On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary

We construct two-term asymptotics λk = ε m−2(M + εμk + O(ε )) of eigenvalues of a mixed boundary-value problem in Ω ⊂ R with many heavy (m > 2) concentrated masses near a straight part Γ of the boundary ∂Ω. ε is a small positive parameter related to size and periodicity of the masses; k ∈ N. The main term M > 0 is common for all eigenvalues but the correction terms μk, which are eigenvalues of a limit problem with the spectral Steklov boundary conditions on Γ, exhibit the effect of asymptotic splitting in the eigenvalue sequence enabling the detection of asymptotic forms of eigenfunctions. The justification scheme implies isolating and purifying singularities of eigenfunctions and leads to a new spectral problem in weighed spaces with a “strongly” singular weight.


Introduction and setting of the problem
In this paper we introduce unaccustomed splitting asymptotic procedure for eigenvalues, purifying singularities of eigenfunctions. The spectral problem under consideration is a mixed boundary value problem for the Laplace operator in a domain Ω ⊂ R 2 with heavy concentrated masses periodically distributed along a straight part Γ of the boundary. The density of the concentrated masses is of order ε −m , with m > 2, and the period is ε ≪ 1. This problem has remained unsolved for a long time and has led to results that are currently the subject of discussion (cf. Remark 1.1 and Section 1.3). Moreover, numerical computations also fail: we refer to [4] for instability effects when approaching numerically the principal mode in close-range problems and for further references. A primary asymptotic analysis (cf. [37]) shows that all the eigenvalues in the low-frequency range of the spectrum have the same main asymptotic term which does not provide a characterization of the corresponding eigenfunctions. We construct the two-term asymptotics, that is, the main term and the first correction term, which gives a much more precise information on the behavior of the eigenvalues as ε → 0 + and allows us to describe the asymptotic structure of the corresponding eigenfunctions which exhibit a strongly oscillatory character.
The setting of the problem and some background are outlined in Sections 1.1-1.3 while Section 1.4 of this introduction summarizes the structure of the paper. We emphasize that the strong oscillations of the eigenfunctions detected in this paper along with the singularities coming from the boundary conditions make it difficult to obtain the convergence results. As a consequence, we claim that all the results and proofs that we present are necessary to show the approach of the eigenvalues, and of the eigenfunctions in the natural space of the setting of the problem: cf. the simple statements of Theorems 6.1 and 6.4.
In Ω we consider the eigenvalue problem

4)
∂ ν u ε = 0 on Γ \ T ε , (1.5) where ∂ ν is the directional derivative along the outward normal, ∂ ν = − ∂ ∂x 2 on Γ, and χ ε is the characteristic function of the set Θ ε , see (1.1), χ ε (x) = 1 for x ∈ Θ ε and χ ε (x) = 0 for x ̸ ∈ Θ ε . Finally, m ∈ R is a positive number and in the sequel we assume m > 2. (1.7) The singularly perturbed problem (1.3)-(1.5) can be associated with time dependent harmonic oscillations of a membrane which is fixed over its sides Γ ⊓ and is clamped by a periodic set of small "clips" θ ε n , n ∈ Z. The flexibility of the clips is the same as that of the membrane material but, in view of our assumption (1.7), the weight of each θ ε n is much greater than that of the whole membrane. In other words, θ ε −N , . . . , θ ε N are heavy concentrated masses distributed periodically at the lower flat part Γ of the boundary ∂Ω and they are fixed over their sides τ ε n . Restriction (1.7) and a special shape of the domains are chosen to reduce the required technicalities to the necessary minimum while preserving all disclosed effects (cf. Remark 3.5).
In Section 3, we will construct the two-term asymptotics Λ ε k = M + εµ k + . . . (1.16) of the rescaled eigenvalues (1.11) and demonstrate that the number µ k in (1.16) and the function u k appearing in the asymptotic form for the eigenfunction (cf. Section 3.5) u ε k (x) = W (ε −1 x)u k (x) + . . . (1.17) are an eigenpair of the Steklov spectral problem −∆u = 0 in Ω, (1.18) − ∂u ∂x 2 = µbu on Γ, (1.20) which involves a well determined coefficient b > 0 (cf. (2.11) and (3.15)). In (1.17), W stands for the normalized eigenfunction associated with the principal eigenvalue M of (1.13)- (1.15) in the half-strip ϖ that we extend by periodicity in the x 1 -direction. The type of asymptotic expansions (1. 16) and (1.17) have been announced without proofs in [32]. This, together with estimates of the asymptotic remainders, which for simplicity we denote in this section by ellipsis points, will be proved and evaluated further in Theorems 6.1 and 6.4. In addition, a new focus of the justification scheme provides a powerful and novel contribution in our paper.
One of the greatest difficulties faced by the authors is in the verification of the uniform boundedness for the L 2 (Ω)-norm of the gradient of the fractional function u ε (1.21) due to the singularities of both eigenfunctions W and u ε k at the collision points. Nevertheless, in Section 2.4 we will prove that u ε k is sufficiently smooth: in particular, it is continuously differentiable at these points.

