(4.4) if and only if 𝑘(𝜀)=𝜆𝜀.

We are looking for 𝑢in𝜀(𝜉), which solves the equation:Δ𝑥𝑢in𝜀(𝜉)=𝜆𝜀𝑢in𝜀(𝜉).(4.5) If (4.4) and (4.5) are satisfied, then 𝑢𝜀 from (4.1) is the solution of Δ𝑥𝑢𝜀=𝜆𝜀𝑢𝜀+𝐹,(4.6) where 𝐹=𝑢in𝜀Δ𝑥𝜒2𝑥𝑢in𝜀𝑥𝜒. Our aim is to construct 𝑢in𝜀 so that 𝐹 will be of small order as 𝜀0. This is the reason why we need to have 𝑣𝑖 as exponentially decaying functions.

Now we derive the formula for the Laplacian in 𝜉-variables:Δ𝑥=𝜕2𝜕𝑟2+1𝑟𝜕+1𝜕𝑟𝑟2𝜕2𝜕𝜃2=1𝜀2𝜕2𝜕𝜉22+1𝜀𝜀𝜉2𝜕1𝜕𝜉2+1𝜀2𝜀𝜉212𝜕2𝜕𝜉21=1𝜀2Δ𝜉+1𝜀𝜀𝜉2𝜕1𝜕𝜉2+1𝜀21𝜀𝜉212𝜕12𝜕𝜉21.(4.7)

By substituting the Taylor series for the functions 1𝜀𝜀𝜉2,11𝜀21𝜀𝜉2121(4.8) in (4.7), we get the final formula for Δ𝑥:Δ𝑥=1𝜀2Δ𝜉+𝑛=0(𝑛+1)𝜀𝑛2𝜉𝑛2𝜕2𝜕𝜉21𝑛=0𝜀𝑛1𝜉𝑛2𝜕𝜕𝜉2.(4.9)

Substituting (2.4) and (4.3) in (4.5) and taking into account (4.9), we deduce the following formula:𝑖=1𝜀𝑖Δ𝜉𝑣𝑖=𝜀+𝜀2𝜉2++𝜀𝑛+1𝜉𝑛2+𝑖=1𝜀𝑖𝜕𝑣𝑖𝜕𝜉22𝜀𝜉2+3𝜀2𝜉22++(𝑛+1)𝜀𝑛𝜉𝑛2+𝑖=1𝜀𝑖𝜕2𝑣𝑖𝜕𝜉21𝜀2𝜆0+𝜀3𝜆1++𝜀𝑛+2𝜆𝑛𝑖=1𝜀𝑖𝑣𝑖.(4.10)

By equating terms of the same power of 𝜀, we obtain that𝜀1Δ𝜉𝑣1𝜀=0,2Δ𝜉𝑣2=𝜕𝑣1𝜕𝜉22𝜉2𝜕2𝑣1𝜕𝜉21,𝜀,𝑘Δ𝜉𝑣𝑘=𝑘1𝑗=1𝜉2𝑗1𝜕𝑣𝑘𝑗𝜕𝜉2(𝑗+1)𝜉𝑗2𝜕2𝑣𝑘𝑗𝜕𝜉21𝑘3𝑗=0𝜆𝑗𝑣𝑘𝑗2,.(4.11)

Consider now the boundary conditions from (2.2). According to the property of 𝜒,𝑢𝜀(𝑥)=𝑢ex𝜀(𝑥)+𝑢in𝜀1𝑟𝜀,𝜃𝜀=𝒥0(𝑘(𝜀)𝑟)+𝑖=1𝜀𝑖𝑣𝑖(𝜉),(4.12) in a small neighborhood of 𝜕Ω. Moreover, on 𝜕Ω, it yields that 0=𝜕𝑢𝜀𝜕𝑟=𝑘(𝜀)𝒥0(𝑘(𝜀))𝑖=1𝜀𝑖1𝜕𝑣𝑖𝜕𝜉2||||𝜉2=0.(4.13)

