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This article is cited in 1 scientific paper (total in 1 paper) “Far-field interaction” of concentrated masses in two-dimensional Neumann and Dirichlet problems S. A. Nazarov Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg
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     § 1. Introduction1.1. Statement of the problemLet $\Omega$ and $\omega_1,\dots,\omega_J$ be domains in the plane $\mathbb{R}^2$ enveloped by simple closed smooth Lipschitz contours $\partial \Omega$ and $\partial \omega_1,\dots,\partial \omega_J$. We fix pairwise different points $P^1,\dots,P^J\,{\in}\,\Omega$, and, given a small positive parameter $\varepsilon$, introduce the petty sets (Fig. 1)
$$
\begin{equation}
\omega_j^\varepsilon=\{x\colon\xi^j:=\varepsilon^{-1}(x-P^j)\in\omega_j\},\qquad j=1,\dots,J.
\end{equation}
\tag{1.1}
$$
Let $X^\varepsilon$ be the characteristic function of their union $\omega^\varepsilon= \omega^\varepsilon_1\cup\dots\cup\omega^\varepsilon_J$ (that is, $X^\varepsilon(x)=1$ for $x\in{\overline{\omega^\varepsilon}}$, and $X^\varepsilon(x)=0$ for $x\in\Omega^\varepsilon:= \Omega\setminus \overline{\omega^\varepsilon}$ ); moreover, $X^\varepsilon_j$ and $X_j$ are analogous functions of the isolated sets $\omega_j^\varepsilon$ and $\omega_j$. The bound $\varepsilon_0>0$ for parameter $\varepsilon$ is chosen so that ${\overline{\omega^\varepsilon}}\subset \Omega$ as $\varepsilon\in(0,\varepsilon_0]$. We assume that $\varepsilon_0<1$ but, if necessary, we reduce the value $\varepsilon_0\in(0,1)$ (with the same notation). We will be mostly concerned with the spectral Neumann problem
$$
\begin{equation}
-\Delta_xu^\varepsilon(x)=\lambda^\varepsilon (1+\varepsilon^{-\gamma} X^\varepsilon(x))u^\varepsilon(x),\qquad x\in\Omega,
\end{equation}
\tag{1.2}
$$
$$
\begin{equation}
\partial_nu^\varepsilon(x)=0,\qquad x\in\partial \Omega.
\end{equation}
\tag{1.3}
$$
Here, $\Delta_x$ is the Laplace operator, $\lambda^\varepsilon$ is the spectral parameter, and $\partial_n$ is the outward normal derivative defined almost everywhere on the Lipschitz boundary. The presence of the large parameter $\varepsilon^{-\gamma}$ with the exponent in the “density” $1+\varepsilon^{-\gamma} X^\varepsilon$ means that the “mass” $\varepsilon^{-\gamma}|\omega^\varepsilon_j|= \varepsilon^{2-\gamma}|\omega_j|$ of each inclusion in (1.1) is not smaller than that $|\Omega^\varepsilon|$ of the perforated domain $\Omega^\varepsilon$; here $|\omega_j|$ stands for the area of $\omega_j$. In view of possible singularities of eigenfunctions $u^\varepsilon$ at peculiar points of the boundary (see Fig. 1, (a)), the weak statement [1] of problem (1.2), (1.3) is required, namely
$$
\begin{equation}
(\nabla_xu^\varepsilon,\nabla_x\psi)_\Omega=\lambda^\varepsilon\bigl( (u^\varepsilon,\psi)_{\Omega^\varepsilon}+\varepsilon^{-\gamma} (u^\varepsilon,\psi)_{\omega^\varepsilon}\bigr)\quad \forall\,\psi\in H^1(\Omega),
\end{equation}
\tag{1.5}
$$
where $(\,{\cdot}\,,{\cdot}\,)_\Omega$ is the standard inner product in the Lebesgue space $L^2(\Omega)$, and $H^1(\Omega)$ is Sobolev space with standard norm. Due to compactness of the embedding $H^1(\Omega)\subset L^2(\Omega)$, the variational problem (1.5) (or the boundary-value problem (1.2), (1.3) in the smooth case (see Fig. 1, (b))) has unbounded monotone sequence of eigenvalues
$$
\begin{equation}
0=\lambda^\varepsilon_1<\lambda^\varepsilon_2\leqslant\lambda^\varepsilon_3 \leqslant\dots\leqslant\lambda^\varepsilon_k\leqslant\cdots \to +\infty.
\end{equation}
\tag{1.6}
$$
The corresponding eigenfunctions $u^\varepsilon_k\in H^1(\Omega)$ can be subject to the orthogonality and normalization conditions
$$
\begin{equation}
\varepsilon^{\gamma-2}(u^\varepsilon_k,u^\varepsilon_m)_{\Omega^\varepsilon}+ \varepsilon^{-2}(u^\varepsilon_k,u^\varepsilon_m)_{\omega^\varepsilon}=\delta_{k,m},\qquad k,m\in \mathbb{N}:=\{1,2,3,\dots\},
\end{equation}
\tag{1.7}
$$
where $\delta_{k,m}$ is the Kronecker delta. The first eigenvalue $\lambda^\varepsilon_1=0$ corresponds to a constant eigenfunction. If the boundary $\partial\Omega$ is smooth, then $u^\varepsilon_k\in H^2(\Omega)$. The pairs $\{\lambda^\varepsilon_k;u^\varepsilon_k\}$ are called eigenpairs. 1.2. PrehistoryThe problem under consideration is a so-called problems on “concentrated masses”. For the first time, a similar problem for $\gamma=3$ in a domain $\Omega\subset\mathbb{R}^3$ with unique ($J=1$) inclusion $\omega^\varepsilon=\omega^\varepsilon_1$ and the Dirichlet condition on the outer boundary was studied in [2], where it was shown there that the ordered normalized eigenvalues $\varepsilon^{2-\gamma}\lambda^\varepsilon_k=\varepsilon^{-1}\lambda^\varepsilon_k$ of problem (1.2), (1.8) converge as $\varepsilon\to+0$ to the corresponding entries of the sequence
$$
\begin{equation}
\mu_1<\mu_2\leqslant\mu_3 \leqslant\dots\leqslant\mu_k\leqslant\cdots \to +\infty
\end{equation}
\tag{1.9}
$$
of eigenfunctions of the spectral equation in the entire space ($d=3$ and $j=1$)
$$
\begin{equation}
-\Delta_\xi w_j(\xi^j)=\mu_j X_j(\xi^j)w_j(\xi^j),\qquad \xi^j\in\mathbb{R}^d.