State-of-the-art in the literature and new challenges
The problem under consideration in this paper belongs to a series of problems known with the name of "vibrating systems with concentrated masses" in the literature of applied mathematics (cf. [39,35,25,14] for the first works). Further specifying, we deal with very many concentrated masses near the boundary and strongly alternating boundary conditions. In this framework we refer to [16] and [14] for problems in two and three dimensional domains respectively, the size of the masses being much smaller than the period of the structure. Different relations between sizes of masses and distance between them (cf. terms such as critical sizes and extreme relations), and different values of the parameter m have been considered in [14,16] where the authors were concerned with the localization of eigenvalues giving rise to local vibrations of the concentrated masses or global vibrations of the system; also many questions were formulated in these papers which have been partially solved in further publications such as [15,17,18,36,38]: see [18,38] for a long list of references on the subject.
The terms light or heavy concentrated masses were introduced in the literature to distinguish between the different ranges of the parameter m, namely m ∈ (0, 2) or m > 2 since the asymptotic behavior of the eigenvalues is qualitatively different for m in one of these ranges or m = 2. All the cases have been considered in the above mentioned papers but the structure of the eigenfunctions associated with the very low frequencies have remained as open questions in the case where m > 2, namely, associated with the eigenvalues λ ε i = O(ε m−2 ), for fixed i = 1, 2, · · · (cf. (1.11) and (1.12)). Even the determination of the structure of the first eigenmode of vibration was an open problem. The same can be said for m = 2 and the low frequencies which in this case read λ ε i = O(1). In fact, we note that for the precise distribution of masses (1.1), size of masses O(ε) and boundary conditions (1.4) and (1.5), the cases where m ≤ 2 have not been considered in the literature. However, when m < 2 (light concentrated masses), the limit problem for the eigenpairs of (1.1)-(1.5) is the Dirichlet eigenvalue problem in Ω. This can be easily proved by standard homogenization techniques: cf. closer results for different problems in [5,16,32]. In contrast, when m = 2, to determine the asymptotic forms of the eigenfunctions remains an open problem to be considered. Now the limit points of the eigenvalues in the sequence (1.9) can be the eigenvalues the Dirichlet problem in Ω, {β j } ∞ j=1 , and the eigenvalues of the cell problem in ϖ. For fixed k, the convergence of λ ε k , as ε → 0, depends strongly on the position of M , cf. (1.13)-(1.15), in the sequence and on the possibility for M to coincide with some β j : we refer to [32] for asymptotic expansions in a close problem. Also the dimension n = 3 remains under examination, cf. [14] in the case where the size of the masses is much smaller than the period. Light and heavy concentrated masses with different boundary conditions, and low and high frequencies, have also been considered in [14,15,16,17,36,38] for dimensions 2 and 3 of the space, the boundary of the masses touching ∂Ω. In this respect, we mention [32] for a different geometry of the masses θ ε which do not touch ∂Ω and for very different boundary conditions on ∂Ω. In spite of this, there is a big gap in the research on these kinds of problems since the structure of the first eigenmode of vibration has only been glimpsed by means of asymptotic expansions in [32]. However, the existing results in the literature (cf. [18,36,37] and references therein) allow us to obtain information on the structures of certain eigenfunctions associated with eigenvalues λ ε i(ε) = ε m−2 M j + · · · , when j > 1 and M j is an eigenvalue of the cell problem (1.13)-(1.15) in the sequence (2.6), but i(ε) cannot be fixed, (see (1.16) to compare), it converges towards ∞ as ε → 0. In addition, the structure of the corresponding eigenfunctions is described by quasimodes which approach "groups of eigenfunctions" (cf. Lemma 4.1). Obviously, these results deal with the highfrequency range, and they complement those in this paper, which becomes essential for the description of asymptotics of eigenpairs in the low-frequency range.
As outlined in previous works (cf., e.g., [13,17,18]) when searching for eigenvalues giving rise to certain kinds of vibrations, the question of the normalization of the corresponding eigenfunctions is crucial. In addition, here we need a reformulation of the problem in weighted Sobolev spaces and a thorough analysis of solutions near the points where the strongly alternating boundary conditions change, the so-called collision points (cf. Lemmas 2.1, 2.4 and Proposition A.1). We note that in the asymptotics for eigenfunctions (1.17), fast and slow variables are involved together, and the function u k (which corresponds to µ k , the second term of the asymptotic expansion of the rescaled eigenvalues (1.16)) act as an envelope for the fast oscillations of the eigenfunction u ε k of our spectral problem (1.8). It should also be noted that the factorization principle here used has been detected in the literature of homogenization problems: we refer to [40] for the first work, related with perforated media, where weighted Sobolev spaces are also used. However, here we deal with a very singular "weight" W (ε −1 x) near the boundary and we need to obtain smoothness properties for u ε (x)W (ε −1 x) −1 in the neighborhoods of the points of ∂Ω where both functions u ε and W vanish. We use a technique of localization for u ε (x)W (ε −1 x) −1 near the concentrated masses (cf. figure 3) which allows us to derive its convergence in H 1 (Ω)-weak, avoiding approaches with norms in the weighted Sovolev spaces. As a consequence, we obtain sharp bounds for convergence rates of eigenpairs (cf. also Remark 6.2) and, what is very important, this approach is obtained in the norms of the space of the setting of the original problem (cf. (6.1) and (6.5)).
In this connection, it may be worthy mentioning that [22] considers an eigenvalue problem in a planar domain of the dense-comb type which differs from our problem (1.3)-(1.5) both in the geometry and distribution of the heavy masses, and in the boundary conditions, but meets technical difficulties similar to those mentioned in Remark 1.1. However, an analysis of the singularities of the eigenfunctions at the corner points (cf. [10]) and the proof of the equivalence of problems outlined in Remark 1.1 are absent.
Also, it is worthy mentioning that other very different problems in the framework of the vibrating systems with many concentrated masses have been considered in the literature recently. Let us mention, e.g., [28,30], where asymptotics for eigenvalues are described by means of the spectrum of a certain integral (elliptic pseudo-differential) operator on the torus axis. The geometry and the justification schemes in these papers differ in all aspects from the content of our present paper.
Finally, notice that the Steklov problem (1.18)-(1.20) appears here associated with the second order approach of the eigenvalues. Let us mention references [1,7,8,9,12,14] which make it clear how Steklov type boundary conditions can appear associated with the first order approach of eigenvalues of singularly perturbed spectral problems which present a high mass concentration along a part of the boundary or at points along the boundary. See [6] for further recent bibliography on Steklov problems.

Structure of the paper
The organization of the paper is marked by the asymptotics (1.16) and (1.17) of the eigenpairs of (1.3)-(1.5), and by the tools which we need to justify these asymptotics. We gather the final results in Theorems 6.1 and 6.4 in a simplified way, while other important results appear throughout the paper.
Taking into account that the first term M in the asymptotics (1.16) and that the multiplying function W in (1.17) turn out to be the dominant eigenpair of the cell problem (2.2), Section 2 contains the setting of this problem (the first limit problem) along with a detailed study of the dominant eigenmode: properties which allow us to show that u ε k (x)W (ε −1 x) −1 belongs to H 1 (Ω) (cf. (2.31)). This seems to be consistent with (1.17) (u k ∈ H 1 (Ω)) but requires some smoothness results for solutions of boundary value problems near corners (cf. [10] and Ch. 2 in [31] for the general theory), namely, in our case near the points where the boundary condition changes from Dirichlet to Neumann or viceversa. Appendix complements these smoothness results.
Section 3 contains asymptotic expansions for the eigenpairs (λ ε k , u ε k ) of (1.3)-(1.5), and the Steklov spectral problem (3.17) (the second limit problem ). The compound asymptotic expansion for the eigenfunctions (3.2) includes terms of the outer expansion and boundary layer functions whose properties prove essential for these justifications. These expansions and functions are in Sections 3.1-3.4 (cf. also Remark 3.2).
Sections 4-6 contain justifications of asymptotic expansions providing precise bounds for convergence rates in terms of the eigenvalue number.
The hard computations in Section 4 rely on results about "near eigenvalues and eigenfunctions" from the spectral perturbation theory (cf. Lemma 4.1). The main result (cf. Theorem 4.2) establishes that for each eigenvalue µ k of (3.17), µ k of multiplicity κ k , and for sufficiently small ε, there are at least κ k eigenvalues of λ ε j satisfying with a constant c k independent of ε. The result already improves the convergence (1.12). In addition, in Section 4, a certain approach to the eigenfunctions is stated (cf. (4.29)), which is provided by the asymptotic expansions constructed in Section 3 (cf. (3.2), (3.6)), with the boundary layer functions suitably adapted in such a way that the new function (4.10) belongs to the same space of definition of the eigenfunctions u ε k . Nevertheless, these approaches do not provide the convergence expected (cf. (1.11) and (1.16)).
To obtain this convergence, which in some way implies k = j in (1.22), we need to reformulate the original spectral problem (1.8) by introducing a new spectral parameter and a corresponding eigenfunction as follows: (1.23) These pairs (µ ε , u ε ) prove to be eigenpairs of a new spectral problem (5.2) which is formulated in the suitable weighted Sobolev spaces and it turns out to have a dicrete spectrum (cf. Proposition 5.1). Theorem 5.6 states the result of convergence of the renormalized eigenpairs (ε −1 µ ε , u ε ) towards those of the second limit problem (3.17). The spectral convergence for (5.2) holds with conservation of the multiplicity (see (5.51) and (5.53)), and to derive the convergence in a space independent of ε we use a technique that allows a localization of the new eigenfunctions u ε in small teeth near the concentrated masses (see figure 3). This is done in Section 5.2, the main results of this section being summarized in Proposition 5.5. Finally, we need to combine the partial results in Sections 4 and 5 to derive the desired approach for eigenvalues and eigenfunctions of the original problem along with precise bounds for convergence rates in terms of the eigenvalue number k and the perturbation parameter ε (cf. (6.1) and (6.5)): j becomes k, k + 1, · · · κ k − 1 in (1.22).