We assume that the function 𝑘(𝜀) has asymptotics:𝑘(𝜀)=𝑘0+𝜀𝑘1++𝜀𝑛𝑘𝑛+,(4.14) and since 𝜆𝜀=𝑘2(𝜀), we can derive the following formulas for 𝜆𝑖:𝜆0=𝑘20,𝜆1=2𝑘0𝑘1,,𝜆𝑖=𝑖𝑗=0𝑘𝑗𝑘𝑖𝑗.(4.15) Rewriting 𝒥0(𝑘(𝜀)) as a Taylor series with respect to 𝜀, we have𝒥0(𝑘(𝜀))=𝒥0𝑘0+𝒥0𝑘0𝑘1𝜀+𝒥1!0𝑘0𝑘21+𝒥0𝑘0𝑘2𝜀22!+.(4.16)

Substituting (4.16) in (4.13), using (4.14), and equating the terms with the same powers of 𝜀, we get the following boundary condition for 𝑣𝑖,𝑖=1,2,:𝜕𝑣𝑖𝜕𝜉2=𝑔𝑖𝑘1,,𝑘𝑖1onΓ,(4.17) where 𝑔1=𝑘0𝒥0𝑘0,𝑔2=𝑘1𝒥0𝑘0+𝑘0𝑘1𝒥0𝑘00.(4.18)

Consider now the boundary conditions on small holes. Analogously, 𝑢𝜀(𝑥)=𝒥0(𝑘(𝜀)𝑟)+𝑖=1𝜀𝑖𝑣𝑖(𝜉)=𝒥0𝑘(𝜀)1𝜀𝜉2+𝑖=1𝜀𝑖𝑣𝑖(𝜉).(4.19)

Substituting the Taylor series for 𝒥0(𝑘(𝜀)(1𝜀𝜉2)) with respect to 𝜀 in the last formula, using (4.14), and equating the terms with the same powers of 𝜀 in equation 𝑢𝜀=0 on Γ𝜀, we get the following boundary condition for 𝑣𝑖,𝑖=1,2,, on 𝜕𝐵: 𝑣𝑖=𝑘𝑖𝒥0𝑘0+𝑓𝑖𝜉2;𝑘0,𝑘1,,𝑘𝑖1on𝜕𝐵,(4.20) where 𝑓𝑖 are polynomials of power 𝑖 with respect to 𝜉2 with coefficients which depend on (𝑘0,𝑘1,,𝑘𝑖1). The precise formula for 𝑓𝑖 can be derived for each fixed 𝑖. For example, we have that𝑓1=𝑘0𝒥0𝑘0𝜉2,𝑓2=𝑘1𝒥0𝑘0𝜉212𝒥0𝑘0𝑘1𝑘0𝜉22.(4.21) The following Lemma is useful for our analysis. For the proof see for example,[3].

Lemma 4.1. Suppose that 𝐹 and 𝑣 satisfy the conditions of Lemma 3.2. (a) If 𝐹 is even with respect to 𝜉1, then 𝑣 is even; (b) if 𝐹 is odd with respect to 𝜉1 and 𝐴1=𝐴2=0, then 𝑣 is odd with respect to 𝜉1 and decays exponentially as 𝜉2.