\end{equation}
\tag{1.10}
$$
Miscellaneous variants of similar problems of mathematical physics on small isolated inclusions were also examined (see [3]–[9], among many others). Furthermore, the tools of singular perturbations of spectral problems, as proposed in [2], gave rise to other kind of problems in which small petty inclusions, heavy or light, concentrate near the domain boundary or near interior smooth submanifolds without boundary. Such problems are quite different from those considered in our paper not only in their setting, but even in formulation of results. This is why we will omit the details, and mention only one of the latest papers [10] on this topic, which gives a detailed list of references and and rather a completed survey. Among diverse objects of the analysis and the results in these directions, we mention the spacial ($d=3$) Neumann problem (1.2), (1.3) with exponent $\gamma\geqslant3$ due to the effect of “far-field interaction” of petty inclusions observed in [11] (see also [12], [13]); namely, the limits of positive eigenvalues (1.6) are eigenvalues of the integro-differential equations
$$
\begin{equation*}
\begin{aligned} \, &-\Delta_\xi w_j(\xi^j) \\ &\quad=\mu X_j(\xi^j) \biggl(w_j(\xi^j) -\biggl(\sum_{k=1}^J|\omega_k|\biggr)^{-1}\sum_{k=1}^J |\omega_k|\langle w_k\rangle\biggr),\qquad \xi^j\in\mathbb{R}^3,\quad j=1,\dots,J, \end{aligned}
\end{equation*}
\notag
$$
composing a united system, because their right-hand sides involve the sum of mean values of all eigenfunctions
$$
\begin{equation*}
\langle w_k\rangle=\frac{1}{|\omega_k|}\int_{\omega_k}w_k(\xi^k)\, d\xi^k, \qquad k=1,\dots,J.
\end{equation*}
\notag
$$
Here and in what follows, $|\omega_k|$ is the volume or the area of a domain $\omega_k$. If the Dirichlet condition (1.8) is imposed on the boundary of a spacial domain $\Omega$ with several inclusions, the effect of interaction disappears, and so the limit sequence (1.9) is just the union of the spectra of independent equations (1.10) with indexes $j=1,\dots,J$. If inequality (1.4) is violated (that is, for “light” inclusions), the effect under discussion is absent in both problems (1.2), (1.3) and (1.2), (1.8), and the algorithm for construction of the asymptotics of eigenpairs simplifies considerably. The two-dimensional Neumann (1.2), (1.3) and Dirichlet (1.2), (1.8) problems to be considered later differ substantially from the three-dimensional ones because of the logarithmic growth of the fundamental solution of the Laplace operator on the plane the growth modifies substantially the properties of eigenpairs of equations (1.10) for $d=2$. First of all, the interaction phenomenon cannot be detected in both problems by passing to the limit, that is, at the level $1=\varepsilon^0$ (see § 2.3, § 5.3 and § 7.2) — this result is obtained via the approach developed in [4], [7] for the case $\gamma>2$ and $J=1$, but requires an additional analysis for $\gamma=2$ (see § 8). At the same time, we further demonstrate that the phenomenon manifests itself asymptotically at the levels $|{\ln \varepsilon}|^{-1}$ and $|{\ln \varepsilon}|^{-2}$ (or at higher levels, but which are still logarithmic) in both problems again (see § 5.1, § 5.2, § 7.1, § 7.2, and a description of various interactions in § 7.3; cf. the analysis in [14] of the Steklov problem in a planar domain with small holes).The second peculiarity of the two-dimensional problem (1.2), (1.3) is the logarithmic dependence of the coefficients in asymptotic eigenpairs $\{\lambda^\varepsilon;u^\varepsilon\}$ on the parameter $|{\ln\varepsilon}|$, which results from the logarithmic singularity of the fundamental solution (1.11). An analogous dependence also occurs in the Dirichlet problem for the Laplace operator in a planar domain $\Omega^\varepsilon$ with one or several small holes [15] and in the problem with spectral Steklov condition [14], and in many others (cf. [16], Ch. 4, 5). In particular, for solutions to the Poisson equation
$$
\begin{equation}
-\Delta_xu^\varepsilon(x)=f^\varepsilon(x),\qquad x\in\Omega^\varepsilon=\Omega \setminus\overline{\omega^\varepsilon}\subset\mathbb{R}^2,
\end{equation}
\tag{1.12}
$$
or to the spectral equation
$$
\begin{equation}
-\Delta_xu^\varepsilon(x)=\lambda^\varepsilon u^\varepsilon(x),\qquad x\in\Omega^\varepsilon,
\end{equation}
\tag{1.13}
$$
with the boundary conditions (1.8) and
$$
\begin{equation}
u^\varepsilon(x)=0,\qquad x\in\partial\omega^\varepsilon,
\end{equation}
\tag{1.14}
$$
it is not difficult to construct formal asymptotic series in powers of the small parameter $|{\ln\varepsilon}|^{-1}$ (cf. § 5 and § 7). At the same time, it is quite doubtful to accept their partial sums as a reasonable approximate solutions, because the growth of the logarithmic function is very slow: simple calculations
$$
\begin{equation}
\begin{alignedat}{5} |{\ln\varepsilon}| &\approx 2.3&\quad &\text{for} &\quad \varepsilon &=0.1, \\ |{\ln\varepsilon}| &\approx 6.9 &\quad &\text{for} &\quad \varepsilon &=0.001, \\ |{\ln\varepsilon}| &\approx 11.5 &\quad &\text{for} &\quad \varepsilon &=0.00001 \end{alignedat}
\end{equation}
\tag{1.15}
$$
show that the quantity $|{\ln\varepsilon}|$ can be regarded as a large parameter only for incredibly small diameters of inclusions (1.1). At the same time, in [17], [18], Ch. 3, and in [15], [16], Ch. 9, for the static (1.12), (1.10), (1.14) and spectral (1.13), (1.10), (1.14) problems, “summation” of these series was carried out, and asymptotic expansions of the solutions and eigenpars with remainders of power-law smallness $\varepsilon^N$ were derived. However, the coefficients of such series for solutions of the Dirichlet problem in a singularly perturbed domain $\Omega^\varepsilon$ are rational functions on $|{\ln\varepsilon}|$, and the coefficients of the series for eigenvalues and eigenfunctions are analytic functions of the variable $1/|{\ln\varepsilon}|^{-1}$. It is worth here to mention the papers [19], [20], where by means of other (non-asymptotic) methods, the analytic dependence of simple eigenvalues on the parameter $\varepsilon$ in a multidimensional domain and on pair of the parameters $\varepsilon$ and $1/|{\ln\varepsilon}|$ in a planar domain was proved. 1.3. Preliminary description of the resultsIt is also not difficult to construct series in $|{\ln\varepsilon}|^{-1}$ for problems (1.2), (1.3) and (1.2), (1.8) under consideration (see § 4.1, § 5.1, § 5.2, § 7.2, where explicit expressions for initial terms of the series are found). However, the main trick in our paper is based on the technique of weighted spaces with detached asymptotics (cf. [21], [22]); it will be applied as in [23], where the Dirichlet problem for formally self-adjoint systems of second-order differential equations in a domain with a small hole, but only in dimension $d\geqslant3$ (which excludes appearance of logarithms). In [23], a close relationship between the above technique and the method of matched asymptotic expansions was revealed (see [24], [18], [16], Ch. 2, etc.). In our case, the proposed approach is as follows: first, one finds the behaviour at infinity of the solutions to the limit equations (1.10) and near the fixed points $P^1,\dots,P^J$ of the solution to the Neumann problem