The first limit problem in the cell
This section is devoted to the cell problem, namely, the limit problem involved with the first term M of the asymptotics for the eigenvalues (1.16) and the properties of the corresponding eigenfunction W (cf. Sections 2.2-2.3). Some of these properties deal with the required smoothness for (1.23) (cf. Section 2.4) that we need in Sections 4-6.
This limit problem is (1.13)-(1.15), where we note that we have used M to denote the spectral parameter, and X the characteristic function of the set θ. The variational formulation of problem (1.13)-(1.15) reads: to find a number M and non-trivial function W ∈ H such that (2. 2) The Hilbert space H in the integral identity (2.2) is determined by completing the linear space C ∞ c,per (ϖ, τ ) of infinitely differentiable functions, vanishing on τ , with compact support and 1-periodics in ξ 1 , with respect to the norm ∥∇ ξ V ; L 2 (ϖ)∥. It consists of functions in H 1 loc (ϖ) which have a finite gradient norm and satisfy the stable boundary conditions, namely the first relations in (1.14) and (1.15). Since the Dirichlet condition on τ is included in the space, the classical one-dimensional Hardy inequality proves that the norm introduced in H is equivalent to the following one: Indeed, the above-mentioned Hardy inequality dt, (2.4) in particular, requires V(0) = 0 (to verify (2.4) make the change y → t = 1/y in (5.10)).
To fulfil this condition, we multiply V with the cut-off function ξ → X (ξ 2 ), and apply the Friedrichs inequality which is valid due to the Dirichlet condition V = 0 on τ . Setting V(t) = X (t)V (ξ 1 , t) in (2.4), we integrate the obtained inequality in ξ 1 ∈ (−l, l), take into account (2.5) to estimate the L 2 -norm of the last term in the formula X ∇ ξ V = ∇ ξ V − V ∂X /∂ξ 2 , and finally observe that ξ −2 As a result, the weighted Lebesgue norm in (2.3) is bounded by the gradient norm, and this shows the mentioned equivalence of norms.
Owing to the compact embedding H ⊂ L 2 (θ), problem (2.2) possesses the unbounded positive increasing sequence of eigenvalues where eigenvalues are repeated according to their multiplicities. We also choose the corresponding eigenfunctions W k ∈ H satisfying the orthogonality and normalization conditions In what follows we address only the principal eigenpair {M 1 , W 1 } of the problem and omit the subscript 1 in the notation. Due to the strong maximum principle, the eigenvalue M = M 1 is simple and the eigenfunction W = W 1 can be chosen positive in ϖ \ τ .

Properties of the principal eigenfunction
The 1-periodic function W is a harmonic function in ϖ \ θ and becomes infinitely differentiable outside the set θ as well as inside θ where X = 1. Hence, the Fourier series (2.8) converges for ξ 2 > h with any fixed h, there are constants c q,h such that there hold the estimates To compute the constant B, we insert the functions W and ξ 2 into the Green formula on the rectangle ϖ(R) = (−1/2, 1/2) × (0, R) (cf. (1.13)), and we take limits as R → +∞, to obtain Thus, where the strict inequality is inherited from the positivity of W and the last integral can be reduced to (−1/2, −l) ∪ (l, 1/2).

Asymptotics near collision points
As was mentioned in Section 2.2, the eigenfunction W is smooth everywhere, except at the curve τ ⊓ , in particular, at the points P ± = (±l, 0) where the Dirichlet and Neumann conditions (1.14) meet each other. Here, we show that the latter brings the worst singularity O ( dist (ξ, P ± ) 1/2 ) into W (ξ) and leads W out from the space H 2 loc (ϖ). At the same time, jumps of the second-order derivatives at ∂θ \ τ keep W in H 2 loc (ϖ \ (P − ∪ P + )). Our justification scheme relies upon asymptotic formulas for W near the collision points P ± that are obtained below by means of the Kondratiev theory [10]; also, the necessary information about behavior of solutions to the Poisson equation near corner and collision points can be found, e.g., in Ch. 2 in [31].
We need the polar coordinate systems (ρ ± , φ ± ) ∈ R × [0, 2π] centered at P ± , see figure  2, and a cut-off function ς ∈ C ∞ [0, +∞) such that Lemma 2.1. Let W be the principal mode of (2.2). There holds the decomposition where K ± and K 1 ± are some coefficients, the remainder W satisfies the estimates

14)
with ρ(ξ) = min{ρ ± }, and h > 0 can be taken to be arbitrary but the constants c h p depend on h.
Proof. The detached terms in (2.13) are obtained by the Fourier method which, owing to the separation of variables in the Laplace equation, provides the problem (see figure 2 for the orientation of the angular variables φ ± ) for the exponent κ and the angular part W κ ± in the harmonics Solving (2.15), we get 17) and include in the asymptotic form (2.13) only functions (2.16) with j = 0 and j = 1. However, these functions bring a discrepancy O(ρ 1/2 ± ) into equation (2.13) because its right-hand side M X(ξ)W (ξ) has been ignored in the above consideration as a lower-order term. According to the general procedure in [10], see also § 3.3 in [31] and § 3.5 in [31], the main part of the discrepancy is compensated by the next term in the decomposition of W , which must be searched in the form ) . (2.18) Inserting (2.18) into problem (1.13), (1.14) and collecting expressions of order ρ Since the homogeneous problem (2.19) has the unique eigenfunction cos(5φ/2) (up to a multiplicative constant), the Fredholm alternative gives the compatibility condition (2.20) The formal procedure performed above was worked out and justified in [10] where estimates of remainders are derived in weighted Sobolev norms. Estimates in weighted Hölder norms are obtained in [20] (see also § 3.6 in [31]). We apply these estimates and then join the detected terms (2.18) to the remainder W (ξ). Since we will prove below that K ± ̸ = 0 (cf. (2.21)), the coefficients (2.20) do not vanish and, therefore, the bound in (2.14) is optimal, and this ends the proof of the lemma.
It should be noted that the results in Lemma 2.1 hold for any eigenfunction of (2.2), and the coefficients K ± , K 1 ± in (2.13) and K 2 ± in (2.18) depend on the whole data in problem (1.13), (1.14). However, for the principal eigenfunction, a method in [21] (see also Ch. 2 in [31]) establishes an integral representation of K ± which guarantees the above-mentioned inequality (2.21).

Lemma 2.2.
Under the hypotheses of Lemma 2.1, the coefficients in (2.13) satisfy cos φ ± 2 and the eigenfunction W into the Green formula on the semi-annulus In these calculations, we have used the exact formula for Z ± and the asymptotic decomposition (2.13) of W while computing the last limit as δ → 0 + . Thus, we can write Then, inequality (2.21) follows from the relations and the lemma is proved.
Formulas (2.21), (2.11) together with the consequence of the strong maximum principle, help us to study the behavior of the eigenfunction W in the whole domain ϖ, obtaining the following result.
Note that, obviously, inequalities (2.24) and (2.25) hold for ξ 1 in other larger intervals which do not contain the collisions points; the constants arising in the inequalities depend on the endpoints of these intervals.

Analysis of eigenfunctions in the original problem
The considerations in Section 2.3 can be applied to problem (1.3)-(1.5) for an examination of its eigenfunctions in the vicinity of the collision points P ± (nε) = (εn ± εl, 0), n ∈ Z(N ). The polar coordinates systems centered at these points are denoted by (r n± , φ n± ) and we can state the following result.
For further use, in Section 5, we provide with some results on the fractional function where the weight multiplier W ε (x) = W (ε −1 x) is the principal eigenfunction of problem (1.13)-(1.15) written in the rapid variables (2.1) and extended periodically over the halfplane R 2 + . Function (2.29) still belongs to H 1 (Ω) because in the vicinity of each collision point, the numerator and denominator in (2.29) have very similar asymptotic forms (2.27) and (2.13), respectively. Indeed, for a small r n± , we obtain with the smooth trigonometric function and the remainder u ε k (x) having a faster decay as r n± → 0 + . Thus, function (2.29) as well as its first-order derivatives, are bounded at the collision points.
Inside smooth, actually flat, parts of the base Γ with either Dirichlet (1.4), or Neumann (1.5) conditions, both u ε k and W are smooth while u ε k (x 1 , 0) = 0 for |x 1 − εn| < l and W enjoys properties (2.24)-(2.26). These properties, together with (2.30), demonstrate that u ε k falls into the Hölder class C 0,α near the base with any α ∈ (0, 1). At the same time, according to square-root singularities, cf. (2.27), the function u ε k belongs to the class C 0,α under the restriction α < 1/2 only. In other words, fraction (2.29) achieves much better differential properties than the eigenfunction u ε k itself. Consequently, from Lemmas 2.1, 2.2 and 2.4 and Corollary 2.3, we have proved the following result.
Proposition 2.5. The function u ε k defined by (2.29) belongs to C(Ω), and It proves useful to comment several points of the above considerations. First, the function W ε is positive and differentiable outside a neighborhood of ∂θ ε so that a "bad" behavior of u ε k at corner points of the arc Γ ⊓ is not able to disturb the confirmed inclusion (2.31).
Second, the factors r bring into the calculation singularities at r n± = 0 and φ n± = π, respectively. The radial singularity is readily compensated by infinitesimal bounds in estimates (2.28) and (2.14) but the angular singularity O(|φ n± − π| −1 ) requires further discussion, see Appendix.
Finally, we emphasize that the coefficients K ε kn± and K 1ε kn± in (2.27), the bound in (2.28) and other characteristics of u ε k and u ε k depend on the small parameter ε, and the derivation in Section 5.2 of the estimate for the norm ∥∇u ε k ; L 2 (Ω)∥, uniformly in ε ∈ (0, ε 0 ], turns out to be the most intriguing issue in the paper.