Theorem 4.2. There exist numbers 𝑘𝑖 and 𝜋-periodic in 𝜉1 functions 𝑣𝑖 with finite Dirichlet integral in Π and exponential decay as 𝜉2, such that these functions are solutions of the following boundary-value problems: Δ𝑣𝑖=𝐹𝑖𝑖1𝑗=1𝜉2𝑗1𝜕𝑣𝑖𝑗𝜕𝜉2(𝑗+1)𝜉𝑗2𝜕2𝑣𝑖𝑗𝜕𝜉21𝑖3𝑗=0𝜆𝑗𝑣𝑖𝑗2𝑖𝑛Π𝑣𝐵,𝑖=𝑘𝑖𝒥0𝑘0+𝑓𝑖𝜉2;𝑘0,𝑘1,,𝑘𝑖1𝑜𝑛𝜕𝐵,𝜕𝑣𝑖𝜕𝜉2=𝑔𝑖𝑘1,,𝑘𝑖1𝑜𝑛Γ.(4.22) Moreover, the constants are defined by the formula: 𝑘𝑖1=𝜋𝒥0𝑘0Π𝐵𝑌𝐹𝑖𝑑𝜉+𝜕𝐵𝑓𝑖𝜉2;𝑘0,𝑘1,,𝑘𝑖1𝜕𝑌𝜕𝜈𝐵𝑑𝑆𝐵+𝑔𝑖𝑘1,,𝑘𝑖1Γ𝑌𝑑𝜉1(4.23) In particular, 𝑘1=𝜋𝐶(𝐵)𝑘0𝒥20𝑘0,(4.24)𝑘2=𝑘212𝑘0.(4.25)

Proof. Let 𝑣 be the solution of boundary-value problem (3.15). It can be easily verified that 𝑣1=𝑘0𝒥0𝑘0𝑣(4.26) is a solution of (4.22), (4.20), (4.17) for 𝑓1,𝑔1, and 𝑘1 defined by (4.21), (4.18), and (4.24). For any 𝑘2 boundary-value problem (4.22), (4.20), (4.17) for 𝑣2 has a 𝜋-periodic solution with finite Dirichlet integral. By Lemma 3.2 and (3.27), 𝑣2 has exponential decay as 𝜉2 if and only if 𝑘2 is given by (4.23) for 𝑖=2. Let us verify formula (4.25) without applying the general (4.23). It is obvious that 𝜕2𝑣1𝜕𝜉21𝜕=2𝑣1𝜕𝜉22.(4.27) By using that fact one can write the boundary-value problem for 𝑣2 as Δ𝑣2=𝜕𝑣1𝜕𝜉22𝜉2𝜕2𝑣1𝜕𝜉21=𝜕𝑣1𝜕𝜉2+2𝜉2𝜕2𝑣1𝜕𝜉22inΠ𝑣𝐵,2=𝑘2𝒥0𝑘0+𝑘1𝒥0𝑘0𝜉212𝒥0𝑘0𝑘1𝑘0𝜉22on𝜕𝐵,𝜕𝑣2𝜕𝜉2=0onΓ.(4.28) It can be verified that the function 𝑣2=12𝜉22𝜕𝑣1𝜕𝜉2(4.29) is 𝜋-periodic with finite Dirichlet integral in Π, has exponential decay as 𝜉2, and satisfies problem (4.28) for 𝑘2 defined by (4.25). We can use the induction process to finalize the proof.

Since 𝑘𝑖 are defined by (4.23), we can calculate 𝜆𝑖 by using (4.15). Denote 𝑢𝜀,𝑁=𝒥0𝜆𝜀,𝑁𝑟+𝜒(1𝑟)𝑣𝜀,𝑁,(4.30) where 𝜆𝜀,𝑁 and 𝑣𝜀,𝑁 are the partial sums of (2.4) and (4.3), respectively.

Theorem 4.2 implies the validity of the following useful result.

Theorem 4.3. For any integer 𝑁>0, the function 𝑢𝜀,𝑁 is the solution of the boundary-value problem Δ𝑢𝜀,𝑁=𝜆𝜀,𝑁𝑢𝜀,𝑁+𝐹𝜀,𝑁𝑖𝑛Ω𝜀,𝑢𝜀,𝑁=𝑎𝜀,𝑁(𝜃)𝑜𝑛Γ𝜀,𝜕𝑢𝜀,𝑁𝜕𝑟=𝑏𝜀,𝑁(𝜃)𝑜𝑛𝜕Ω,(4.31) where 𝑎𝜀,𝑁𝐿2(Γ𝜀)=𝑂(𝜀𝑁1),𝑏𝜀,𝑁𝐿2(𝜕Ω)=𝑂(𝜀𝑁1),𝐹𝜀,𝑁𝐿2(Ω𝜀)=𝑂(𝜀𝑁1), and 𝑁1 as 𝑁.