$$
\begin{equation}
\begin{gathered} \, -\Delta_xw_0(x)=\mu_0w_0(x),\qquad x\in\Omega, \\ \partial_nw_0(x)=0,\qquad x\in\partial\Omega, \end{gathered}
\end{equation}
\tag{1.16}
$$
after which the inner and outer expansions are matched in the intermediate zones, that is, at the distance $O(\sqrt{\varepsilon}\,)$ from $P^j$, where these formulas, different in their nature, imply asymptotics of the same solution to the original singularly perturbed problem (1.2), (1.3). The matching procedure, as applied to above-mentioned expansions, imposes linear relations for free coefficients, and the main question in realization of this formal approach is how interpret the obtained system of differential and algebraic equations as a regularly perturbed Fredholm operator in a certain Hilbert space. At this stage, there appear weighted spaces with detached asymptotics and the linear pencil (3.15) involving operators of the limit problems and numeral matrices depending on the parameter $1/|{\ln\varepsilon}|$ (see § 3.2). The spectrum of the pencil belongs to the closed positive real axis $\overline{\mathbb{R}_+} =[0,+\infty)$ and implies unbounded monotone sequence of normal eigenvalues (see Theorem 2 for details). In § 6, we derive an error estimate in the constructed model, namely, in Theorem 8, we obtain an inequality for the difference between the eigenvalues $\lambda^\varepsilon_\kappa$ of the original problem (1.2), (1.3) and the normalized eigenvalues $\varepsilon^{\gamma-2}\mu_k(1/|{\ln\varepsilon}|)$ of the operator pencil (3.15). Besides, in contrast to the discussed series in inverse powers of logarithm, the bound in the above inequality is of power-law order $\varepsilon^\vartheta$ with any exponent $\vartheta\in(0,1)$. This is the main advantage of the elaborated model, which is even more manifested than in a similar model from [23] for $d\geqslant3$, because the appearance of $|{\ln\varepsilon}|$ in the operator pencil does not bring additional difficulties, because formulas (1.15) demonstrate that, for realistic values of the primary parameter $\varepsilon$, the logarithm is not too large, so that the fraction $1/|{\ln\varepsilon}|$ is not too small. Moreover, the pencil is realized as a regular perturbation of the family of operators of the limit problems, and asymptotic formulas are derived from classical results of perturbation theory of linear operators (see, for example, [25]). Finally, the model serves for both the cases and for which the asymptotic procedures are substantially different, because the families of the limit problems do not coincide with each other; namely, in situation (1.17), equations (1.10), $j=1,\dots,J$, are augmented with the spectral Neumann problem (1.16) s(or with the integral identity (2.8) for non-smooth boundary). This difference can be easily visualized in examples from § 5, where we specify asymptotic expansions for the infinitesimal eigenvalues of problems (1.2), (1.3) and (1.2), (1.8).Related questions are considered in the concluding sections. First, in § 7, the results obtained are adapted for the planar Dirichlet problem (1.2), (1.8) – there being no serious changes in comparison with the Neumann problem (1.2), (1.3). At the same time, in § 7.4 one principal difference between these problems is indicated in the case $J=1$: perturbation of eigenvalues in the Dirichlet problem becomes usually of order $1/|{\ln\varepsilon}|$, while in the Neumann problem this order is $\varepsilon^2$. Finally, for the reader convenience, we verify in § 8 the convergence Theorem 1, which differs from those in [4], [7] in the method of the proof due to the change of the boundary condition (1.2) by (1.8) and by the possibility of applying some results of the performed asymptotic analysis. In particular, this approach has allowed us to verify the conclusion of Theorem 1 also for $\gamma=2$. § 2. The known results2.1. Limit spectral equationsAs usual, the change of the spectral parameter
$$
\begin{equation}
\lambda^\varepsilon \mapsto \mu^\varepsilon=\varepsilon^{2-\gamma}\lambda^\varepsilon,
\end{equation}
\tag{2.1}
$$
the introduction of the dilated coordinates $\xi^j$ (cf. definition (1.1)), and the formal transition to $\varepsilon=0$ converts (1.2) into the differential equation (1.10) on the plane without any boundary condition, because the boundary $\{\xi^j\colon P^j+\varepsilon\xi^j\in\partial \Omega\}$ is sent to infinity by making $\varepsilon\to+0$. The variational statement of the spectral equation (1.10),
$$
\begin{equation}
(\nabla_\xi w_j,\nabla_\xi\psi_j)_{\mathbb{R}^2}=\mu_j(w_j,\psi_j)_{\omega_j}\quad \forall\,\psi_j\in\mathcal{H}_j
\end{equation}
\tag{2.2}
$$
is posed in the Hilbert space $\mathcal{H}_j$, which is the completion of the linear space $C^\infty_{\mathrm{c}}(\mathbb{R}^2)$ (of infinitely differentiable compactly supported functions) in the “energy” norm
$$
\begin{equation}
\|\psi;\mathcal{H}_j\|=\bigl(\|\nabla_\xi\psi;L^2(\mathbb{R}^2)\|^2 +\|\psi;L^2(\omega_j)\|^2\bigr)^{1/2}.
\end{equation}
\tag{2.3}
$$
The one-dimensional Hardy inequality “with logarithm” [26],
$$
\begin{equation}
\int_0^1|U(r)|^2\, \frac{dr}{r|\ln r|^2}\leqslant4\int_0^1\biggl| \frac{dU}{dr}(r)\biggr|^2 r\, dr\quad\text{for } U\in H^1(0,1)\text{ with }U(1)=0,
\end{equation}
\tag{2.4}
$$
shows that the norm (2.3) is equivalent to the weighted norm1
$$
\begin{equation}
\bigl(\|\nabla_\xi\psi;L^2(\mathbb{R}^2)\|^2 +\|(1+|\xi|)^{-1}(1+|{\ln(1+|\xi|)}|)^{-1}\psi;L^2(\mathbb{R}^2)\|^2\bigr)^{1/2}.
\end{equation}
\tag{2.5}
$$
The constant functions are known to lie in the energy space $\mathcal{H}_j$, because they can be approximated by $C^\infty_{\mathrm{c}}(\mathbb{R}^2)$-functions with respect to norm (2.3), and since their weighted norms (2.5) are finite. Lemma 1. The spectrum of problem (2.2) is discrete and forms unbounded monotone non-negative2 sequence while the corresponding eigenfunctions $w_{jk}\in\mathcal{H}_j$ can be subject to the orthogonality and normalization conditions The null eigenvalue $\mu_{j1}$ corresponds to the constant eigenfunction $w_{j1}(\xi^j)=|\omega_j|^{-1/2}$. 2.2. The problem in a bounded domainIn case (1.17), the change (2.1) does not affect the spectral parameter, that is, $\lambda^\varepsilon=\mu^\varepsilon$, and, therefore, there appears additional limit problem, namely the spectral Neumann problem (1.16) in the domain $\Omega$. The variational statement of this problem is as follows:
$$
\begin{equation}
(\nabla_x w_0,\nabla_x\psi_0)_\Omega=\mu_0(w_0,\psi_0)_\Omega\quad\forall\, \psi_0\in H^1(\Omega).