Two scales asymptotic expansion and related issues
In this section, we obtain the first two terms of the asymptotic expansion for the eigenvalues of problem (1.3)-(1.5) (with variational formulation (1.8)) and the composite asymptotic expansion for the corresponding eigenfunctions. We determine the terms arising in these expansions from the eigenpairs of two spectral problems posed either in Ω (cf. problem (1.18)-(1.20) and Section 3.4) or in the half-strip ϖ (cf. problem (2.2) and Sections 3.2-3.3). We show that both slow and rapid variables are essential to define the first term of the asymptotic expansions for eigenfunctions (cf. Section 3.5 for dominant terms). Section 3.6 contains a two-scale convergence result.

The second limit problem: a problem in Ω
We introduce the following asymptotic ansätze for an eigenpair of the singularly perturbed where the dots stand for lower-order terms of the approximations , M = M 1 is the principal eigenvalue of the first limit problem (1.13), (1.14), (1.15), and u k , u ′ k are terms of of the regular asymptotic expansion (see Section 3.4 for regularity results). Moreover, ς 0 is a smooth cut-off function such that and w k , w ′ k are boundary layer terms, namely periodic functions in the half-strip ϖ with an exponential decay at infinity.
Here, and in Sections 3.2-3.3, we successively determine the asymptotic terms in (3.2) and derive the second limit problem which reads (1.18)-(1.20), involving a coefficient b > 0 and the correction term µ = µ k in (3.1).
Firstly, we note that the Laplace equation (1.18) asymptotically follows from the differential equation (1.3) because the parameter (3.1) is infinitesimal and the support of the function with the big coefficient λ ε ε −m χ ε is located in the cε-neighborhood of Γ, and hence, it does not appear disappear in Ω when ε → 0. The Dirichlet condition (1.19) is directly inherited from the boundary condition (1.4) on Γ ⊓ . The Steklov spectral condition (1.20) will be found by examining the natural decay property of the boundary layer terms.

The first term of the boundary layer
In this and the next section we omit the subscript k in the notation, cf. (1.18)-(1.20).
We insert ansätze (3.2) and (3.1) into problem (1.3)-(1.5), consider the rapid variables (2.1) and apply the obvious formulas and we gather the coefficients of the same powers of the small parameter ε, then, we arrive at the problem Notice that the variable x 1 ∈ (−1, 1) remains as a parameter in this problem.
Evidently, a solution of (3.5), with the exponential decay as ξ 2 → +∞ takes the form where W = W 1 is the principal eigenfunction of the first limit problem (1.13)-(1.15) (cf. norm (2.7)) and B, the first coefficient of the Fourier series (2.8), is defined by (2.11) in its representation (2.8).
Since M is a simple eigenvalue, the Fredholm alternative brings the only compatibility condition in problem (3.10) Assuming that (3.11) is satisfied, one solution w ′ ∈ H is defined up to an additive function , we deduce that a particular solution w ′ 0 of (3.10) admits the representation with an exponentially decaying remainder w ′ 0 (x 1 , ξ) and with B ′ 0 (x 1 ) the constant function in the ξ variable describing the behaviour of yields the unique solution of (3.7) with the exponential decay Consequently, it suffices to guarantee condition (3.11) for the above defined data f ′ and g ′ . Let us examine this condition in further detail. First of all, according to the 1-periodicity of W (ξ) in ξ 1 and formula (3.9), we have Then, recalling (2.11), we obtain Finally, the relation (3.6) together with the normalization condition (2.7) yield where the functions w ′ q are certain 1-periodic functions in ξ 1 which are smooth everywhere in ϖ, except at the arc τ ⊓ and the collision points P ± where, respectively, jumps of second derivatives and singularities O(ρ 1/2 ± ) occur. The assertion on the formula (3.16) is due to the form of the nonhomogeneus term f ′ (see (3.8) and (3.9)); here u reads u k in the case where µ = µ k in the sequence (3.18).
gives the solution of (3.7) that we have obtained above (cf. (3.12)) and this completely solves the problem (3.7). Also, note that we have defined the second term w ′ (x 1 , ξ) of the boundary layer type without imposing any condition on the second term u ′ (x) of the regular type; hence, we could put u ′ = 0 in ansatz (3.2). Nevertheless, we note that the term u ′ together with the replacement in the boundary layer term, are needed to determine lower-order terms which are, however, omitted in our present study.

Eigenpairs of the second limit problem: the Steklov problem
The variational formulation of problem (1.18)-(1.20) reads: to find a number µ and a non- Here, H 1 0 (Ω; Γ ⊓ ) is the Sobolev space of functions vanishing at the arc Γ ⊓ . Since the trace operator: H 1 (Ω) → L 2 (Γ) is compact, the following assertion becomes evident.
which repeat according to their multiplicities. The corresponding eigenfunctions u k ∈ H 1 0 (Ω; Γ ⊓ ) can be subject to the orthogonality and normalization conditions The eigenfunction u k ∈ H 1 0 (Ω; Γ ⊓ ) has additional smoothness near the base Γ in spite of the corner points Q ± = (±L, 0) where the Dirichlet and Steklov conditions meet each other. Indeed, these corners have the angle π/2, and applying the Kondratiev theory [10] again and performing a simple calculation (cf. Ch. 2 in [31]), one may verify the representation ) .
In this way, in a neighborhood of Γ the function u k falls into the classes H 4 and C 3,α with any α ∈ (0, 1). At the same time, we have that u k ∈ H 4 (Γ).

Transforming the asymptotic expansions of eigenfunctions
Ansatz (3.2) with boundary layer terms proves to be convenient in Section 4 for an estimation of asymptotic remainders in (3.1) and (3.2). However, in order to highlight our approach in Section 6.2, we rewrite the main asymptotic term of the eigenfunction u ε k in a different form.
Using formula (3.6) for the boundary layer term w k (x 1 , ξ) in (3.2), we have (3.20) and, therefore (see (2.10)), the last product in (3.20) can be bounded by the infinitesimal value cε everywhere in Ω.
It should be emphasized that representation (3.20) along with the above-estimates was the main reason to introduce the asymptotic ansatz (1.17) in Section 1.2 and to consider the quotient function (2.29) in Sections 2.4 and 5.1.

A two scale convergence result
For convenience, we introduce the following result which provides bounds for convergence rates of 1-periodic functions when they satisfy a certain exponential decay in the ξ 2 direction.
Proposition 3.4. Assume that z ∈ H 1 (Ω), and Z ∈ L 2 (ϖ) is a function which is extended 1-periodically in ξ 1 over the half-plane R 2 + and has the exponential decay as ξ 2 → +∞, namely ∥e βξ2 Z; L 2 (ϖ)∥ < ∞ with some β > 0. Then Proof. Due to the exponential decay of Z, we can restrict the first integral in (3.21) on the rectangle Ω 0 = (−1, 1) × (0, l 0 ) ⊂ Ω with some fixed l 0 ∈ (0, min{l ± , d}], the committed error being exponentially small in ε. We have and, furthermore, Making the coordinate change x → ξ, it can be easily seen that the expression I ε 2 satisfies Another expression will be estimated by means of the Cauchy inequality Indeed, recalling the Slobodetskii norm in the trace space H 1/2 (Γ) for H 1 (Ω), and applying the Hölder inequality, we obtain where 2N + 1 = O(ε −1 ) is the number of cells (see Section 1.1). We complete the proof with the following estimate using the Newton-Leibnitz formula and the integral Hölder inequality: Thus, gathering the estimates for I ε i , i = 1, 2, 3, the result of the proposition holds. Remark 3.5. Note that the geometry of the domain chosen (cf. figure 1) provides certain properties for solutions useful mainly to simplify technical proofs, and avoid introducing more cut-off functions which bring more cumbersome computations. However, formal computations hold for a more general geometry of Ω and the proofs can be extended.