Proof. According to the definition of 𝑢𝜀,𝑁, we have that Δ𝑥𝑢𝜀,𝑁=Δ𝑥𝒥0𝜆𝜀,𝑁𝑟+𝜒(1𝑟)𝑣𝜀,𝑁=Δ𝑥𝒥0𝜆𝜀,𝑁𝑟Δ𝑥𝜒𝑣𝜀,𝑁2𝑥𝜒𝑥𝑣𝜀,𝑁𝜒Δ𝑥𝑣𝜀,𝑁=𝜆𝜀,𝑁𝒥0𝜆𝜀,𝑁𝑟+𝜆𝜀,𝑁𝜒(1𝑟)𝑣𝜀,𝑁𝜆𝜀,𝑁𝜒(1𝑟)𝑣𝜀,𝑁Δ𝑥𝜒𝑣𝜀,𝑁2𝑥𝜒𝑥𝑣𝜀,𝑁𝜒Δ𝑥𝑣𝜀,𝑁=𝜆𝜀,𝑁𝑢𝜀,𝑁+𝐹𝜀,𝑁,(4.32) where 𝐹𝜀,𝑁=𝑣𝜀,𝑁Δ𝑥𝜆𝜒𝜒𝜀,𝑁𝑣𝜀,𝑁+Δ𝑥𝑣𝜀,𝑁2𝑥𝜒𝑥𝑣𝜀,𝑁=𝐼1+𝐼2+𝐼3.(4.33) Passing from (𝑥1,𝑥2) variables to polar coordinates (𝑟,𝜃), we get that 𝜕𝜕𝑥1𝜕=cos𝜃𝜕𝑟sin𝜃𝑟𝜕,𝜕𝜕𝜃𝜕𝑥2𝜕=sin𝜃+𝜕𝑟cos𝜃𝑟𝜕𝜕𝜃,(4.34)Δ𝑥=1𝑟𝜕𝑟𝜕𝜕𝑟+1𝜕𝑟𝑟2𝜕2𝜕𝜃2.(4.35)
By using the fact that lim𝑥𝑥𝑒𝛼𝑥=0 and due to the result of Theorem 4.2, we have that, for any 1𝑖𝑁, 𝜀𝑖𝑣𝑖=𝜀𝑖𝑂𝑒𝛼𝜉2=𝜀𝑁𝑂𝜀𝑖𝑁𝑒𝛼(1𝑟)/𝜀=𝜀𝑁𝑂(𝜀𝑚𝜀)=𝑂𝑁+𝑚,(4.36) where 𝑚 is fixed. Hence, 𝑣𝜀,𝑁=𝑂(𝜀𝑁+𝑚). Similarly, taking into account (4.34) and (4.35), we can deduce that 𝑥𝑣𝜀,𝑁𝜀=𝑂𝑁+𝑚𝛼cos𝜃𝜀sin𝜃𝑟,𝛼sin𝜃𝜀+cos𝜃𝑟,𝑥𝜒=cos𝜃𝜒,sin𝜃𝜒.(4.37) Consequently, 𝑥𝑣𝜀,𝑁𝑥𝜀𝜒=𝑂𝑁+𝑚𝑂1𝜀𝑟.(4.38) Furthermore, Δ𝑥𝑣𝜀,𝑁𝛼=𝑂𝜀𝜀𝑟𝑁+𝑚+𝛼2𝜀2𝑂𝜀𝑁+𝑚+1𝑟2𝑂𝜀𝑁+𝑚1=𝑂𝜀2𝑟2𝑂𝜀𝑁+𝑚,Δ𝑥1𝜒=𝑟𝜒+𝜒1=𝑂𝑟.(4.39)