\end{equation}
\tag{2.8}
$$
The next result can be found in any textbook on partial differential equations or mathematical physics. Lemma 2. Problem (2.8) (or (1.16), in the case of smooth boundary) has the discrete spectrum and the corresponding eigenfunctions $w_{0k}\in H^1(\Omega)$ can be subject to the orthogonality and normalization conditions The null eigenvalue $\mu_{01}$ corresponds to the constant eigenfunction $w_{01}(x)=|\Omega|^{-1/2}$. In case (1.18), the limit problem in the domain $\Omega$ loses the spectral parameter, and its properties are, of course, known. We will need the generalized Green function [27], which, in the framework of the distribution theory (see, for example, [28]), is a solution of the problem
$$
\begin{equation}
\begin{gathered} \, -\Delta_xG(x,x')=\delta(x-x')-|\Omega|^{-1},\qquad x\in\Omega, \\ \partial_nG(x,x')=0,\qquad x\in\partial\Omega. \nonumber \end{gathered}
\end{equation}
\tag{2.11}
$$
Here, $\delta$ is the Dirac mass, and the subtrahend $|\Omega|^{-1}$ in equation (2.11) is required to satisfy the compatibility condition of the problem (the mean value of the right-hand side vanishes). We set Note that
$$
\begin{equation}
\begin{gathered} \, G_j(x)=\Phi(x-P^j)+\mathbf{G}^0_{jj}+O(r_j),\qquad r_j=|x-P^j|\to+0, \\ G_k(x)=\mathbf{G}^0_{jk}+O(r_k),\qquad r_k=|x-P^k|\to+0, \quad k\ne j, \quad k=1,\dots,J. \end{gathered}
\end{equation}
\tag{2.13}
$$
The matrix $\mathbf{G}^0$ of size $J\times J$, which is composed of the coefficients $\mathbf{G}^0_{jk}$ from (2.13), is symmetric. Functions (2.12) are defined up to additive constants, which can be fixed to satisfy
$$
\begin{equation}
\mathbf{G}^0\mathbf{e}=0\quad\text{with}\quad \mathbf{e}=(1,\dots,1)^\top\in \mathbb{R}^J.
\end{equation}
\tag{2.14}
$$
An interpretation of the Green functions (2.12) as solutions of the Neumann problem in the punctured domain $\Omega\setminus\{P^1,\dots,P^J\}$ will be given in § 2.4. 2.3. Limit passageLet us combine sequences (2.6), $j=1,\dots,J$, in a common sequence (1.9) (that is, $M=\{\mu_k\}_{k\in\mathbb{N}}$), and order it. In the critical case (1.17), we augment it with sequence (1.9) of eigenvalues of the Neumann problem (2.8) (or (1.16), for the smooth contour $\partial\Omega$). In what follows, we regard $M$ and (1.9) as the union of the spectra of limit problems constructed in the above way. Each of spectra (2.6) and (2.9) includes null, and hence, in situations (1.18) and (1.17), we have, respectively,
$$
\begin{equation}
\gamma>2\quad\Longrightarrow\quad\mu_1=\dots=\mu_J=0
\end{equation}
\tag{2.15}
$$
and
$$
\begin{equation}
\gamma=2\quad \Longrightarrow\quad\mu_1=\dots=\mu_J=\mu_{J+1}=0.
\end{equation}
\tag{2.16}
$$
The next result can be obtained for $\gamma>2$ by a slight modification of the arguments from [4], [7], but in case $\gamma=2$ it will be verified in § 8 in a roundabout way. Theorem 1. The entries of sequences (1.6) and (1.9) of eigenvalues of the original and the limit problems are related by 2.4. The Kondratiev theoryThe weighted Sobolev space $V^1_\beta(\Omega;\mathcal{P})$ (the Kondratiev space [29]) is defined as the completion of the linear space $C^\infty_{\mathrm{c}}(\overline{\Omega}\setminus \mathcal{P})$ in the norm
$$
\begin{equation}
\|\psi_0;V^1_\beta(\Omega;\mathcal{P})\|=\bigl( \|\mathbf{r}^\beta \nabla_x\psi_0;L^2(\Omega)\|^2+\|\mathbf{r}^{\beta-1}\psi_0;L^2(\Omega)\|^2\bigr)^{1/2}.
\end{equation}
\tag{2.18}
$$
Here, $\mathcal{P}=\{P^1,\dots,P^J\}$ is the set of fixed points in the domain $\Omega$ and $\mathbf{r}(x)=\operatorname{dist}(x,\mathcal P)$ is the distance to it. The space $V^1_\beta(\Omega;\mathcal{P})$ consists of functions $\psi_0\in H^1_{\mathrm{loc}} (\overline{\Omega}\setminus \mathcal{P})$ with finite norm (2.18). In the case $\beta<0$, these functions decay as $x\to P^j$, but in the case $\beta>0$, they can grow at a certain rate, and the rate of decay or growth is controlled by the the weight exponent $\beta\in\mathbb{R}$. The weighted Lebesgue space $L^2(\Omega)$ is endowed with the norm
$$
\begin{equation*}
\|\psi_0;L^2_\beta(\Omega)\|=\|\mathbf{r}^\beta\psi_0;L^2(\Omega)\|.
\end{equation*}
\notag
$$
The Hardy inequality (2.4) is responsible for another weight multiplier in the inequality
$$
\begin{equation}
\|\mathbf{r}^{-1}(1+|{\ln\mathbf{r}}|)^{-1} \psi_0;L^2(\Omega)\|\leqslant c_{\Omega,\mathcal{P}} \|\psi_0;H^1(\Omega)\|,
\end{equation}
\tag{2.19}
$$
that is, the spaces $V^1_0(\Omega;\mathcal{P})$ and $H^1(\Omega)$ are different for $d=2$, but they coincide in a multidimensional domain $\Omega\subset\mathbb{R}^d$, $d\geqslant3$ (this being another difference from the planar problem under consideration). The integral identity
$$
\begin{equation}
(\nabla_x w_0,\nabla_x\psi_0)_\Omega-\mu_0(w_0,\psi_0)_\Omega= f_0(\psi_0)\quad\forall\, \psi_0\in V^1_{-\beta}(\Omega;\mathcal{P})
\end{equation}
\tag{2.20}
$$
applies to the limit problem in a domain $\Omega$ in a weighted space, where it is required to find a function $w_0\in V^1_\beta(\Omega;\mathcal{P})$ from a linear continuous functional $f_0\in V^1_{-\beta}(\Omega;\mathcal{P})^\ast$ in the space $V^1_{-\beta}(\Omega;\mathcal{P})$; for instance,
$$
\begin{equation*}
f_0(\psi_0)=(\mathbf{f}_0,\psi_0)_\Omega\quad\text{with}\quad \mathbf{f}_0\in L^2_{\beta+1}(\Omega)
\end{equation*}
\notag
$$
because $V^1_{-\beta}(\Omega;\mathcal{P})\subset L^2_{-\beta-1}(\Omega)$ by definition (2.18). Besides, $(\,{\cdot}\,,{\cdot}\,)_\Omega$ is an extension of the inner product in $L^2(\Omega)$ to duality between suitable weighted spaces $L^2_\tau(\Omega)$ and $L^2_{-\tau}(\Omega)$. Problem (2.20) generates the mapping
