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

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, 4–11]. 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 𝐻^∘1. 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 𝐻^∘1 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 [12–16]. 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,𝑁≫1, and consider the open set 𝐵_𝜀 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 Ω_𝜀=Ω⧵𝐵_𝜀 and 𝜕𝐵_𝜀=Γ_𝜀, see Figure 1. Let us describe the geometry of 𝐵_𝜀 in details. Consider the semi-strip:Π=𝜉∶−𝜋2<𝜉¹<𝜋2,𝜉²>0,Γ∶=𝜉∶−𝜋2<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 𝐵_𝑎 be the union of the 𝜋-integer translations of 𝐵 along the axis 𝜉₁. Then we define 𝐵_𝜀 as the image of 𝐵_𝑎 under the mapping 𝜃=𝜀𝜉₁,𝑟=1−𝜀𝜉₂.

Figure 1 

Perforated circle.

Figure 2 

Cell of periodicity.

Consider the following spectral problem:−Δ𝑢𝑢_𝜀𝜀=𝜆_𝜀𝑢_𝜀inΩ_𝜀,𝜕𝑢_𝜀=0onΓ_𝜀,𝜕𝑟=0on𝜕Ω.(2.2)

The problem,−Δ𝑢𝑢₀=𝜆₀𝑢₀inΩ,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(Ω_𝜀,Γ_𝜀) (see the definition of 𝐻¹(Ω_𝜀,Γ_𝜀) in Section 7). One of the main results of this paper is the following asymptotics for 𝜆_𝜀 converging to 𝜆₀:𝜆_𝜀=𝜆₀+^∞_𝑖=1𝜀^𝑖𝜆_𝑖,(2.4) where 𝜆_𝑖 are some fixed constants which can be calculated according to (4.23) and (4.15) in the case of simple 𝜆_𝜀 and according to (5.10) and (4.15) when 𝜆_𝜀 is of multiplicity two. In particular, 𝜆₁<0 which implies that 𝜆_𝜀<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 𝜆_𝜀 converging to 𝜆₀ as well as 𝑢_𝜀 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:𝑢𝜆_𝜀=𝜆₀+𝜀𝜆₁+⋯,_𝜀(𝑥)=𝑢₀(𝑥)+𝜀𝑢₁(𝑥)+𝜀𝛼₀(𝜃)𝑣(𝜉)+⋯,(3.1) where 𝜉=(𝜉₁,𝜉₂),𝜉₁=𝜃/𝜀,𝜉₂=(1−𝑟)/𝜀, and𝑢₀(𝑥)=𝛼₀(𝜃)(1−𝑟)+𝑂(1−𝑟)²as𝑟⟶1,𝛼₀(𝜃)=−𝜕𝑢⁰𝜕𝑟||||_𝑟=1𝑢₁,(𝑥)=𝑢₁∣_𝑟=1+𝛼₁(𝜃)(1−𝑟)+𝑂(1−𝑟)²as𝑟⟶1,𝛼₁(𝜃)=−𝜕𝑢¹𝜕𝑟||||_𝑟=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: −Δ_𝑥𝑢₁=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)^02𝜋𝛼²⁰𝑑𝜃⁻¹on𝜕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=𝜆¹_Ω𝑢²⁰𝑑𝑥+𝜆₀_Ω𝑢₁𝑢₀𝑑𝑥.(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𝜕𝑢₀𝜕𝑟𝑢¹𝑑𝜃=−_𝜕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 Γ_𝜀. 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 that−Δ_𝑥𝑢₀+𝜀𝑢₁+𝜀𝛼₀𝑣+⋯=𝜆₀+𝜀𝜆₁+⋯𝑢₀+𝜀𝑢₁+𝜀𝛼₀𝑣+⋯.(3.8) Taking into account (2.3) and (3.3), we see that 𝑣 has to satisfy the equation−Δ_𝑥𝛼₀𝑣=𝜆₀𝛼₀𝑣.(3.9)

Rewrite Δ_𝑥 in polar coordinates and pass to the 𝜉-variables in the argument of 𝑣: Δ_𝑥𝛼₀𝑣=1𝑟𝜕𝜕𝑟𝑟𝜕𝜕𝑟𝛼⁰𝑣+1𝑟²𝜕²𝛼₀𝑣𝜕𝜃²=𝛼₀𝜕²𝑣𝜕𝑟²+𝛼⁰𝑟𝜕𝜕𝑣𝑟+𝑟1²𝑣𝜕²𝛼₀𝜕𝜃²+2𝜕𝛼⁰𝜕𝜃𝜕𝜕𝜃𝑣+𝛼⁰𝜕²𝑣=𝛼₀𝜕𝜃²𝜀²𝜕²𝑣𝜕𝜉²²−𝛼⁰𝜀−𝜀²𝜉₂𝜕𝜕𝜉𝑣²+11−𝜀𝜉₂²𝑣𝜕²𝛼₀𝜕𝜃²+2𝜀𝜕𝜕𝛼𝜃⁰𝜕𝜕𝜉𝑣₁+𝛼⁰𝜀²𝜕²𝑣𝜕𝜉²¹.(3.10) Finally, replacing formulas 1/(𝜀−𝜀²𝜉₂) and 1/(1−𝜀𝜉₂)² with Taylor series with respect to 𝜀, substituting the obtained formula for Δ_𝑥(𝛼₀𝑣) in (3.9), and equating terms at 𝜀⁻², we deduce thatΔ_𝜉𝑣=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 00==𝑢𝜕_𝜀𝑢_𝜀=𝑢₀+𝜀𝑢₁+𝜀𝛼₀𝑣+⋯=𝜀𝛼₀𝜉₂+𝑢₁∣_𝑟=1+𝛼₀𝑣+𝑂𝜀²,𝜕𝑟=𝜕𝜕𝑢𝑟⁰+𝜀𝜕𝜕𝑢𝑟¹+𝜀𝛼⁰𝜕𝑣𝜕𝑟+⋯=−𝛼⁰−𝜀𝛼₁−𝛼₀𝜕𝑣𝜕𝜉₂+⋯,(3.12) which implies that 𝛼₀𝜉−₂+𝛼₀𝑢₁∣_𝑟=1+𝛼₀𝑣=0,−𝛼₀𝜕𝑣𝜕𝜉₂=0.(3.13)