According to the definition of 𝜒, the support of 𝑥𝜒 and Δ𝑥𝜒 is the set {1/4𝑟1/2}. Summarizing, we have that 𝐼1𝜀=𝑂𝑁+𝑚𝑂1𝑟,𝐼2𝜀=𝑂𝑁+𝑚𝑂1𝜀2𝑟2,𝐼3𝜀=𝑂𝑁+𝑚𝑂1𝜀𝑟,(4.40) and we can derive that 𝐹𝜀,𝑁2𝐿2Ω𝜀=Ω𝐹2𝜀,𝑁=𝑟𝑑𝑟𝑑𝜃Ω{1/4𝑟1}𝐼22𝑟𝑑𝑟𝑑𝜃+Ω{1/4𝑟1/2}𝐼1+𝐼32𝑟+2𝐼2𝐼1+𝐼3𝑟𝜀𝑑𝑟𝑑𝜃=𝑂2𝑁+2𝑚𝑂1𝜀3𝜀+𝑂2𝑁+2𝑚𝑂1𝜀4𝜀=𝑂2𝑁+2𝑚𝑂1𝜀4.(4.41) Therefore, 𝐹𝜀,𝑁𝐿2(Ω𝜀)𝜀=𝑂𝑁+𝑚2𝜀=𝑂𝑁1,𝑁1as𝑁.(4.42) Consider now 𝑢𝜀,𝑁 on Γ𝜀: 𝑢𝜀,𝑁=𝒥0𝜆𝜀,𝑁𝑟+𝑣𝜀,𝑁=𝜀𝑁+1𝛽𝑁+1+𝜀𝑁+2𝛽𝑁+2+,(4.43) where 𝛽𝑗 are the coefficients of the Taylor series of the function 𝒥0(𝜆𝜀,𝑁𝑟). Hence, 𝑎𝜀,𝑁=𝑂(𝜀𝑁+1) and 𝑎𝜀,𝑁2𝐿2Γ𝜀=2𝜀𝜕𝐵𝜀𝑎2𝜀,𝑁2𝑑𝜃𝜀𝜀2𝜋𝑎𝜀𝑂2𝑁+2𝜀=𝑂2𝑁+2,(4.44) which yields that 𝑎𝜀,𝑁𝐿2(Γ𝜀)=𝑂(𝜀𝑁+1)=𝑂(𝜀𝑁1),𝑁1 as 𝑁. Analogously, one can verify that 𝑏𝜀,𝑁𝐿2(𝜕Ω)=𝑂(𝜀𝑁1),𝑁1 as 𝑁. The proof is complete.