Estimation of asymptotic remainders
Throughout this section, we justify up to a certain degree asymptotics (3.1) and (3.2) (cf. (3.20)) for eigenvalues and eigenfunctions.We obtain bounds for discrepancies between the eigenvalues of the original problem and the first two terms of the asymptotic expansions, and similarly for the corresponding eigenfunctions. However, this still does not imply the approach of the k-th eigenvalue in the sequence (1.9) through the k-th eigenvalue in the sequence (3.18) with the same k. We set some preliminaires in Section 4.1, while we gather the main results in Section 4.4 (cf. Theorem 4.2 and estimate (4.29)).

Abstract formulation of the ε-dependent eigenvalue problem
In the Hilbert space H ε = H 1 0 (Ω; Γ ε ⊓ ) we introduce the scalar product and an operator K ε by the identity Operator K ε is compact, positive, continuous and symmetric. Therefore, it has a positive monotone infinitesimal sequence of eigenvalues counted according to their multiplicity while κ ε = 0 is the only point of the essential spectrum of K ε . In view of (4.2) and (4.1), the integral identity (1.8) is equivalent to the abstract equation where, in addition, a simple calculation shows that sequences (4.3) and (1.9) satisfy the relationship In what follows, to show the above mentioned proximity of the asymptotic formulas (3.1) and (3.2), we use the following simple consequence of the spectral decomposition of the resolvent, also known as result on "near eigenvalues and eigenvectors", see, e.g. [42] and Ch. 6 in [3]. (4.5) Then, there exists an eigenvalue κ ε q of the operator K ε such that Moreover, for any δ • ∈ (δ, k ε ), one finds some coefficients a ε Q ε , . . . , a ε Q ε +X ε −1 satisfying

the corresponding eigenvectors subject to the orthogonality and normalization conditions
Comparing (4.7) and (1.10), we recall formulas (4.2) and (1.8) to arrive at the relation k ε and U ε arising in (4.5) are the so-called near eigenvalue and eigenvector respectively for the operator K ε ; we perform a suitable choice in Section 4.2, and we compute δ in Section 4.3.
Let us consider the discrepancy δ ε = δ ε k in (4.5), namely where the supremum is taken over all v ∈ H ε such that ∥v; H ε ∥ = 1. We postpone computing δ ε k to Section 4.3, although some of the bounds below will be used in this computation. We proceed by calculating the scalar products see (4.1), for the functions u ε k and u ε j in (4.10) corresponding to the same eigenvalue µ k of multiplicity κ k , (4.14) For κ k = 1, we have j = k in (4.13) but in Section 4.4 we will also need to deal with the case j ̸ = k. In fact, our aim in the rest of the section is to show the relationships for certain constants c j,k . First, we observe that the last term in (4.13) meets the estimate Second, considering the first two terms on the right-hand side of (4.13), we use Proposition 3.4 and we proceed by evaluating scalar products for the part u ε0 for the first term on the right-hand side, we introduce the functions z and Z in Proposition 3.4 as follows: z(x) = u k (x)u j (x) either for x ∈ Ω or x ∈ Γ, and Z(ξ) = (W (ξ) − B) 2 when ξ ∈ Θ, Z(ξ) = 0 outside. We proceed similarly for the rest of the terms, and we derive that Finally, in view of (4.16), to conclude with the sought-for relationship (4.15), it suffices to mention that (4.17) and (4.18) lead to the formulas as can be easily checked, while a similar calculation shows that the rest u ε′ k = u ε k − u ε0 k = εζ 0 ζ ′ ε w ′ admits the estimates ∥∇u ε′ k ; L 2 (Ω)∥ ≤ c k ε 1/2 and ε −1 ∥u ε′ k ; L 2 (Θ ε )∥ ≤ c k ε 1/2 .

Estimating discrepancies
Here, we obtain an estimate for δ ε k given by (4.12), cf. (4.24). Inequality (4.15), in particular, means that the first factor on the right-hand side of (4.12) is less than cε 1/2 for a certain constant c. Let us evaluate the scalar products under the sign sup. The last one evidently meets the estimate By definition (4.10), the entry of the first scalar product becomes respectively. In the set Ω(d) the functions W and w ′ are exponentially small so that (see (2.10) and (3.3)). The two thin vertical strips touching the sides Γ ± and composing the set Ω ′ ε , do not contain the collision points and therefore w ′ is twice differentiable in Ω ′ ε . Hence, on account of the bounds for the derivatives of ς ′ ε and the Fourier expansion of w ′ similar to (2.8) with B = 0, we have Here, we have used the normalization condition ∥v; H ε ∥ = 1 in (4.12) as well as the consequence of the Hardy inequality (2.4) in the variable t = 1∓x 1 together with the Dirichlet condition v = 0 on Γ ± . Also the integral in Ω ′ ε of the function e −4πx2/ε has been computed, and Ω 0 denotes the set {x ∈ Ω : 0 < x 2 < d}.
Some of the terms in (4.19) can be treated as follows: These estimates are obtained by Proposition 3.4 as a consequence of the following: in the first two estimates, the subtrahend on the left-hand side of inequality (3.21) has been considered for the corresponding bound. In the third estimate, similarly to (3.13), the formula ∫ ϖ ∂w ′ ∂ξ 1 (ξ) dx = 0 for the 1-periodic function has been taken into account. We emphasize that the function Z in Proposition 3.4 is a product of the test function v with the trace on Γ of ∂ j u k /∂x j 1 , with j = 0, . . . , 3 (cf. Remark 3.1), which belongs to H 1 0 (Ω; Γ ⊓ ) according to the differential properties of the eigenfunction u k described in Section 3.4.
With the help of equations (3.5) and (3.7), (3.8), we rewrite the scalar products with the remaining terms in (4.19) in the form: These scalar products will be considered together with the other term in (4.12) In (4.21) and (4.22), we have taken into account that, by definitions (3.3) and (4.11), both the cut-off functions ς 0 and ς ′ ε equal 1 on the union Θ ε of the concentrated masses. The first term on the right-hand side of (4.22) becomes The first and second scalar products can be readily found in (4.21) so that all of them cancel each other under the last sign sup in (4.12). By definition (4.1) and the Taylor formula, the modulo of the third scalar product does not exceed the expression Furthermore, the coefficient of ε −1 in (4.22) coincides with while the first and third scalar products have also appeared in (4.21) and thus are canceled in (4.12). Finally, we obtain the estimate which is quite similar to (4.23). It remains to consider the scalar product in (4.12) with the derivative ∂ x2 u ε k , namely Here, we took into account that ∂W ∂ξ 2 ( x 1 ε , 0 ) v(x 1 , 0) = 0 on Γ \ T ε due to the Neumann boundary condition (1.14) for W as well as the second formula in (3.7) for w ′ , and formulas (4.11) for ς ′ ε and (4.20) for v. Finally, note that all the constants c arising in the bounds throughout the section depend on k and they are bounded by c k ε for some constant c k .
Thus, gathering our calculations through the section, we observe that all the terms of the expression under the last sign sup in (4.12) are either canceled out, or bounded by c k ε. This together with formula (4.15), which gives the estimate from below of the norm ∥u ε k ; H ε ∥, while k ε k from (4.9), lead to the inequality δ ε k ≤ c k ε 3/2 .