Taking into account (3.4), we derive the boundary conditions for 𝑣 on 𝜕𝐵 and on Γ:𝑣=−𝜉₂+𝜆₁^02𝜋𝜕𝑣𝛼²⁰𝑑𝜃⁻¹on𝜕𝐵,𝜕𝜉₂=−1onΓ.(3.14)

Summing up (3.11) and (3.14), we get the following boundary-value problem for 𝑣∶𝑣=−𝜉₂Δ_𝜉𝑣=0Π⧵𝐵,+𝜆₁^02𝜋𝜕𝑣𝛼²⁰𝑑𝜃⁻¹on𝜕𝐵,𝜕𝜉₂=−1onΓ.(3.15) Define the function 𝑌 as the solution of the following boundary-value problem in the cell of periodicity:𝜕Δ𝑌𝑌𝑌==00ionnΠ𝜕⧵𝐵𝐵,,𝜕𝜉₁𝜕𝑌=0on𝜕Π⧵Γ,𝜕𝜉₂𝜕𝑌=0onΓ,𝜕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)=𝜉₂+𝐶(𝐵)+𝑂𝑒^−𝛼𝜉2as𝜉₂⟶∞,(3.17) where𝐶(𝐵)=_Π⧵𝐵||∇𝑌−𝜉₂||²𝑑𝜉+||𝐵||>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,𝜉∈Γ;(3.19) with finite Dirichlet integral in Π. Then there exists the unique weak solution, which has asymptotics 𝑣=𝐶+𝑂(𝑒^−𝛼𝜉2),𝛼>0. To obtain 𝑣 as a function with exponential decay as 𝜉₂→∞, it is necessary and sufficient to have _Π⧵𝐵𝑌𝐹𝑑𝜉+_𝜕𝐵𝐴₁𝜕𝑌𝜕𝜈_𝐵𝑑𝑆_𝐵+_Γ𝐴₂𝑌𝑑𝜉₁=0.(3.20)

Proof. The existence of the solution with asymptotics 𝑣=𝐶+𝑂(𝑒^−𝛼𝜉2) 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 Π_𝑅=Π∩{0<𝜉₂<𝑅} and Γ_𝑅={𝜉∶−𝜋/2<𝜉₁<𝜋/2,𝜉₂=𝑅}. By multiplying the equation from (3.15) by 𝑌, integrating it over Π_𝑅⧵𝐵, and using the property of 𝑌, we get that _{Π𝑅⧵𝐵}𝐹𝑌𝑑𝜉=−_{Π𝑅⧵𝐵}∇𝑣∇𝑌𝑑𝜉+_Γ𝑅𝜕𝑣𝜕𝜉₂𝑌𝑑𝜉₁−_Γ𝜕𝑣𝜕𝜉₂𝑌𝑑𝜉₁=_{Π𝑅⧵𝐵}𝑣Δ𝑌𝑑𝜉−_Γ𝑅𝑣𝜕𝑌𝜕𝜉₂𝑑𝜉₁+_Γ𝑣𝜕𝑌𝜕𝜉₂𝑑𝜉₁−_𝜕𝐵𝑣𝜕𝑌𝜕𝜈𝑑𝑆^𝐵+_Γ𝑅𝜕𝑣𝜕𝜉₂𝑌𝑑𝜉₁−_Γ𝜕𝑣𝜕𝜉₂𝑌𝑑𝜉₁=−_Γ𝑅𝑣𝜕𝑌𝜕𝜉₂𝑑𝜉₁−_𝜕𝐵𝐴₁𝜕𝑌𝜕𝜈𝑑𝑆^𝐵+_Γ𝑅𝜕𝑣𝜕𝜉₂𝑌𝑑𝜉₁−_Γ𝐴₂𝑌𝑑𝜉₁.(3.21) Passing to the limit as 𝑅→∞, we obtain that _Π⧵𝐵𝐹𝑌𝑑𝜉=−𝜋𝐶−_𝜕𝐵𝐴₁𝜕𝑌𝜕𝜈𝑑𝑆^𝐵−_Γ𝐴₂𝑌𝑑𝜉₁.(3.22) This can be rewritten as 𝐶=1𝜋−_Γ𝐴₂𝑌𝑑𝜉₁−_𝜕𝐵𝐴₁𝜕𝑌𝜕𝜈𝑑𝑆^𝐵−_Π⧵𝐵𝐹𝑌𝑑𝜉.(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=−_𝜕𝐵−𝜉₂+𝐾𝜕𝑌𝜕𝜈_𝐵𝑑𝑆_𝐵+_Γ𝑌𝑑𝜉₁,(3.24) where we denote 𝐾=𝜆₁(∫^02𝜋𝛼²⁰𝑑𝜃)⁻¹. However, (3.24) implies that𝐾=_𝜕𝐵𝜉₂𝜕𝑌𝜕𝜈_𝐵𝑑𝑆_𝐵+_Γ𝑌𝑑𝜉₁_𝜕𝐵𝜕𝑌𝜕𝜈_𝐵𝑑𝑆_𝐵⁻¹.(3.25) Integrate the identities 0=∫_{Π𝑅⧵𝐵}Δ𝑌𝑑𝜉,0=∫_{Π𝑅⧵𝐵}𝜉₂Δ𝑌𝑑𝜉: 0=_{Π𝑅⧵𝐵}Δ𝑌𝑑𝜉=_{𝜕Π𝑅⧵𝐵}𝜕𝑌𝜕𝑛𝑑𝑆=_𝜕𝐵𝜕𝑌𝜕𝜈_𝐵𝑑𝑆_𝐵+_Γ𝑅𝜕𝑌𝜕𝜉₂0=_{Π𝑅⧵𝐵}𝑑𝜉₁,𝜉₂Δ𝑌−𝑌Δ𝜉₂𝑑𝜉=_{𝜕Π𝑅⧵𝐵}𝜉²𝜕𝑌=𝜕𝑛−𝑌𝜕𝜕𝜉𝑛²𝑑𝑆_𝜕𝐵𝜉₂𝜕𝑌𝜕𝜈_𝐵𝑑𝑆_𝐵+_Γ𝑌𝑑𝜉₁+_Γ𝑅𝜉²𝜕𝑌𝜕𝜉₂−𝑌𝑑𝜉₁.(3.26)