5. Complete Expansion in the Case of Multiple Eigenvalue 𝜆0

In this section we consider the case when 𝜆0 is of multiplicity two. The asymptotics of the eigenvalue were constructed in the form (2.4) and𝑢𝜀(𝑥)=𝑢ex𝜀(𝑥)+𝜒(1𝑟)𝑢in𝜀1𝑟𝜀,𝜃𝜀,𝑢,𝜃(5.1)ex𝜀(𝑥)=cos(𝑛𝜃)𝒥𝑛𝑢(𝑘(𝜀)𝑟),(5.2)in𝜀(𝑥)=cos(𝑛𝜃)𝑖=1𝜀𝑖𝑣even𝑖(𝜉)+sin(𝑛𝜃)𝑖=2𝜀𝑖𝑣odd𝑖(𝜉).(5.3)

In this case, 𝑣even𝑖=𝑘𝑖𝒥𝑛𝑘0+𝑓𝑖(𝑛)𝜉2;𝑘0,𝑘1,,𝑘𝑖1on𝜕𝐵,(5.4) where 𝑓𝑖(𝑛) are polynomials of power 𝑖 with respect to 𝜉2 with coefficients which depend on (𝑘0,𝑘1,,𝑘𝑖1). Moreover, 𝜕𝑣even𝑖𝜕𝜉2=𝑔𝑖(𝑛)𝑘1,,𝑘𝑖1onΓ,(5.5) where𝑓1(𝑛)=𝑘0𝒥𝑛𝑘0𝜉2,𝑓2(𝑛)=𝑘1𝒥0𝑘0𝜉212𝒥0𝑘0𝑘1𝑘0𝜉22,𝑔1(𝑛)=𝑘0𝒥𝑛𝑘0,𝑔2(𝑛)=𝑘1𝒥𝑛𝑘0+𝑘0𝑘1𝒥𝑛𝑘0𝑣0,(5.6)odd𝑖=0,𝜉𝜕𝐵,𝜕𝑣odd𝑖𝜕𝜉2=0,𝜉Γ.(5.7)

Substituting (5.3) and (2.4) in (4.5), passing to the variables 𝜉 and (𝜃,𝜌), and collecting all the terms with equal order of 𝜀, we get two systems of equations for 𝑣even𝑖 and 𝑣odd𝑖: Δ𝑣even𝑖=𝑖1𝑗=1𝜉2𝑗1𝜕𝑣even𝑖𝑗𝜕𝜉2(𝑗+1)𝜉𝑗2𝜕2𝑣even𝑖𝑗𝜕𝜉21𝑛𝑖3𝑗=0(𝑗+1)𝜉𝑗2𝜕𝑣odd𝑖𝑗1𝜕𝜉1𝑛2𝑖3𝑗=0(𝑗+1)𝜉𝑗2𝑣even𝑖𝑗2𝑖3𝑗=0𝜆𝑗𝑣even𝑖𝑗2inΠ𝐵,(5.8)Δ𝑣odd𝑖=𝑖2𝑗=1𝜉2𝑗1𝜕𝑣even𝑖𝑗𝜕𝜉2(𝑗+1)𝜉𝑗2𝜕2𝑣odd𝑖𝑗𝜕𝜉21+𝑛𝑖2𝑗=0(𝑗+1)𝜉𝑗2𝜕𝑣even𝑖𝑗1𝜕𝜉1+𝑛2𝑖3𝑗=0(𝑗+1)𝜉𝑗2𝑣odd𝑖𝑗2𝑖3𝑗=0𝜆𝑗𝑣odd𝑖𝑗2inΠ𝐵.(5.9)

Theorem 5.1. There exist numbers 𝑘𝑖 and 𝜋-periodic in 𝜉1 even functions 𝑣even𝑖 and odd functions 𝑣odd𝑖 with finite Dirichlet integral in Π, which have exponential decay as 𝜉2, such that these functions are solutions of the boundary-value problems (5.8), (5.4), (5.5), and (5.9), (5.7), respectively. Moreover, the constants 𝑘𝑖 are defined by the formula: 𝑘𝑖1=𝜋𝒥𝑛𝑘0Π𝐵𝑌𝐹𝑖𝑑𝜉+𝜕𝐵𝑓𝑖(𝑛)𝜉2;𝑘0,𝑘1,,𝑘𝑖1𝜕𝑌𝜕𝜈𝐵𝑑𝑆𝐵+𝑔𝑖(𝑛)𝑘1,,𝑘𝑖1Γ𝑌𝑑𝜉1,(5.10)

Proof. The problems (5.8), (5.5), (5.4) for functions 𝑣even1,𝑣even2 coincide with problems (4.22), (4.17), and (4.20) (if one change 𝒥0(𝑘0) by 𝒥𝑛(𝑘0) and 𝑓𝑖,𝑔𝑖 by the respective 𝑓𝑖(𝑛),𝑔𝑖(𝑛)). Therefore the construction of 𝑣even1,𝑣even2 and 𝑘1,𝑘2 is just the same as the construction from the proof of Theorem 4.2. Due to (5.9), (5.7), the problem for 𝑣odd2 is as follows: Δ𝑣odd2=𝑛𝜉2𝜕𝑣even1𝜕𝜉1inΠ𝑣𝐵,odd2=0on𝜕𝐵,𝜕𝑣odd2𝜕𝜉2=0onΓ.(5.11) The function 𝑣even1 is even (due to (4.26)) and, hence, the right-hand side is odd in (5.11) and is even in (5.8). By Lemma 3.2 and Theorem 4.2, we conclude that there exists the even solution 𝑣odd2 of (5.11) with exponential decay. Then we can use the iteration process to complete the proof.