$$
\begin{equation}
V^1_\beta(\Omega;\mathcal{P})\ni w_0\mapsto f_0=B^0_\beta(\mu)w_0\in V^1_{-\beta}(\Omega;\mathcal{P})^\ast
\end{equation}
\tag{2.21}
$$
which is continuous for any weight exponent $\beta\in\mathbb{R}$, but which has the Fredholm property only for $\beta\notin\mathbb{Z}=\{0,\pm1,\pm2,\dots\}$. Thus, $\beta=0$ is a forbidden weight exponent, and, although (2.18) is a Hilbert norm, the integral identity (2.20) is not a variational statement of any problem. Let us formulate an assertion supported by general results [29] (see also [30], Ch. 2, and [31], Theorem 2.7, (2)). Proposition 1. Let $\beta\in(0,1)$. 1) The operators $B^0_{+\beta}(\mu)$ and $B^0_{-\beta}(\mu)$ are Fredholm, and where $\operatorname{Ind}B=\operatorname{dim}\operatorname{ker}B -\operatorname{dim}\operatorname{co}\operatorname{ker}B$ is the index of an operator $B$. Moreover, $B^0_{\mp\beta}(\mu)$ is the adjoint operator of $B^0_{\pm\beta}(\mu)$.2) If $w_0\in V^1_\beta(\Omega;\mathcal{P})$ is a solution of problem (2.20) with a right-hand side $f_0\in V^1_\beta(\Omega;\mathcal{P})^\ast\subset V^1_{-\beta}(\Omega;\mathcal{P})^\ast$, then
$$
\begin{equation}
w_0(x)=\widetilde{w}_0(x)+\sum_{j=1}^J\chi_j(x)\bigl(a^0_j+b^0_j\Phi(x-P^j)\bigr),
\end{equation}
\tag{2.23}
$$
where $\Phi$ is the fundamental solution (1.11), $\chi_j\in C^\infty_{\mathrm{c}}(\Omega)$ is a cut-off function,
$$
\begin{equation}
\chi_j=1\quad\textit{for}\quad x\in\mathbb{B}_{R/2}(P^j),\qquad \chi_j=0\quad\textit{for}\quad x\notin\mathbb{B}_R(P^j),
\end{equation}
\tag{2.24}
$$
the radius $R>0$ of the disc $\mathbb{B}_R(P^j)=\{x\colon |x-P^j|<R\}\subset\Omega$ is so that where the remainder $\widetilde{w}_0\in V^1_{-\beta}(\Omega;\mathcal{P})$ and the coefficient columns $a^0=(a^0_1,\dots,a^0_J)^\top$, $b^0=(b^0_1,\dots,b^0_J)^\top$ ($\top$ stands for transposition) satisfy the inequality The factor $c_\beta(\mu)$ depends on $\beta$ and $\mu$, but not on $f_0$ and $w_0$, and $c_\beta(\mu)\to+\infty$ as $\beta\to+0$ or $\beta\to1-0$. It is easily checked that any linear combination of the Green functions (2.12) belongs to the Kondratiev space $V^1_\beta(\Omega;\mathcal{P})$, but satisfies the homogeneous ($f_0=0$) problem (2.20) with $\mu_0=0$ if and only if Here, is the orthogonal projector onto the subspace $\mathbb{C}^J_\bot$ of columns which are orthogonal to column $\mathbf{e}$ (see formula (2.14)), that is, $\mathbb{I}_J$ is the unit matrix of size $J\times J$, and $\mathbb{E}_J$ is a ($J\times J$)-matrix composed of $J^2$ $1$’s. We mention that a constant function is also a solution of the same homogeneous problem in $V^1_\beta(\Omega;\mathcal{P})$. Summarizing, we mention that the functions $1$ and $G_1-G_J$, $\dots$, $G_{J-1}-G_J$ form a basis for the subspace $\ker B^0_\beta(0)$.2.5. Equation in the planeThe weighted spaces $L^2_\beta(\mathbb{R}^2)$ and $V^1_\beta(\mathbb{R}^2)$, respectively, are defined as the completions of the linear space $C^\infty_{\mathrm{c}}(\mathbb{R}^2)$ in the norms
$$
\begin{equation*}
\begin{gathered} \, \|f;L^2_\beta(\mathbb{R}^2)\|=\|(1+|\xi|)^\beta f;L^2(\mathbb{R}^2)\|, \\ \|w;V^1_\beta(\mathbb{R}^2)\|=\bigl(\| \nabla_\xi w;L^2_\beta (\mathbb{R}^2)\|^2+\|w;L^2_{\beta-1}(\mathbb{R}^2)\|^2\bigr)^{1/2}. \end{gathered}
\end{equation*}
\notag
$$
A function $w_j\in V^1_\beta(\mathbb{R}^2)$ decays at infinity for $\beta>0$, but can grow for $\beta<0$. The integral identity
$$
\begin{equation}
(\nabla_\xi w_j,\nabla_\xi\psi_j)_{\mathbb{R}^2}-\mu_j(w_j,\psi_j)_{\omega_j}= f_j(\psi_j)\quad\forall\, \psi_j\in V^1_\beta(\mathbb{R}^2)
\end{equation}
\tag{2.29}
$$
(which is similar to (2.20)) with a right-hand side $f_j\in V^1_\beta(\mathbb{R}^2)^\ast$ defines the mapping
$$
\begin{equation}
V^1_{-\beta}(\mathbb{R}^2)\ni w_j\mapsto f_j=B^j_{-\beta}(\mu)w_j\in V^1_\beta(\mathbb{R}^2)^\ast.
\end{equation}
\tag{2.30}
$$
The next result is quite similar to Proposition 1, and its proof can be found in the sources mentioned above. Proposition 2. Let $\beta\in(0,1)$. 1) $B^j_{-\beta}(\mu)$ and $B^j_{+\beta}(\mu)$ are Fredholm operators, and Moreover, $B^j_{\mp\beta}(\mu)$ is the adjoint operator to $B^j_{\pm\beta}(\mu)$.2) If $w_j\in V^1_{-\beta}(\mathbb{R}^2)$ is a solution to problem (2.29) with a right-side $f_j\in V^1_{-\beta}(\mathbb{R}^2)^\ast\subset V^1_\beta(\mathbb{R}^2)^\ast$, then
$$
\begin{equation}
w_j(\xi)=\widetilde{w}_j(\xi)+(1-\chi_\omega^j(\xi))(a_j+b_j\Phi(\xi))
\end{equation}
\tag{2.32}
$$
where $\Phi$ is the fundamental solution (1.11), $\chi_\omega^j\in C^\infty_{\mathrm{c}}(\mathbb{R}^2)$ is a cut-off function, the radius $R^j_\omega>0$ is fixed so that $\mathbb{B}_{R_\omega^j} \supset{\overline{\omega}}_j$, and the remainder $\widetilde{w}_j\in V^1_\beta(\mathbb{R}^2)$, which decays at infinity, satisfies, together with the coefficients $a_j$ and $b_j$, the inequality The multiplier $c_\beta(\mu)$ depends on $\beta$ and $\mu$, but not on $f_j$ and $w_j$, and $c_\beta(\mu)\to+\infty$ as $\beta\to1-0$ or $\beta\to+0$. 2.6. Remarks on smoothnessBeing harmonic with respect to $\xi\notin\overline{\omega_j}$, the eigenfunctions $w_{jk}\in\mathcal{H}_j$ of problem (2.2) become infinitely differentiable outside any neighbourhood of the set $\overline{\omega_j}$, and admit convergent Fourier series. In particular, representations (2.32) hold with the coefficients $b_{jk}=0$ and the remainders $\widetilde{w}_{jk}$ that satisfy
$$
\begin{equation*}
|\nabla_\xi^m\widetilde{w}_{jk}(\xi)|\leqslant c_{mk}|\xi|^{-1-m}, \qquad \xi\in\mathbb{R}^2\setminus\mathbb{B}_{R^j_\omega},\quad m\in\mathbb{N}_0:=\mathbb{N} \cup\{0\}.