The intermediate result on asymptotics
Let us apply Lemma 4.1 with k ε k and U ε k in (4.9) and δ ε = c 0 k ε 3/2 with c 0 k a positive constant. Lemma 4.1 provides us with an eigenvalue κ ε p of the operator K ε such that Now, using (4.4) and (4.9), we obtain for ε ∈ (0, ε k ], from (4.26) we derive the chain of inequalities where c 1 k is expressed through c 0 k , M and ε k µ k . Note that above we have used that Λ ε p is bounded by a constant depending on k. Thus, we have found a rescaled eigenvalue (1. Let us verify that in the case (4.14) a neighborhood of M + εµ k contains at least κ k eigenvalues. We take δ ε • = Sε 3/2 (4.28) in Lemma 4.1 where S > 0 is a big number to be fixed later. Then, considering µ k of multiplicity κ k , for the same k ε k in (4.9) we define U ε j = ∥u ε j ; H ε ∥ −1 u ε j for each j = k, . . . , k + κ k − 1, and obtain (4.12) for δ ε k = c 0 k ε 3/2 . Thus, using Lemma 4.1, we derive that for each j = k, . . . , k which, according to (4.6), is normalized in R X ε and satisfies where c • k = max{c 1 k , . . . , c 1 k+κ k −1 }. Moreover, according to (4.6) and (4.7) we have with j, l = k, . . . , k + κ k − 1 and the summation over q, m = Q ε , . . . , Q ε + X ε − 1. For Q ε and X ε we use the same notation as in Lemma 4.1, X ε being the total number of eigenvalues κ ε On the other hand, formulas (4.9) and (4.15) show that, for sufficiently small ε, Thus, we obtain the inequality In other words, the columns a ε (k) , . . . , a ε (k+κ k −1) are normalized and "almost orthogonal" in R X ε for a small ε and a big S. This may happen for a sufficiently small ε and a sufficiently large S, only under the restriction X ε ≥ κ k , as can be shown by contradiction, and, again by Lemma 4.1, we detect at least κ k eigenvalues of the operator K ε in the Sε 3/2 -neighborhood of the point k ε k in (4.9). Since we can show that the Λ ε p are bounded by a constant depending on k only, the replacement c 0 k → S in (4.25) does not affect our conclusion (4.27) and we formulate the result that we have obtained.
Finally, let us note that the equality p(k) = k will be proved in Section 6.1.

The convergence theorem
In this section, we reformulate the original eigenvalue problem (1.8) in terms of new spectral parameters and eigenfunctions (1.23). The new spectral problem reads (5.2) and we show that it has a discrete spectrum (cf. Sections 5.1 and 5.2). Its rescaled eigenvalues {ε −1 µ ε k } ∞ k=1 converge towards the eigenvalues {µ k } ∞ k=1 of (3.17) with conservation of multiplicity; also a certain convergence of the corresponding eigenfunctions holds. In order to obtain this convergence, we formulate problem (5.2) in weighted Sobolev spaces with the singular weight W ε which is obtained from the principal eigenmode of the cell problem (2.2) (cf. Sections 5.2 and 5.3). The main results of the section are stated in Proposition 5.5 and Theorem 5.6. The above-mentioned convergence, with conservation of the multiplicity, is derived at the beginning of Section 6.1.

Reformulation of the ε-dependent eigenvalue problem
Considering the first eigenvalue M and the corresponding eigenfunction W = W 1 of problem (1.13)-(1.15), we recall the weight multiplier W ε (x) = W (ε −1 x) in (2.29) is positive in R 2 + and ε-periodic in x 1 (cf. Section 2). In this way, function (2.29) is properly defined in the domain Ω for any eigenfunction u ε = u ε k of the original problem (1.3)-(1.5). Considering the inclusion (2.31), we reformulate this spectral problem to get a new eigenvalue problem for the fractional function u ε = u ε /W ε in (2.29). Recalling Hence, function (2.29) verifies the differential equation with the new spectral parameter Multiplying (5.1) by the test function W ε v ε , v ε being any smooth function in Ω vanishing on Γ ⊓ , we take into account the relation and, integrating by parts, derive the integral identity Let us note that, according to (2.29) and (1.4), (1.5), the function u ε satisfies the boundary conditions u ε = 0 on Γ ⊓ , This makes null the line integral over ∂Ω \ T ε while the integral over T ε vanishes because W = 0 on τ and, therefore, W ε = 0 on T ε . In other words, the differential equation (5.1) equipped with the boundary conditions (5.3) and (5.4), does not need any boundary condition on the set T ε where the weight multiplier W ε becomes null. This peculiarity of differential equations with degenerating coefficients is a subject which has been investigated in the literature, cf. [41] and [23]. The eigenvalue problem (5.2) must be posed in the space H ε obtained by completing the space C ∞ 0 (Ω;Γ ⊓ ) with the weighted norm C ∞ 0 (Ω;Γ ⊓ ) denotes the linear space of C ∞ functions vanishing in a neighborhood of Γ ⊓ . We emphasize that, in view of Corollary 2.3, Proposition 2.5 ensures that for the eigenfunctions u ε of the original problem (1.8), the fractional function u ε = u ε /W ε satisfies u ε ∈ C(Ω) and u ε ∈ H ε .
Proposition 5.1. The spectral problem (5.2) in H ε has the monotone unbounded positive sequence of eigenvalues and the corresponding eigenfunctions u ε k ∈ H ε can be subject to the orthogonality and normalization conditions Proof. To prove the conclusion of the statement, we need the inequality which follows from the classical one-dimensional Hardy inequality, namely, from (5.10). Indeed, to derive (5.7) from (5.10), we extend u ε as null over the half-strip (−1, 1)×R + , set U(y) = u ε (x 1 , y), integrate in x 1 ∈ (−1, 1) and observe that, in view of Corollary 2.3, Notice that the constant C Ω and, therefore, c Ω in (5.7) are independent of ε ∈ (0, ε 0 ].
In addition, introducing L ε the weighted Lebesgue space with the norm we show that the embedding H ε ⊂ L ε is compact. To do this, we consider any sufficiently small δ > 0 and we represent the embedding operator I ε as the sum I ε (δ) + I ε δ of the restriction operators onto the sets Ω \ T ε δ and T ε δ , respectively. Here, T ε δ is the union of the small rectangles where W ε (x) ≤ Cδ 1/2 according to (2.26) and (2.25). Hence, the operator I ε δ has a small norm O(δ 1/2 ) due to the weight multiplier W ε in (5.8) while I ε (δ) stays compact because W ε (x) ≥ c(δ, ε) > 0 on Ω \ T ε δ and therefore I ε (δ)H ε = H 1 (Ω \ T ε δ ; Γ ⊓ ). As a matter of fact, since the function W ε (x) is bounded, it suffices to consider that the embedding H ε ⊂ H 1 (Ω\T ε δ ; Γ ⊓ ) holds continuously. This amounts to say that I ε can be approximated by compact operators in the operator norm and thus is compact too, and this shows the compactness of the embedding H ε ⊂ L ε . Furthermore, inequality (5.7) ensures that and, owing to assumption (1.7), on the left hand side of (5.2), the second term is a small perturbation of the first one. Then, we note that the right-hand side of (5.2) can be written as for χ ε in (1.6), and the scalar product here defines a norm equivalent to that of L ε , cf. (5.8). Consequently, problem (5.2) is a standard eigenvalue problem in the framework of sesquilinear, continuous and coercive forms on the couple of Hilbert spaces H ε ⊂ L ε and the result of the proposition holds.
In the next section, we establish the equivalence between both spectral problems, (1.8) and (5.2). In order to do it, we show Proposition 5.3 below which provides properties for eigenfunctions of (5.2) complementing those in Proposition 2.5 for (1.8). Its proof uses the estimate in the next lemma, which readily follows from the classical Hardy inequality