Passing to the limit as 𝑅→∞, we find that0=_𝜕𝐵𝜕𝑌𝜕𝜈_𝐵0=_𝜕𝐵𝑑𝑆_𝐵+𝜋,𝜉₂𝜕𝑌𝜕𝜈_𝐵𝑑𝑆_𝐵+_Γ𝑌𝑑𝜉₁−𝜋𝐶(𝐵).(3.27) Then (3.25) and (3.27) together with Remark 2.2 imply that𝜆₁=−𝐶(𝐵)^02𝜋𝛼²⁰𝑑𝜃=−2𝜋𝐶(𝐵)𝑘²⁰𝒥^0²𝑘₀<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:𝑢_𝜀(𝑥)=𝑢^ex_𝜀(𝑥)+𝜒(1−𝑟)𝑢ⁱⁿ_𝜀1−𝑟𝜀,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:𝑢^ex_𝜀(𝑥)=𝒥₀(𝑘(𝜀)𝑟),(4.2)𝑢ⁱⁿ_𝜀(𝜉)=^∞_𝑖=1𝜀^𝑖𝑣_𝑖(𝜉).(4.3) Here 𝑘(𝜀)=√𝜆_𝜀,𝑣_𝑖(𝜉) are 𝜋-periodic in 𝜉₁ functions with exponential decay as 𝜉₂→∞. One can easily show that (4.2) solves the equation: −Δ_𝑥𝑢^ex_𝜀(𝑥)=𝜆_𝜀𝑢^ex_𝜀(𝑥)(4.4) if and only if 𝑘(𝜀)=√𝜆_𝜀.

We are looking for 𝑢ⁱⁿ_𝜀(𝜉), which solves the equation:−Δ_𝑥𝑢ⁱⁿ_𝜀(𝜉)=𝜆_𝜀𝑢ⁱⁿ_𝜀(𝜉).(4.5) If (4.4) and (4.5) are satisfied, then 𝑢_𝜀 from (4.1) is the solution of −Δ_𝑥𝑢_𝜀=𝜆_𝜀𝑢_𝜀+𝐹,(4.6) where 𝐹=−𝑢ⁱⁿ_𝜀Δ_𝑥𝜒−2∇_𝑥𝑢ⁱⁿ_𝜀∇_𝑥𝜒. Our aim is to construct 𝑢ⁱⁿ_𝜀 so that 𝐹 will be of small order as 𝜀→0. This is the reason why we need to have 𝑣_𝑖 as exponentially decaying functions.

Now we derive the formula for the Laplacian in 𝜉-variables:Δ_𝑥=𝜕²𝜕𝑟²+1𝑟𝜕𝜕𝑟+𝑟1²𝜕²𝜕𝜃²=1𝜀²𝜕²𝜕𝜉²²+1𝜀𝜀𝜉₂−1𝜕𝜕𝜉₂+1𝜀²𝜀𝜉₂−1²𝜕²𝜕𝜉²¹=1𝜀²Δ_𝜉+1𝜀𝜀𝜉₂−1𝜕𝜕𝜉₂+1𝜀²1𝜀𝜉₂−1²−1𝜕²𝜕𝜉²¹.(4.7)

By substituting the Taylor series for the functions 1𝜀𝜀𝜉₂−1,𝜀1²𝜀𝜉₂1−1²−1(4.8) in (4.7), we get the final formula for Δ_𝑥:Δ_𝑥=1𝜀²Δ_𝜉+^∞_𝑛=0(𝑛+1)𝜀^𝑛−2𝜉^𝑛2𝜕²𝜕𝜉²¹−^∞_𝑛=0𝜀^𝑛−1𝜉^𝑛2𝜕𝜕𝜉₂.(4.9)