\end{equation*}
\notag
$$
We emphasize that the eigenfunctions lie in the space $H^2_{\mathrm{loc}}(\mathbb{R}^2)$, because the right-hand side $\mu_{jk}X_jw_{jk}$ of equation (1.10) belongs to $L^2_{\mathrm{loc}}(\mathbb{R}^2)$. Thus, the integral identity (2.2) is needed only to verify Lemma 1. The eigenfunctions $w_{0k}\in H^1(\Omega)$ of problem (2.8) may have singularities on the Lipschitz boundary $\partial\Omega$, but they are infinitely differentiable inside the domain $\Omega$. The same can be said about the Green function $G_j$ in view of its singularity (2.13) at the point $P^j$. § 3. Weighted spaces with detached asymptotics and asymptotic transmission conditions3.1. Operators of limit problemsLet $\beta\in(0,1)$ be as above. We denote by $\mathcal{V}^1_\beta(\Omega)$ the space of functions $w_0$ of the form (2.23) and endow it with the norm
$$
\begin{equation}
\|w_0;\mathcal{V}^1_\beta(\Omega)\|=\bigl(\|\widetilde{w}_0;V^1_{-\beta}(\Omega; \mathcal{P})\|^2+|a^0|^2+|b^0|^2\bigr)^{1/2}.
\end{equation}
\tag{3.1}
$$
Similarly, $\mathcal{V}^1_\beta(\mathbb{R}^2)$ is the space of functions (2.32) with the norm
$$
\begin{equation}
\|w_j;\mathcal{V}^1_\beta(\mathbb{R}^2)\|=\bigl(\|\widetilde{w}_j; V^1_\beta(\mathbb{R}^2)\|^2+|a_j|^2+|b_j|^2\bigr)^{1/2}.
\end{equation}
\tag{3.2}
$$
Note that
$$
\begin{equation*}
w_0\in\mathcal{V}^1_\beta(\Omega)\subset V^1_\beta(\Omega;\mathcal{P}),\qquad B^0_\beta(\mu)w_0\in V^1_\beta(\Omega;\mathcal{P})^\ast
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
w_j\in\mathcal{V}^1_\beta(\mathbb{R}^2)\subset V^1_{-\beta}(\mathbb{R}^2),\qquad B^j_{-\beta}(\mu)w_j\in V^1_{-\beta}(\mathbb{R}^2)^\ast;
\end{equation*}
\notag
$$
here, the norms (3.1) and (3.2) are finite in view of estimates (2.25) and (2.34). We mention that $\mathcal{V}^1_\beta(\Omega)$ is the pre-image of the subspace $V^1_{-\beta}(\Omega;\mathcal{P})$ under the mapping (2.21), and $\mathcal{V}^1_\beta(\mathbb{R}^2)$ is the pre-image of the subspace $V^1_\beta(\mathbb{R}^2)$ under mapping (2.30). Each of the above spaces is a Hilbert space. Since a particular behaviour near the set $\mathcal P$ or at infinity is prescribed for their elements, the spaces are called wieghted spaces with detached asymptotics. In this section, all functions are complex-valued. In view of Propositions 1 and 2, the mapping
$$
\begin{equation}
\mathfrak{B}_\beta\colon \mathcal{V}^1_\beta(\Omega)\times \prod_{j=1}^J\mathcal{V}^1_\beta(\mathbb{R}^2)=:\mathfrak{V}_\beta \to \mathfrak{R}_\beta:=V^1_\beta(\Omega)^\ast\times \prod_{j=1}^JV^1_{-\beta}(\mathbb{R}^2)^\ast,
\end{equation}
\tag{3.3}
$$
which is obtained as the restriction of the vector operator
$$
\begin{equation}
\bigl(B^0_{+\beta}(0),B^1_{-\beta}(0),\dots,B^J_{-\beta}(0) \bigr)
\end{equation}
\tag{3.4}
$$
composed of operators (2.21) and (2.30) with $\mu=0$ onto the subspace
$$
\begin{equation*}
\mathfrak{V}_\beta\subset V^1_\beta(\Omega)\times \prod_{j=1}^JV^1_{-\beta}(\mathbb{R}^2) \text{ with vector elements } \overrightarrow{w}=(w_0,w_1,\dots,w_J),
\end{equation*}
\notag
$$
inherits from (3.4) all the properties, in particular, $\mathfrak{B}_\beta$ is a Fredholm operator, and, in view of (2.22), (2.31), 3.2. Asymptotic transmission conditionsWith the aim at applying the method of matched asymptotic expansions (see [24], [18], [16], Ch. 2, etc.), we impose the following relations on the coefficients of decompositions of components of vector function $\overrightarrow{w}$: Here, the new large parameter appears; whose prototype was mentioned several times in § 1. Note that $\varepsilon\in(0,1)$ and $\ln\varepsilon<0$. Remark 1. In view of (1.11) and (1.1), for the fundamental solution $\Phi$ in the stretched coordinates $\xi^j$, we have Given vectors $\overrightarrow{w}\in\mathfrak{V}_\beta$, consider their projections to the vector complex space $\mathbb{C}^J$
$$
\begin{equation}
\pi^+_\Omega\overrightarrow{w}=a^0,\quad\pi^+_\omega\overrightarrow{w}=a, \qquad \pi^-_\Omega\overrightarrow{w}=b^0,\quad\pi^-_\omega\overrightarrow{w}=b.
\end{equation}
\tag{3.11}
$$
Here, the columns from decompositions (2.23) and (2.32) are used. It is convenient to rewrite (3.6) and (3.7) in the form
$$
\begin{equation}
\pi^-_\Omega\overrightarrow{w}- \pi^-_\omega\overrightarrow{w}=0\in \mathbb{C}^J,
\end{equation}
\tag{3.12}
$$
$$
\begin{equation}
\pi^-_\Omega\overrightarrow{w} +\frac{1}{\mathfrak{z}}(\pi^+_\Omega \overrightarrow{w}- \pi^+_\omega\overrightarrow{w})=0\in \mathbb{C}^J.