Lemma 5.2.
For fixed T 1 and T 2 , 0 < T 1 < T 2 , and for any U ∈ C ∞ [0, +∞), the inequality holds, with C a constant depending on T 1 and T 2 but independent of U .
Proof. On account of the boundedness of W ε it suffices to show that u ε ∇W ε ∈ L 2 (Ω), which holds due to the inequality with some ε-dependent constant c ε . To prove (5.12), we decompose the domain Ω into subdomains which (after changing to variables (2.1)) are contained in regions of the plane where bounds in Corollary 2.3 hold.
Let Ω ε denote Ω∩{x 2 > ε}. Let 0 < δ 1 < min{l, 1 2 −l}. For each n ∈ Z(N ), we consider the intervals  1 and figure 2), we denote by E ε n the half-disk Above, we take δ 1 and r 1 in such a way that Ω = Ω ε ∪ ∪ Note that Ω ε denotes a teeth domain which is nothing but Ω minus small rectangles of height ε and width 2(l + δ 1 )ε containing the collision points (cf. similar domains in (5.9) and figure 3). Let us denote by t ε n the larger, but still small, rectangles In Ω ε , we take into account that u ε vanishes on Γ 0 and apply the Friedrichs inequality to obtain As a consequence of properties (2.23) and (2.24), we get (5.12) in Ω ε , namely, Let us proceed obtaining the desired estimates in the small rectangles d ε n ⊂ t ε n , d ε n := D ε n × (0, ε), as follows. Let x 1 ∈ D ε n , we apply (5.11) taking U = u ε , τ = x 2 , T 1 = ε and T 2 = 2ε, and we deduce .
Then, we take the integral over D ε n , consider the sum for n ∈ Z(N ) and, since u ε vanishes at x 1 = ±L, apply the Friedrichs inequality in the rectangle (−L, L) × (ε, 2ε). As a consequence of the boundedness of |∇W ε | and properties (2.23) and (2.25), we can write .
(5.14) In order to obtain estimates near the collision points (namely, in subdomains of t ε n out of d ε n and Ω ε ), we consider the half-disk E ε n centered at the collision point P = (n + l)ε, where the boundary conditions change from Dirichlet to Neumann, obtaining suitable bounds, and we proceed in the same way with the other collision point (n − l)ε in each half-strip of width ε.
Let us fix n. If no confusion arises we skip indexes ε and n, and denote by (r, φ) the polar coordinates. We write the half-disk E ε n as the union of three sectors which are contained in S D , S and S N , where and the constants φ 0 ∈ (0, π 4 ) and 0 < r 1 < r 2 are chosen to ensure that S ∩ d ε n ̸ = ∅ and S ∩ {r ∈ (εr 1 , εr 2 )} is contained in γ ε n × (ε, 2ε) (cf. Section 1.1 for the definition of γ ε n ). These choices restrict those for δ 0 and r 1 performed above and can be done as follows: let us fix φ 0 < π/4 and choose δ 1 such that S ∩ d ε n is only the point of polar coordinates (εr 1 , φ 0 ), then take r 2 such that S ∩{r ∈ (εr 1 , εr 2 )} ⊂ γ ε n ×(ε, 2ε). Obviously, by symmetry Let us start getting bounds for the integrals of |u ε ∇W ε | 2 over the symmetric sector S. Taking τ = r, T 1 = εr 1 and T 1 = εr 1 in (5.11), we can write ∫ , and consequently, ∫ where we note that when considering sums for n ∈ Z(N ), the second integral on the righthand side can be estimated in terms of (5.12), as has been done in (5.14).
As regards the sector S D , we apply (5.11) with τ = π − φ, The second integral on the right-hand side has been estimated in the previous step (cf. estimates in the sector S), while the first integral can be rewritten and bounded as follows Finally, in the sector S N , we apply the inequality and take integrals over r ∈ (0, r 1 ε). We have .
The second integral on the right-hand side has been estimated above (cf. estimates in the sector S), while the first integral is bounded by c ε ∥W ε ∇u ε ; L 2 (E ε n )∥ 2 . Gathering the bounds on the three sub-sectors S, S N and S D , we can write .

The equivalence of the ε-dependent problems and the crucial estimate for eigenfunctions
Clearly, relations (5.2), (5.6), Proposition 5.3, and properties (2.23)-(2.26) of the weight function W ε show that any eigenpair {µ ε k , u ε k } ∈ R + × H ε of problem (5.2) gives rise to an eigenpair of problem (1.8), together with the estimate for the eigenfunction where C does not depend on ε ∈ (0, ε 0 ], and k ∈ N. The converse follows from the two following observations. First, (2.31) confirms that function (2.29) lives in H 1 (Ω; Γ ⊓ ), therefore, in H ε . Second, taking W ε v ε as a test function in the integral identity (1.8), the latter turns into (5.2) by a simple algebraic transformation which involves applying the Green formula to ∆W ε multiplied by W ε v ε u ε k (see arguments in Section 5.1). Hence, any eigenpair {λ ε k , u ε k } of problem (1.8) generates an eigenpair of problem (5.2), that together with (5.16) means that in both cases, numbers k and K of eigenvalues coincide with each other. In this way, we can assert that the eigenvalue problems (1.8) and (5.2) are equivalent and It should be noted that Proposition 5.1 shows that the values defined by (5.19) are positive. However, we observe that this can also be obtained independently by proving that ε 2−m λ ε 1 − M > 0, and applying the technique in [36], which uses the minimax principle and comparison results for eigenvalues in different domains (cf. Lemma 5.4). At the same time, we note that the eigenfunctions u ε k in (5.16) and u ε k in (5.18) are not normalized according to conditions (1.10) and (5.6), respectively. Besides, although the L 2 (Ω)-norm of u ε k is uniformly bounded in ε ∈ (0, ε 0 ], see (5.17), the gradient norm ∥∇ u ε k ; L 2 (Ω)∥ grows unboundedly as ε → 0 because of the last term in the formula Our immediate objective is to show the uniform estimate for the eigenfunctions u ε k normalized by (5.6) (see statement of Proposition 5.5). This estimate becomes the key point when proving Theorem 5.6 on convergence for eigenpairs of (5.2) in Section 5.3. Since W ε (x) ≥ c δ > 0 for x ∈ Ω \ T ε δ with any δ > 0, see (5.9) and (2.23)-(2.26), and, by virtue of (5.6) and (5.7), we have ∥W ε ∇u ε k ; L 2 (Ω))∥ ≤ c for ε ∈ (0, ε 0 ] and k ∈ N. Then, it is sufficient to derive an appropriate estimate of the gradient ∇u ε k on the union of the small rectangles (5.9) with some fixed δ > 0 to be determined. To this end, we localize the solution u ε k of problem (5.1), (5.3), (5.4) onto a neighborhood of t ε nδ with the help of the cut-off function and h 0 > 0 taken from (1.2) and (2.9), respectively. The function |∇ς ε n | has a support in the set Π ε n = Ξ 2ε n \ Ξ 1ε n , see figure 3, a, and we define where ξ = ε −1 (x 1 − εn, x 2 ) are the stretched coordinates (2.1) and U ε kn is the function u ε k somehow localized. From (5.1), we derive the differential equation where cf. (5.24), and We emphasize that the factor ε 2 comes to the right-hand side of (5.27) from the relationship W ε (x)∆ x = ε −2 W (ξ)∆ ξ , and the additional term in (5.29), that is, involves the commutator (cf. (4.19) for the definition) of the differential operator from the left hand side of (5.27) with the cut-off function ς(ξ) = ς 1 (ξ 1 )ς 2 (ξ 2 ) obtained from (5.23). We finally note that function (5.26) satisfies the boundary conditions The Dirichlet condition (5.31) is due to the definition of the cut-off function (5.23) and the Neumann condition (5.32) is additionally inherited from (1.5) and (1.14). As has been mentioned previously, there is no need to impose any condition on the segment τ = (−l, l) × {0}.
Owing to (5.25), the orthogonality condition ∫ is satisfied. Thus, the Poincaré inequality in Π = Ξ 2 \ Ξ 1 (before the rescaling) ensures that ∫ while the factor ε 2 is due to the small size of the set Π ε n in figure 3, a. Function (5.30) has a support in Π, see figure 3, b, and Here, we have applied inequality (5.33) together with the formulas dξ = ε −2 dx and In what follows, we employ the notation so that, according to (5.22) , ∑ n∈Z(N ) n ε kn ≤ c for ε ∈ (0, ε 0 ] and k ∈ N while estimate (5.34) can be simplified as follows: Let us estimate the constant (5.25). Writing the Newton-Leibnitz formula for the function u ε k extended by zero from Ω onto (−1, 1)×R + (recall the Dirichlet condition As a result, we obtain Thus, taking into account the area mes 2 (Π ε n ) = O(ε 2 ), we conclude that Now, we are in position to evaluate all the terms on the right-hand side of (5.29). First of all, we notice that for any k ∈ N, there exist positive ε k and C k supporting the inequality µ ε k ≤ C k ε for ε ∈ (0, ε k ]. (5.38) This estimate will be proved at the beginning of Section 5.3 (cf. (5.51)) based on results in Section 4, that is, independent of the results of this section. Thus, there is no problem assuming for the rest of this section that (5.38) is satisfied, and we observe that the coefficients in the terms of (5.29) admit the estimates: as a small perturbation of the differential expression on the left hand side of (5.27). From Section 2.3 we know that U ε kn ∈ H 1 (Ξ 2 ), although the corresponding norm is not estimated yet. We insert into problem (5.27), (5.31), (5.32) the representation formula and after long but simple algebraic computations which involve computing ∇(W −1 ) and ∆(W −1 ) we arrive at the differential equation for ξ ∈ Ξ 2 , with the boundary conditions ) .