Substituting (2.4) and (4.3) in (4.5) and taking into account (4.9), we deduce the following formula:^∞_𝑖=1𝜀^𝑖Δ_𝜉𝑣_𝑖=𝜀+𝜀²𝜉₂+⋯+𝜀^𝑛+1𝜉^𝑛2+⋯^∞_𝑖=1𝜀^𝑖𝜕𝑣_𝑖𝜕𝜉₂−2𝜀𝜉₂+3𝜀²𝜉²²+⋯+(𝑛+1)𝜀^𝑛𝜉^𝑛2+⋯^∞_𝑖=1𝜀^𝑖𝜕²𝑣_𝑖𝜕𝜉²¹−𝜀²𝜆₀+𝜀³𝜆₁+⋯+𝜀^𝑛+2𝜆_𝑛^∞_𝑖=1𝜀^𝑖𝑣_𝑖.(4.10)

By equating terms of the same power of 𝜀, we obtain that𝜀𝜀¹∶Δ_𝜉𝑣₁=0,₂∶Δ_𝜉𝑣₂=𝜕𝑣¹𝜕𝜉₂−2𝜉₂𝜕²𝑣₁𝜕𝜉²¹𝜀⋮,_𝑘,∶Δ_𝜉𝑣_𝑘=^𝑘−1_𝑗=1𝜉^2𝑗−1𝜕𝑣_𝑘−𝑗𝜕𝜉₂−(𝑗+1)𝜉^𝑗2𝜕²𝑣_𝑘−𝑗𝜕𝜉²¹−^𝑘−3_𝑗=0⋮.𝜆_𝑗𝑣_{𝑘−𝑗−2},(4.11)

Consider now the boundary conditions from (2.2). According to the property of 𝜒,𝑢_𝜀(𝑥)=𝑢^ex_𝜀(𝑥)+𝑢ⁱⁿ_𝜀1−𝑟𝜀,𝜃𝜀=𝒥⁰(𝑘(𝜀)𝑟)+^∞_𝑖=1𝜀^𝑖𝑣_𝑖(𝜉),(4.12) in a small neighborhood of 𝜕Ω. Moreover, on 𝜕Ω, it yields that 0=𝜕𝑢^𝜀𝜕𝑟=𝑘(𝜀)𝒥^0(𝑘(𝜀))−^∞_𝑖=1𝜀^𝑖−1𝜕𝑣_𝑖𝜕𝜉₂||||_𝜉2=0.(4.13)

We assume that the function 𝑘(𝜀) has asymptotics:𝑘(𝜀)=𝑘₀+𝜀𝑘₁+⋯+𝜀^𝑛𝑘_𝑛+⋯,(4.14) and since 𝜆_𝜀=𝑘²(𝜀), we can derive the following formulas for 𝜆_𝑖:𝜆₀=𝑘²⁰,𝜆₁=2𝑘₀𝑘₁,…,𝜆_𝑖=^𝑖_𝑗=0𝑘_𝑗𝑘_𝑖−𝑗.(4.15) Rewriting 𝒥^0(𝑘(𝜀)) as a Taylor series with respect to 𝜀, we have𝒥^0(𝑘(𝜀))=𝒥^0𝑘₀+𝒥^0𝑘₀𝑘₁𝜀1!+𝒥^0𝑘₀𝑘²¹+𝒥^0𝑘₀𝑘₂𝜀²2!+⋯.(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 𝑣_𝑖,𝑖=1,2,…:𝜕𝑣_𝑖𝜕𝜉₂=𝑔_𝑖𝑘₁,…,𝑘_𝑖−1onΓ,(4.17) where 𝑔₁=𝑘₀𝒥^0𝑘₀,𝑔₂=𝑘₁𝒥^0𝑘₀+𝑘₀𝑘₁𝒥^0𝑘₀≡0.(4.18)

Consider now the boundary conditions on small holes. Analogously, 𝑢_𝜀(𝑥)=𝒥₀(𝑘(𝜀)𝑟)+^∞_𝑖=1𝜀^𝑖𝑣_𝑖(𝜉)=𝒥₀𝑘(𝜀)1−𝜀𝜉₂+^∞_𝑖=1𝜀^𝑖𝑣_𝑖(𝜉).(4.19)