Denote 𝑢𝜀,𝑁=cos(𝑛𝜃)𝒥0𝜆𝜀,𝑁𝑟+𝜒(1𝑟)𝑣𝜀,𝑁,(5.12) where 𝜆𝜀,𝑁 and 𝑣𝜀,𝑁 are the partial sums of (2.4) and (5.3), respectively.

Theorem 5.1 implies the validity of the following result.

Theorem 5.2. For any integer 𝑁>0, the function 𝑢𝜀,𝑁 is the solution of the boundary-value problem: Δ𝑢𝜀,𝑁=𝜆𝜀,𝑁𝑢𝜀,𝑁+𝐹𝜀,𝑁𝑖𝑛Ω𝜀,𝑢𝜀,𝑁=𝑎𝜀,𝑁(𝜃)cos(𝑛𝜃)𝑜𝑛Γ𝜀,𝜕𝑢𝜀,𝑁𝜕𝑟=𝑏𝜀,𝑁(𝜃)cos(𝑛𝜃)𝑜𝑛𝜕Ω,(5.13) where 𝑎𝜀,𝑁𝐿2(Γ𝜀)=𝑂(𝜀𝑁1),𝑏𝜀,𝑁𝐿2(𝜕Ω)=𝑂(𝜀𝑁1),𝐹𝜀,𝑁𝐿2(Ω𝜀)=𝑂(𝜀𝑁1), and 𝑁1 as 𝑁.

Proof. The proof is analogous to the proof of Theorem 4.3. Hence, we omit the details.

6. Verification of the Asymptotics

Consider the boundary-value problem:Δ𝑈𝜀=𝜆𝑈𝜀+𝐹inΩ𝜀,𝑈𝜀=0onΓ𝜀,𝜕𝑈𝜀𝜕𝑟=0on𝜕Ω,(6.1) where 𝐹𝐿2(Ω) and 𝜆𝜆0 is some fixed number.

Similarly to the techniques used in [3, 18], one can show that the boundary-value problem (6.1) has the solution 𝑈𝜀𝐻1(Ω) and the following representation holds: 𝑈𝜀=𝑢𝜀𝜆𝜀𝜆Ω𝑢𝜀𝑈𝐹𝑑𝑥+𝜀,(6.2) for 𝜆 close to the simple eigenvalue 𝜆0 of the problem (2.3) and 𝑈𝜀=1𝜆𝜀𝜆2𝑖=1𝑢𝑖𝜀Ω𝑢𝑖𝜀𝑈𝐹𝑑𝑥+𝜀,(6.3) for 𝜆 close to multiple eigenvalue 𝜆0 of the problem (2.3). Here 𝑢𝜀 is normalized in 𝐿2(Ω) eigenfunctions to (2.2) and 𝑢𝑖𝜀 is orthonormalized in 𝐿2(Ω) eigenfunctions to (2.2). Moreover, 𝑈𝜀𝐻1𝐶𝐹𝐿2,(6.4) where the constant 𝐶 is independent on 𝜀 and 𝜆. It follows from (6.2) and (6.4) that 𝑈𝜀𝐻1𝐶𝜆𝜀𝜆𝐹𝐿2.(6.5) Consider now the case of simple 𝜆0. Define the function: 𝑈𝑁𝜀1(𝑥)=1+𝜀𝑢𝜀,𝑁1(𝑥)1+𝜀𝑎𝜀,𝑁+𝑏𝜀,𝑁𝜒(1𝑟)𝑣(𝜉)+𝜉2+𝐶(𝐵),(6.6) where 𝑢𝜀,𝑁 and 𝑣 are the solutions of (4.31) and (3.15), respectively, and 𝐶(𝐵) is given by (3.18). Then, by Theorem 4.3, 𝑈𝑁𝜀 is the solution of (6.1) if 𝜆=𝜆𝜀,𝑁,𝐹𝐿2𝜀=𝑂𝑁2,𝑁2as𝑁.(6.7) Taking into account (6.5), (6.7), and the fact that 𝑈𝜀𝐻1<, we can conclude that for each fixed 𝑁, 𝜆𝜀𝜆𝜀,𝑁𝜀=𝑂𝑁2𝜀=𝑜𝑁as𝜀0.(6.8) Therefore the asymptotics constructed in Section 4 coincide with the expansion of 𝜆𝜀. For the case of multiple 𝜆0, one can use the same technique. The difference is follows: one should use (6.3) instead of (6.2) and Theorem 5.2 instead of Theorem 4.3. The asymptotics of 𝜆𝜀 are completely verified.