(5.44)
It should be noted that the above-mentioned term f εU kn in (5.41) has moved to the left hand side in (5.43) and indeed exhibits a small perturbation of the differential operator ∆ ξ + M X. The right-hand side (5.39) of (5.43) is a smooth function everywhere in Ξ 2 , except at the arc τ ⊓ , and inherits from W the singularities O(ρ We insert into the Raileigh quotient the principal eigenfunction U 1 of problem (5.45), (5.44) extended by zero from Ξ 2 onto the half-strip ϖ, cf. the first condition in (5.44). Then and the desired relation M 1 > M 1 follows immediately. The strict inequality is due to the fact that the principal eigenfunction W 1 of (1.13)-(1.15) is positive in ϖ and therefore cannot coincide with our test function.
Let us return to problem (5.43), (5.44) whose differential operator is a small perturbation of −∆ ξ − M X. Then, Lemma 5.4 shows that, for a small ε > 0, this problem is uniquely solvable and thus, owing to (5.36) and (5.39), (5.40), there holds the estimate As we have mentioned above, the function f εc kn in (5.43) is "good". At the same time, the other function (5.30) is supported in Π, that is F ε kn vanishes inside Ξ 1 and its L 2 -norm has been properly bounded in (5.36). In this way, using local estimates [2] of solutions to the Neumann problem (compare the last relation in (5.44)) for the Helmholtz equation, we may restrict our consideration on the smaller rectangle Let us review the situation. Inside the bigger rectangle Ξ 1 ⊃ Ξ 1/2 , the function U ε kn satisfies the differential equation (5.43) with F ε kn = 0. According to (5.38) and (1.7), the coefficient M + µ ε k in the differential operator is a small perturbation of M and the other coefficient ε m (M + µ ε k ) is small itself. The remaining right-hand side f εc kn takes the convenient form (5.39) with coefficients estimated in (5.40). Finally, U ε kn meets the homogeneous Dirichlet and Neumann conditions (5.44) at the base {ξ ∈ ∂Ξ 1 : ξ 2 = 0}. These facts allow us to apply the Kondratiev theory and, based on the theorem on asymptotics in weighted Sobolev [10] and Hölder [21] classes (see also Section 2.3, Appendix and, e.g., Ch. 3 in [31]), we conclude that in the smaller rectangle Ξ 1/2 there hold the same asymptotic forms for U ε kn as we have used in Section 2.3 to examine the fractional function (2.29).
The most profitable inference of the performed localization of the problem is undoubtedly the possibility to estimate all the necessary terms of the function U ε kn in Ξ 1/2 by the expression which is nothing but the sum of the "weak norms" of the solution and the right-hand side in a "bigger" domain Ξ 2 and a "strong norm" of the right-hand side in a "intervening" domain Ξ 1 . In this context, the weak norm means the L 2 (Ξ 2 )-norm but the strong norm is a complicated weighted norm which will be minutely explained in the Appendix. Aiming to estimate a strong norm of the solution in the "small domain" Ξ 1/2 , we note that Ξ 1/2 Ξ 1 Ξ 2 , cf. (5.28). Moreover, the strong norm of the right-hand side reduces to the sum of moduli of coefficients in the linear combination (5.39). The constant C k is independent of n ∈ Z(N ) and ε ∈ (0, ε 0 ] because of the cell's identity and the above-mentioned property of coefficients in the differential operator of the problem. As a result, we obtain the desired local estimate which shows that the squared norm ∥∇ ξ (W −1 U ε kn ); L 2 (Ξ 1/2 ∥ 2 does not exceed expression (5.48).
The above-mentioned facts allow us to perform the limit passage as ε • j → 0 in the integral identity (5.2) with µ ε = µ As a result, we get the integral identity which, in view of (3.15), takes the form (3.17). By a completion argument, we can take any test function v ∈ H 1 0 (Ω; Γ ⊓ ). Proof. From (5.55), it suffices to verify that the limit u • k in (5.53) is not trivial. To this end, cf. (5.55), we will prove the relation for any k, q ∈ N bearing in mind the case k = q. This formula also shows that the limits u • k and u • q with k ̸ = q are orthogonal in L 2 (Γ) and therefore differ from each other. First of all, we write the immediate consequence of formulas (5.2), (5.6) and we observe that, according to (5.51), (5.52) and (1.7), Then, obtaining (5.56) follows the idea in Proposition 3.4. We apply the Hardy inequality (2.4) with t = x 2 and V(t) = u ε j (x 1 , t) − u ε j (x 1 , 0) in order to obtain the relation (cf. for the trace u εΓ j of u ε j on Γ. In addition, from (5.57)-(5.58) we get Furthermore, recalling formulas (2.7) and (3.22), we proceed similarly to (3.23) using Z = W 2 and that ∫ ε Consequently, from the above estimates, after subtracting ε −1 µ ε k M −1 ( u ε k , u ε q ) Γ on the right and left-hand sides of (5.59), it remains to mention that the limit of this subtrahend equals nothing else but µ Γ . This shows (5.56) and the theorem holds.

Asymptotics of eigenvalues and eigenfunctions
This section contains the main results related to the asymptotics of eigenpairs of the original problem (1.8). We combine results obtained in Section 4 and 5 to conclude the convergence rates for the discrepancies between the eigenvalues and eigenfunctions and the terms on the right-hand side of (1.16) and (1.17) respectively (cf. (6.1) and (6.5)).

A Appendix
The material of this appendix complements Section 2 and supports the estimate (5.49) which lead us to Proposition 5.5.

A.1 The homogeneous Kondratiev norms
Let us consider the model mixed boundary-value problem in the half-plane where (r, φ) is the polar coordinate system centered at the coordinate origin O, the collision point, and ∇ j u denotes all partial derivatives of u of order j. It is known, see [10] and, e.g., Ch. 2 in [31], that, for any l ∈ N, the operator where the coefficients K, K 1 and the remainder u ∈ V l+1 β−2 (R 2 + ) satisfy the estimate We emphasize that mapping (A.3) becomes an isomorphism under restriction (A.4) and the inclusion f ∈ V l−1 β−2 (R 2 + ) implies a faster decay rate as r → 0 + than the decay rate of ∆v ∈ V l−1 β (R 2 + ) prescribed by the original inclusion v ∈ V l+1 β (R 2 + ). In the same way, formula (A.6) gives the asymptotics of the solution v in the radial variable r.
(A. 15) Roughly speaking, to compose from functions (A.12) a weighted space with detached asymptotics by means of a procedure in Ch. 12 of [31] requires setting the coefficient functions K(r), K 1 (r) in a certain weighted Kondratiev space and incorporating their norms together with norm (A.13) into the norm of the whole function v. Additional difficulties originate in insufficient smoothness properties of the coefficients: according to (A.15) none of the traces v R± and ∂v ∂x 2 R± and, therefore, none of K and K 1 belongs to the proper space H l+1 (R ± ). The latter requires the introduction of special extension operators into the asymptotic forms of type (A.12) (cf. [11,34] and Ch. 12 in [31]). To avoid unnecessary complications, we consider a particular case with an infinitely differentiable right-hand side f vanishing near the coordinate origin, we deal with the model differential equation ) where the remainder v and the coefficients K(r) = K + K(r), K 1 (r) = K 1 + K 1 (r) ) +|K| + |K 1 | ≤ c