Substituting the Taylor series for 𝒥₀(𝑘(𝜀)(1−𝜀𝜉₂)) with respect to 𝜀 in the last formula, using (4.14), and equating the terms with the same powers of 𝜀 in equation 𝑢_𝜀=0 on Γ_𝜀, we get the following boundary condition for 𝑣_𝑖,𝑖=1,2,…, on 𝜕𝐵: 𝑣_𝑖=−𝑘_𝑖𝒥^0𝑘₀+𝑓_𝑖𝜉₂;𝑘₀,𝑘₁,…,𝑘_𝑖−1on𝜕𝐵,(4.20) where 𝑓_𝑖 are polynomials of power 𝑖 with respect to 𝜉₂ with coefficients which depend on (𝑘₀,𝑘₁,…,𝑘_𝑖−1). The precise formula for 𝑓_𝑖 can be derived for each fixed 𝑖. For example, we have that𝑓₁=𝑘₀𝒥^0𝑘₀𝜉₂,𝑓₂=𝑘₁𝒥^0𝑘₀𝜉₂−12𝒥^0𝑘₀𝑘₁−𝑘₀𝜉₂².(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 𝑘_𝑖 and 𝜋-periodic in 𝜉₁ functions 𝑣_𝑖 with finite Dirichlet integral in Π and exponential decay as 𝜉₂→∞, such that these functions are solutions of the following boundary-value problems: Δ𝑣_𝑖=𝐹_𝑖≡^𝑖−1_𝑗=1𝜉^2𝑗−1𝜕𝑣_𝑖−𝑗𝜕𝜉₂−(𝑗+1)𝜉^𝑗2𝜕²𝑣_𝑖−𝑗𝜕𝜉²¹−^𝑖−3_𝑗=0𝑣𝜆_𝑗𝑣_{𝑖−𝑗−2}𝑖𝑛Π⧵𝐵,_𝑖=−𝑘_𝑖𝒥^0𝜕𝑣𝑘_0𝑖+𝑓_𝑖𝜉₂;𝑘₀,𝑘₁,…,𝑘_𝑖−1𝑜𝑛𝜕𝐵,𝜕𝜉₂=𝑔_𝑖𝑘₁,…,𝑘_𝑖−1𝑜𝑛Γ.(4.22) Moreover, the constants are defined by the formula: 𝑘_𝑖=−1𝜋𝒥^0𝑘₀^Π⧵𝐵𝑌𝐹_𝑖𝑑𝜉+_𝜕𝐵𝑓_𝑖𝜉₂;𝑘₀,𝑘₁,…,𝑘_𝑖−1𝜕𝑌𝜕𝜈_𝐵𝑑𝑆_𝐵+𝑔_𝑖𝑘₁,…,𝑘_𝑖−1_Γ𝑌𝑑𝜉₁(4.23) In particular, 𝑘₁=−𝜋𝐶(𝐵)𝑘₀𝒥^²⁰𝑘₀,(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=−𝑘₀𝒥^0𝑘₀𝑣(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𝜕𝜉₂−2𝜉₂𝜕²𝑣₁𝜕𝜉²¹=𝜕𝜕𝑣𝜉¹₂+2𝜉₂𝜕²𝑣₁𝜕𝜉²²inΠ⧵𝑣𝐵,₂=−𝑘₂𝒥^0𝑘₀+𝑘₁𝒥^0𝑘₀𝜉₂−12𝒥^0𝑘₀𝑘₁𝜕𝑣₂−𝑘₀𝜉₂²on𝜕𝐵,𝜕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 𝑘_𝑖 are defined by (4.23), we can calculate 𝜆_𝑖 by using (4.15). Denote 𝑢_𝜀,𝑁=𝒥₀𝜆_𝜀,𝑁𝑟+𝜒(1−𝑟)𝑣_𝜀,𝑁,(4.30) where 𝜆_𝜀,𝑁 and 𝑣_𝜀,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 𝑢_𝜀,𝑁 is the solution of the boundary-value problem −Δ𝑢𝑢_{𝜀,𝑁𝜀,𝑁}=𝜆_𝜀,𝑁𝑢_𝜀,𝑁+𝐹_𝜀,𝑁𝑖𝑛Ω_𝜀,𝜕𝑢_𝜀,𝑁=𝑎_𝜀,𝑁(𝜃)𝑜𝑛Γ_𝜀,𝜕𝑟=𝑏^𝜀,𝑁(𝜃)𝑜𝑛𝜕Ω,(4.31) where ‖𝑎_𝜀,𝑁‖_{𝐿2(Γ𝜀)}=𝑂(𝜀^𝑁1),‖𝑏_𝜀,𝑁‖_{𝐿2(𝜕Ω)}=𝑂(𝜀^𝑁1),‖𝐹_𝜀,𝑁‖_{𝐿2(Ω𝜀)}=𝑂(𝜀^𝑁1), and 𝑁₁→∞ as 𝑁→∞.