\end{equation}
\tag{3.13}
$$
We also require the mapping
$$
\begin{equation}
\mathfrak{M}_\beta=(\delta_{\gamma,2},X_1,\dots,X_J)\colon \mathfrak{V}_\beta \to \mathfrak{R}_\beta
\end{equation}
\tag{3.14}
$$
which involves 0 and 1 at the first position in cases (1.18), (2.15) and (1.17), (2.16), respectively, and the characteristic functions for $\omega_1,\dots,\omega_J$ in the next positions. It is clear that operator (3.14) is continuous and compact. 3.3. The operator pencilFrom operators (3.3), (3.14) of limit problems and the right-hand sides of algebraic equations (3.12) and (3.13), we compose the mapping
$$
\begin{equation}
\mathfrak{A}^\mathfrak{z}_\beta(\mu):=\bigl(\mathfrak{B}_\beta-\mu\mathfrak{M}_\beta, \pi^-_\Omega- \pi^-_\omega ,\pi^-_\Omega+\mathfrak{z}^{-1}(\pi^+_\Omega- \pi^+_\omega)\bigr) \colon \mathfrak{V}_\beta\to \mathfrak{R}_\beta\times \mathbb{C}^J\times \mathbb{C}^J.
\end{equation}
\tag{3.15}
$$
Since (3.15) is a compact perturbation of operator (3.3) including $2J$ additional restrictions on projections (3.11), we have, in view of (3.5),
$$
\begin{equation*}
\operatorname{Ind}\mathfrak{A}^\mathfrak{z}_\beta(\mu)=\operatorname{Ind}\mathfrak{B}_\beta-2J=0.
\end{equation*}
\notag
$$
Thus, the polynomial (linear) operator pencil $\mu\mapsto\mathfrak{A}^\mathfrak{z}_\beta(\mu)$ has all the properties required for application of fundamental results of the theory of non-selfadjoint operators [32], and, in particular, of the analytic Fredholm alternative (see [32], Theorem 1.5.1). 3.4. Auxiliary assertions and generalized Green formulaTo verify the basic properties of the introduced pencil, we mention some simple facts. Proposition 3. For any $R>0$, there exists $\mathfrak{z}_R>0$ such that, for $\mathfrak{z}\in[\mathfrak{z}_R,+\infty)$ (that is, for small $\varepsilon>0$), the set Proof. 1) We first consider case (1.17). Let $\overrightarrow{w}^{\,\mathfrak{z}}\in\mathfrak{V}_\beta$ be the eigenvector of the pencil corresponding to an eigenvalue $\mu\in\mathbb{C}\setminus\overline{\mathbb{R}_+}$. Since the point $\mu$ does not belong to spectra (2.8) and (2.6), the inhomogeneous limit problems
$$
\begin{equation}
(\nabla_x v_0,\nabla_x\psi_0)_\Omega-\mu(v_0,\psi_0)_\Omega= f_0(\psi_0)\quad\forall\, \psi_0\in H^1(\Omega)
\end{equation}
\tag{3.17}
$$
and
$$
\begin{equation}
(\nabla_\xi v_j,\nabla_\xi\psi_j)_{\mathbb{R}^2}-\mu(v_j,\psi_j)_{\omega_j}= f_j(\psi_j)\quad\forall\, \psi_j\in \mathcal{H}_j
\end{equation}
\tag{3.18}
$$
with functionals $f_0\in H^1(\Omega)^\ast$ and $f_j\in \mathcal{H}_j^\ast$, are uniquely solvable in the spaces $H^1(\Omega)$ and $\mathcal{H}_j$, respectively, and, hence, column (3.6) of the coefficients in representations (2.23) and (2.32) of components of the vectors $\overrightarrow{w}^{\,\mathfrak{z}}$ is non-zero (otherwise the vector $\overrightarrow{w}^{\,\mathfrak{z}}$ would vanish). Moreover, the homogeneous ($f_0=0$) problem (3.17) has the solutions (generalized Green functions)
$$
\begin{equation*}
G^\mu_j(x)=\chi_j(x)\Phi(x-P^j)+{\widehat{G}}^{\,\mu}_j(x),\qquad {\widehat{G}}^{\,\mu}_j \in H^1(\Omega),
\end{equation*}
\notag
$$
for which decompositions (2.23) take the form (analogous to (2.13))
In the same way, the homogeneous ($f_j=0$) problems (2.23), $j=1,\dots,J$, has the solutions
$$
\begin{equation*}
W^\mu_j(\xi)=(1-\chi_\omega^j(\xi))\Phi(\xi)+\widehat{W}^{\,\mu}_j(\xi),
\end{equation*}
\notag
$$
where $\widehat{W}^{\,\mu}_j\in\mathcal{H}_j^\ast$ and
$$
\begin{equation}
\widehat{W}^{\,\mu}_j(\xi)=\mathbf{W}^\mu_j+\widetilde{W}^{\,\mu}_j(\xi), \qquad \widetilde{W}^{\,\mu}_j\in V^1_\beta(\mathbb{R}^2).
\end{equation}
\tag{3.20}
$$
Since the eigenvector satisfies $\pi_\Omega\overrightarrow{w}^{\,\mathfrak{z}}= \pi_\omega\overrightarrow{w}^{\,\mathfrak{z}}$ (see (3.15)), by the above there is only one possibility for the components of the vector $\overrightarrow{w}^{\,\mathfrak{z}}$, namely
$$
\begin{equation}
w^{\,\mathfrak{z}}_0(x)=\sum_{j=1}^Jb_jG^\mu_j(x)=:G^\mu(x)b,
\end{equation}
\tag{3.21}
$$
$$
\begin{equation}
w^{\,\mathfrak{z}}_j(\xi)=b_jW_j^\mu(\xi),\qquad j=1,\dots,J.
\end{equation}
\tag{3.22}
$$
Here, $b\ne 0\in\mathbb{R}^J$. Thus, the coefficient columns get the form where the ($J\times J$)-matrix $\mathbf{G}^\mu$ is composed of the coefficients of expansions (3.19) and the diagonal matrix $\mathbf{W}^\mu=\operatorname{diag}\{\mathbf{W}^\mu_1,\dots,\mathbf{W}^\mu_J\}$ includes coefficients of expansions (3.20). As a result, the transmission conditions (3.7) from the last component of the pencil (3.15) become the algebraic system Since $|\mu|<R$ and since the point $\mu$ is at the distance $1/R>0$ from the set $M$ of eigenvalues of the differential equations (1.10) and the Neumann problem (1.16), the norms of the matrices $\mathbf{W}^\mu$ and $\mathbf{G}^\mu$ are uniformly bounded on the set (3.16). Thus, the matrix $\mathfrak{z}\,\mathbb{I}_J+\mathbf{G}^\mu-\mathbf{W}^\mu$ is invertible for large $\mathfrak{z}$ (small $\varepsilon>0$). Hence, $b=0\in\mathbb{C}^J$, s and so we have a contradiction expressed by the absurd formula $\overrightarrow{w}^{\,\mathfrak{z}}=0$ for an eigenvector. Finally, we find $\mathfrak{z}_R>0$ such that, for $\mathfrak{z}\in[\mathfrak{z}_R,+\infty)$, any point $\mu$ from the set (3.16) cannot be an eigenvalue of the pencil $\mu\mapsto\mathfrak{A}^\mathfrak{z}_\beta(\mu)$, that is, mapping (3.15) is an isomorphism for $\mu\in\mathbb{B}_{\circledR}\setminus\mathbb{M}_{\circledR}$. This verifies the required assertion in the case $\gamma=2$.