7. Application to a Friedrichs-Type Inequality

Consider the sets Ω𝜀,Γ𝜀, which were defined in Section 2.

Definition 7.1. The Sobolev class 𝐻1(Ω𝜀,Γ𝜀) is the class of functions from 𝐻1(Ω𝜀) having zero trace on Γ𝜀.

Theorem 7.2. Let 𝑢𝐻1(Ω𝜀,Γ𝜀). Then a Friedrichs-type inequality Ω𝜀𝑢2(𝑥)𝑑𝑥𝐾𝜀Ω𝜀||||𝑢(𝑥)2𝑑𝑥(7.1) holds, where the best constant 𝐾𝜀 has the asymptotics 𝐾𝜀=1𝑘20+𝒥4𝜋𝐶(𝐵)02𝑘0𝑘20𝜀+𝑜(𝜀),(7.2) as 𝜀0. Here 𝑘0 is the smallest root of the Bessel function 𝒥0 and the constant 𝐶(𝐵) is given by (3.18).

Proof. The geometric approach developed in [5, 9] allows us to state that there is a constant 𝐾>0 such that Ω𝜀𝑢2(𝑥)𝑑𝑥𝐾Ω𝜀||||𝑢(𝑥)2𝑑𝑥.(7.3) The idea and method of proof are exactly similar to the ones which were used in the mentioned papers. We are interested in the behavior of the best possible constant as 𝜀0. Clearly, the best constant 𝐾𝜀=1/𝜆1𝜀, where 𝜆1𝜀 is the smallest eigenvalue of the boundary-value problem (2.2) (due to the variational formulation of the smallest eigenvalue). Therefore, we can apply (2.4) and (3.28) to derive the asymptotic expansion for 𝐾𝜀: 𝐾𝜀=𝜆1𝜀1=𝜆10+𝜀𝜆11+𝑜(𝜀)1=1𝜆102𝜆11𝜆102𝜀+𝑜(𝜀).(7.4) Since we are interested in the smallest eigenvalue 𝜆10, we have to choose the smallest positive root of 𝒥0(𝑘0)=0 as 𝑘0, precisely, 𝑘0=2,405. Then, we get, after some simple calculations and using (4.15) and (3.28), 𝐾𝜀=1𝑘20+𝒥4𝜋𝐶(𝐵)02𝑘0𝑘20𝜀+𝑜(𝜀).(7.5) The proof is complete.

Acknowledgments

This paper was completed during the stay of the second author as PostDoc at Luleå University of Technology in 2010-2011. The second author wants to express many thanks to Luleå University of Technology (Sweden) for financial support and wonderful conditions to work. The work of the first and the second authors was supported in part by RFBR (Project no. 09-01-00353) and the work of the forth author was supported by a grant from the Swedish Research Council (Project no. 621-2008-5186). The authors also thank referee for several suggestions, which have improved the final version of this paper.

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