Proof. According to the definition of 𝑢_𝜀,𝑁, we have that −Δ_𝑥𝑢_𝜀,𝑁=−Δ_𝑥𝒥⁰𝜆_𝜀,𝑁𝑟+𝜒(1−𝑟)𝑣_𝜀,𝑁=−Δ_𝑥𝒥₀𝜆_𝜀,𝑁𝑟−Δ_𝑥𝜒𝑣_𝜀,𝑁−2∇_𝑥𝜒∇_𝑥𝑣_𝜀,𝑁−𝜒Δ_𝑥𝑣_𝜀,𝑁=𝜆_𝜀,𝑁𝒥₀𝜆_𝜀,𝑁𝑟+𝜆_𝜀,𝑁𝜒(1−𝑟)𝑣_𝜀,𝑁−𝜆_𝜀,𝑁𝜒(1−𝑟)𝑣_𝜀,𝑁−Δ_𝑥𝜒𝑣_𝜀,𝑁−2∇_𝑥𝜒∇_𝑥𝑣_𝜀,𝑁−𝜒Δ_𝑥𝑣_𝜀,𝑁=𝜆_𝜀,𝑁𝑢_𝜀,𝑁+𝐹_𝜀,𝑁,(4.32) where 𝐹_𝜀,𝑁=−𝑣_𝜀,𝑁Δ_𝑥𝜒−𝜒𝜆_𝜀,𝑁𝑣_𝜀,𝑁+Δ_𝑥𝑣_𝜀,𝑁−2∇_𝑥𝜒∇_𝑥𝑣_𝜀,𝑁=∶𝐼₁+𝐼₂+𝐼₃.(4.33) Passing from (𝑥₁,𝑥₂) variables to polar coordinates (𝑟,𝜃), we get that 𝜕𝜕𝑥₁=cos𝜃𝜕𝜕𝑟−sin𝑟𝜃𝜕𝜕𝜃,𝜕𝜕𝑥₂=sin𝜃𝜕𝜕𝑟+co𝑟s𝜃𝜕𝜕𝜃,(4.34)Δ_𝑥=1𝑟𝜕𝜕𝑟𝑟𝜕𝜕𝑟+𝑟1²𝜕²𝜕𝜃².(4.35)
By using the fact that lim_𝑥→∞𝑥𝑒^−𝛼𝑥=0 and due to the result of Theorem 4.2, we have that, for any 1≤𝑖≤𝑁, 𝜀^𝑖𝑣_𝑖=𝜀^𝑖𝑂𝑒^−𝛼𝜉2=𝜀^𝑁𝑂𝜀^𝑖−𝑁𝑒^{−𝛼(1−𝑟)/𝜀}=𝜀^𝑁𝑂(𝜀^𝑚)=𝑂𝜀^𝑁+𝑚,(4.36) where 𝑚 is fixed. Hence, 𝑣_𝜀,𝑁=𝑂(𝜀^𝑁+𝑚). Similarly, taking into account (4.34) and (4.35), we can deduce that ∇_𝑥𝑣_𝜀,𝑁=𝑂𝜀^𝑁+𝑚𝛼cos𝜃∇_𝑥𝜀−sin𝑟𝜃,𝛼s𝜀in𝜃+co𝑟s𝜃,𝜒=−cos𝜃𝜒^,−sin𝜃𝜒^.(4.37) Consequently, ∇_𝑥𝑣_𝜀,𝑁∇_𝑥𝜒=𝑂𝜀^𝑁+𝑚𝑂1𝜀𝑟.(4.38) Furthermore, Δ_𝑥𝑣_𝜀,𝑁=−𝛼𝜀𝑟𝑂𝜀^𝑁+𝑚+𝛼𝜀²²𝑂𝜀^𝑁+𝑚+1𝑟²𝑂𝜀^𝑁+𝑚=𝑂1𝜀²𝑟²𝑂𝜀^𝑁+𝑚Δ_𝑥,𝜒=−1𝑟𝜒′+𝜒^=𝑂1𝑟.(4.39)
According to the definition of 𝜒, the support of ∇_𝑥𝜒 and Δ_𝑥𝜒 is the set {1/4≤𝑟≤1/2}. Summarizing, we have that 𝐼₁=𝑂𝜀^𝑁+𝑚𝑂1𝑟,𝐼²=𝑂𝜀^𝑁+𝑚𝑂1𝜀²𝑟²,𝐼³=𝑂𝜀^𝑁+𝑚𝑂1𝜀𝑟,(4.40) and we can derive that ‖‖𝐹_𝜀,𝑁‖‖^2𝐿_{2Ω𝜀}==_Ω𝐹^2𝜀,𝑁𝑟𝑑𝑟𝑑𝜃_{Ω∩{1/4≤𝑟≤1}}=𝑂𝜀_2𝑁+2𝑚𝐼²²𝑟𝑑𝑟𝑑𝜃+_{Ω∩{1/4≤𝑟≤1/2}}𝐼¹+𝐼₃²𝑟+2𝐼₂𝐼₁+𝐼₃𝑟𝑑𝑟𝑑𝜃𝑂1𝜀³+𝑂𝜀^2𝑁+2𝑚𝑂1𝜀⁴=𝑂𝜀^2𝑁+2𝑚𝑂1𝜀⁴.(4.41) Therefore, ‖‖𝐹_𝜀,𝑁‖‖_{𝐿2(Ω𝜀)}=𝑂𝜀^{𝑁+𝑚−2}=𝑂𝜀^𝑁1,𝑁₁⟶∞as𝑁⟶∞.(4.42) Consider now 𝑢_𝜀,𝑁 on Γ_𝜀: 𝑢_𝜀,𝑁=𝒥₀𝜆_𝜀,𝑁𝑟+𝑣_𝜀,𝑁=𝜀^𝑁+1𝛽_𝑁+1+𝜀^𝑁+2𝛽_𝑁+2+⋯,(4.43) where 𝛽_𝑗 are the coefficients of the Taylor series of the function 𝒥₀(√𝜆_𝜀,𝑁𝑟). Hence, 𝑎_𝜀,𝑁=𝑂(𝜀^𝑁+1) and ‖‖𝑎_𝜀,𝑁‖‖^2𝐿_{2Γ𝜀}=2𝜀_𝜕𝐵𝜀𝑎^2𝜀,𝑁𝑑𝜃∼2𝜀2𝜋𝑎𝜀𝑂𝜀^2𝑁+2=𝑂𝜀^2𝑁+2,(4.44) which yields that ‖𝑎_𝜀,𝑁‖_{𝐿2(Γ𝜀)}=𝑂(𝜀^𝑁+1)=𝑂(𝜀^𝑁1),𝑁₁→∞ as 𝑁→∞. Analogously, one can verify that ‖𝑏_𝜀,𝑁‖_{𝐿2(𝜕Ω)}=𝑂(𝜀^𝑁1),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) and𝑢_𝜀(𝑥)=𝑢^ex_𝜀(𝑥)+𝜒(1−𝑟)𝑢ⁱⁿ_𝜀1−𝑟𝑢𝜀,𝜃𝜀,𝜃,(5.1)_ex𝜀𝑢(𝑥)=cos(𝑛𝜃)𝒥_𝑛(𝑘(𝜀)𝑟),(5.2)_in𝜀(𝑥)=cos(𝑛𝜃)^∞_𝑖=1𝜀^𝑖𝑣^even_𝑖(𝜉)+sin(𝑛𝜃)^∞_𝑖=2𝜀^𝑖𝑣^odd_𝑖(𝜉).(5.3)

In this case, 𝑣^even_𝑖=−𝑘_𝑖𝒥^𝑛𝑘₀+𝑓^𝑖(𝑛)𝜉₂;𝑘₀,𝑘₁,…,𝑘_𝑖−1on𝜕𝐵,(5.4) where 𝑓^𝑖(𝑛) are polynomials of power 𝑖 with respect to 𝜉₂ with coefficients which depend on (𝑘₀,𝑘₁,…,𝑘_𝑖−1). Moreover, 𝜕𝑣^even_𝑖𝜕𝜉₂=𝑔^𝑖(𝑛)𝑘₁,…,𝑘_𝑖−1onΓ,(5.5) where𝑓^1(𝑛)=𝑘₀𝒥^𝑛𝑘₀𝜉₂,𝑓^2(𝑛)=𝑘₁𝒥^0𝑘₀𝜉₂−12𝒥^0𝑘₀𝑘₁𝑔_1(𝑛)−𝑘₀𝜉₂²,=𝑘₀𝑣𝒥^𝑛𝑘₀,𝑔^2(𝑛)=𝑘₁𝒥^𝑛𝑘₀+𝑘₀𝑘₁𝒥^𝑛𝑘₀≡0,(5.6)_odd𝑖=0,𝜉∈𝜕𝐵,𝜕𝑣^odd𝑖𝜕𝜉₂=0,𝜉∈Γ.(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 𝑣^even_𝑖 and 𝑣^odd_𝑖: Δ𝑣^even_𝑖=^𝑖−1_𝑗=1𝜉^2𝑗−1𝜕𝑣^even_𝑖−𝑗𝜕𝜉₂−(𝑗+1)𝜉^𝑗2𝜕²𝑣^even_𝑖−𝑗𝜕𝜉²¹−𝑛^𝑖−3_𝑗=0(𝑗+1)𝜉^𝑗2𝜕𝑣^odd_{𝑖−𝑗−1}𝜕𝜉₁−𝑛^2𝑖−3_𝑗=0(𝑗+1)𝜉^𝑗2𝑣^even_{𝑖−𝑗−2}−^𝑖−3_𝑗=0Δ𝑣_odd𝑖𝜆_𝑗𝑣^even_{𝑖−𝑗−2}inΠ⧵𝐵,(5.8)=^𝑖−2_𝑗=1𝜉^2𝑗−1𝜕𝑣^even_𝑖−𝑗𝜕𝜉₂−(𝑗+1)𝜉^𝑗2𝜕²𝑣^odd_𝑖−𝑗𝜕𝜉²¹+𝑛^𝑖−2_𝑗=0(𝑗+1)𝜉^𝑗2𝜕𝑣^even_{𝑖−𝑗−1}𝜕𝜉₁+𝑛^2𝑖−3_𝑗=0(𝑗+1)𝜉^𝑗2𝑣^odd_{𝑖−𝑗−2}−^𝑖−3_𝑗=0𝜆_𝑗𝑣^odd_{𝑖−𝑗−2}inΠ⧵𝐵.(5.9)

Theorem 5.1. There exist numbers 𝑘_𝑖 and 𝜋-periodic in 𝜉₁ even functions 𝑣^even_𝑖 and odd functions 𝑣^odd_𝑖 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 𝑘_𝑖 are defined by the formula: 𝑘_𝑖=−1𝜋𝒥^𝑛𝑘₀^Π⧵𝐵𝑌𝐹_𝑖𝑑𝜉+_𝜕𝐵𝑓^𝑖(𝑛)𝜉₂;𝑘₀,𝑘₁,…,𝑘_𝑖−1𝜕𝑌𝜕𝜈_𝐵𝑑𝑆_𝐵+𝑔^𝑖(𝑛)𝑘₁,…,𝑘_𝑖−1_Γ𝑌𝑑𝜉₁(5,.10)

Proof. The problems (5.8), (5.5), (5.4) for functions 𝑣^even₁,𝑣^even₂ coincide with problems (4.22), (4.17), and (4.20) (if one change 𝒥^0(𝑘₀) by 𝒥^𝑛(𝑘₀) and 𝑓_𝑖,𝑔_𝑖 by the respective 𝑓^𝑖(𝑛),𝑔^𝑖(𝑛)). Therefore the construction of 𝑣^even₁,𝑣^even₂ 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 𝑣^odd₂ is as follows: Δ𝑣^odd₂=𝑛𝜉₂𝜕𝑣^even₁𝜕𝜉₁𝑣inΠ⧵𝐵,_odd2𝜕𝑣_odd2=0on𝜕𝐵,𝜕𝜉₂=0onΓ.(5.11) The function 𝑣^even₁ 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 𝑣^odd₂ of (5.11) with exponential decay. Then we can use the iteration process to complete the proof.

Denote 𝑢_𝜀,𝑁=cos(𝑛𝜃)𝒥₀𝜆_𝜀,𝑁𝑟+𝜒(1−𝑟)𝑣_𝜀,𝑁,(5.12) where 𝜆_𝜀,𝑁 and 𝑣_𝜀,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 𝑢_𝜀,𝑁 is the solution of the boundary-value problem: −𝑢Δ𝑢_{𝜀,𝑁𝜀,𝑁}=𝜆_𝜀,𝑁𝑢_𝜀,𝑁+𝐹_𝜀,𝑁𝑖𝑛Ω_𝜀,𝜕𝑢_𝜀,𝑁=𝑎_𝜀,𝑁(𝜃)cos(𝑛𝜃)𝑜𝑛Γ_𝜀,𝜕𝑟=𝑏^𝜀,𝑁(𝜃)cos(𝑛𝜃)𝑜𝑛𝜕Ω,(5.13) where ‖𝑎_𝜀,𝑁‖_{𝐿2(Γ𝜀)}=𝑂(𝜀^𝑁1),‖𝑏_𝜀,𝑁‖_{𝐿2(𝜕Ω)}=𝑂(𝜀^𝑁1),‖𝐹_𝜀,𝑁‖_{𝐿2(Ω𝜀)}=𝑂(𝜀^𝑁1), 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:−Δ𝑈𝑈_𝜀𝜀=𝜆𝑈_𝜀+𝐹inΩ_𝜀,𝜕𝑈_𝜀=0onΓ_𝜀,𝜕𝑟=0on𝜕Ω,(6.1) where 𝐹∈𝐿₂(Ω) and 𝜆≠𝜆₀ 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 𝑈_𝜀∈𝐻¹(Ω) and the following representation holds: 𝑈_𝜀=𝑢^𝜀𝜆_𝜀−𝜆_Ω𝑢_𝜀𝐹𝑑𝑥+𝑈_𝜀,(6.2) for 𝜆 close to the simple eigenvalue 𝜆₀ of the problem (2.3) and 𝑈_𝜀=1𝜆_𝜀−𝜆²_𝑖=1𝑢^𝑖𝜀_Ω𝑢^𝑖𝜀𝐹𝑑𝑥+𝑈_𝜀,(6.3) for 𝜆 close to multiple eigenvalue 𝜆₀ of the problem (2.3). Here 𝑢_𝜀 is normalized in 𝐿₂(Ω) eigenfunctions to (2.2) and 𝑢^𝑖𝜀 is orthonormalized in 𝐿₂(Ω) eigenfunctions to (2.2). Moreover, ‖‖𝑈_𝜀‖‖_𝐻1≤𝐶‖𝐹‖_𝐿2,(6.4) where the constant 𝐶 is independent on 𝜀 and 𝜆. It follows from (6.2) and (6.4) that ‖‖𝑈_𝜀‖‖_𝐻1≤𝐶𝜆_𝜀−𝜆‖𝐹‖^𝐿2.(6.5) Consider now the case of simple 𝜆₀. Define the function: 𝑈^𝑁𝜀(𝑥)=1+1𝜀𝑢^𝜀,𝑁(𝑥)−1+1𝜀𝑎^𝜀,𝑁+𝑏_𝜀,𝑁𝜒(1−𝑟)𝑣(𝜉)+𝜉₂+𝐶(𝐵),(6.6) where 𝑢_𝜀,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, 𝑈^𝑁𝜀 is the solution of (6.1) if 𝜆=𝜆_𝜀,𝑁,‖𝐹‖_𝐿2=𝑂𝜀^𝑁2,𝑁₂⟶∞as𝑁⟶∞.(6.7) Taking into account (6.5), (6.7), and the fact that ‖𝑈_𝜀‖_𝐻¹<∞, we can conclude that for each fixed 𝑁, 𝜆_𝜀−𝜆_𝜀,𝑁=𝑂𝜀^𝑁2=𝑜𝜀^𝑁as𝜀⟶0.(6.8) Therefore the asymptotics constructed in Section 4 coincide with the expansion of 𝜆_𝜀. 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 𝜆_𝜀 are completely verified.

7. Application to a Friedrichs-Type Inequality

Consider the sets Ω_𝜀,Γ_𝜀, which were defined in Section 2.

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

Theorem 7.2. Let 𝑢∈𝐻¹(Ω_𝜀,Γ_𝜀). Then a Friedrichs-type inequality _Ω𝜀𝑢²(𝑥)𝑑𝑥≤𝐾_𝜀_Ω𝜀||∇𝑢(𝑥)||²𝑑𝑥(7.1) holds, where the best constant 𝐾_𝜀 has the asymptotics 𝐾_𝜀=1𝑘²⁰+4𝜋𝐶(𝐵)𝒥^0²𝑘₀𝑘²⁰𝜀+𝑜(𝜀),(7.2) as 𝜀→0. 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)𝑑𝑥≤𝐾_Ω𝜀||∇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 𝜀→0. Clearly, the best constant 𝐾_𝜀=1/𝜆^1𝜀, where 𝜆^1𝜀 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 𝐾_𝜀: 𝐾_𝜀=𝜆^1𝜀⁻¹=𝜆¹⁰+𝜀𝜆¹¹+𝑜(𝜀)⁻¹=1𝜆¹⁰−2𝜆¹¹𝜆¹⁰²𝜀+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), 𝐾_𝜀=1𝑘²⁰+4𝜋𝐶(𝐵)𝒥^0²𝑘₀𝑘²⁰𝜀+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.