2) In situation (1.18), the limit problem (3.17) loses the parameter $\mu$ and according to comments to (2.26) and (2.27), the general solution of the homogeneous problem in the space $V^1_\beta(\Omega;\mathcal{P})$ assumes the form where $a^0_0\in \mathbb{C}$ and $b\in\mathbb{C}_\bot^J$. Now formulas (3.25) and replace formulas (3.21) and (3.23), so that the transmission conditions (3.7) and relations (3.24), (3.26) result in the system of linear algebraic equations Applying the orthogonal projector $\mathbb{P}$ and observing that, as above, the mapping
$$
\begin{equation*}
\mathfrak{z}\mathbb{P}+\mathbf{G}^0-\mathbb{P}\mathbf{W}^\mu\mathbb{P}\colon \mathbb{C}_\bot^J\,\to\,\mathbb{C}_\bot^J,
\end{equation*}
\notag
$$
is invertible for large $\mathfrak{z}$ (small $\varepsilon>0$), we first derive from condition (2.14) that
$$
\begin{equation*}
\mathbf{G}^0b+\mathfrak{z}b=\mathbb{P}\mathbf{W}^\mu b\quad\Longleftrightarrow\quad b=0\in \mathbb{C}_\bot^J
\end{equation*}
\notag
$$
and then obtain the equation $a^0_0=J^{-1}\mathbf{e}^\top\mathbf{W}^\mu b=0$, which leads to the required contradiction $\overrightarrow{w}^{\,\mathfrak{z}}=0$. This completes the proof of Proposition 3. We next need a way of integration by parts, which is usually called the generalized Green formula (cf. [30], Ch. 6, [23], etc.). Lemma 3. The vector functions $\overrightarrow{w}^{(1)},\overrightarrow{w}^{(2)} \in\mathfrak{V}_\beta$ satisfy Proof. Relation (3.27) is verified by a calculation based on the Green formulas in the domain with small holes $\Omega(\alpha)=\Omega\setminus \bigcup_{j=1}^J\mathbb{B}_\alpha(P_j)$ and in a large disc $\mathbb{B}_{1/\alpha}\subset\mathbb{R}^2$ with subsequent limit passage as $\alpha\to+0$. The left-hand side of (3.27) is equal to
$$
\begin{equation}
\begin{aligned} \, &\lim_{\alpha\to+0}\sum_{j=1}^J \bigl( \bigl(\partial_{r_j}w^{(1)}_0,w^{(2)}_0\bigr)_{\partial\mathbb{B}_\alpha(P_j)}- \bigl(w^{(1)}_0,\partial_{r_j}w^{(2)}_0\bigr)_{\partial\mathbb{B}_\alpha(P_j)}\bigr) \nonumber \\ &\ \qquad-\lim_{\alpha\to+0}\sum_{j=1}^J \bigl(\bigl(\partial_{\rho_j}w^{(1)}_j,w^{(2)}_j\bigr)_{\partial\mathbb{B}_{1/\alpha}}- \bigl(w^{(1)}_j,\partial_{\rho_j}w^{(2)}_j\bigr)_{\partial\mathbb{B}_{1/\alpha}}\bigr) \nonumber \\ &\ =\lim_{\alpha\to+0}\sum_{j=1}^J\int_0^{2\pi}\biggl(-\frac{b^{0(1)}_j}{2\pi} \biggl({\overline{\frac{b^{0(2)}_j}{2\pi}}} \ln\frac{1}{\alpha}+ \overline{a^{0(2)}_j}\biggr)+\overline{\frac{b^{0(2)}_j}{2\pi}}\biggl( \frac{b^{0(1)}_j}{2\pi}\ln\frac{1}{\alpha}+a^{0(1)}_j\biggr)\biggr)\, d\varphi_j \nonumber \\ &\ \qquad+\lim_{\alpha\to+0}\sum_{j=1}^J\int_0^{2\pi} \biggl(-\frac{b^{(1)}_j}{2\pi} \biggl(\overline{\frac{b^{(2)}_j}{2\pi}} \ln\frac{1}{\alpha}+ \overline{a^{(2)}_j}\biggr) -\overline{\frac{b^{(2)}_j}{2\pi}}\biggl( \frac{b^{(1)}_j}{2\pi} \ln\frac{1}{\alpha}+a^{(1)}_j\biggr)\biggr)\, d\varphi_j \nonumber \\ &\ =\sum_{j=1}^J\bigl(\overline{b^{0(2)}_j}a^{0(1)}_j-b^{0(1)}_j\overline{a^{0(2)}_j}- \overline{b^{(2)}_j}a^{(1)}_j+b^{(1)}_j\overline{a^{(2)}_j}\bigr). \end{aligned}
\end{equation}
\tag{3.28}
$$
It remains to recall the definition of projections (3.11). The lemma is proved. We state another property of the pencil $\mathfrak{A}_\beta^\mathfrak{z}$. Proposition 4. The eigenvalues of the pencil (3.15) are real. Proof. Let $\{\mu,\overrightarrow{w}\}$ be an eigenvalue of the pencil. Applying the generalized Green formula (3.27), we obtain
$$
\begin{equation}
\begin{aligned} \, \mu\biggl((w_0,w_0)_\Omega+\sum_{j=1}^J (w_j,w_j)_{\omega_j}\biggr) &=-(\Delta_xw_0,w_0)_\Omega-\sum_{j=1}^J (\Delta_\xi w_j,w_j)_{\mathbb{R}^2} \nonumber \\ &= -(w_0,\Delta_xw_0)_\Omega-\sum_{j=1}^J (w_j,\Delta_\xi w_j)_{\mathbb{R}^2}+\mathfrak{J}. \end{aligned}
\end{equation}
\tag{3.29}
$$
The last term, which involves inner products in $\mathbb{C}^J$ of projections of the vector $\overrightarrow{w}$ from the right-hand side of equation (3.27) is transformed via the algebraic relations (3.12) and (3.13) as follows:
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{J} &=(\pi^+_\Omega\overrightarrow{w},\pi^-_\Omega\overrightarrow{w})_{\mathbb{C}^J}- (\pi^-_\Omega\overrightarrow{w},\pi^+_\Omega\overrightarrow{w})_{\mathbb{C}^J}- (\pi^+_\omega\overrightarrow{w},\pi^-_\omega\overrightarrow{w})_{\mathbb{C}^J}+ (\pi^-_\omega\overrightarrow{w},\pi^+_\omega\overrightarrow{w})_{\mathbb{C}^J} \\ &=(\pi^-_\Omega\overrightarrow{w},\pi^+_\Omega\overrightarrow{w}- \pi^+_\omega\overrightarrow{w})_{\mathbb{C}^J}- (\pi^+_\Omega\overrightarrow{w}-\pi^+_\omega\overrightarrow{w}, \pi^-_\Omega\overrightarrow{w})_{\mathbb{C}^J} \\ &=-\mathfrak{z}^{-1}(\pi^+_\Omega\overrightarrow{w}- \pi^+_\omega\overrightarrow{w},\pi^+_\Omega\overrightarrow{w}- \pi^+_\omega\overrightarrow{w})_{\mathbb{C}^J}{+}\, \mathfrak{z}^{-1}(\pi^+_\Omega\overrightarrow{w}- \pi^+_\omega\overrightarrow{w},\pi^+_\Omega\overrightarrow{w}- \pi^+_\omega\overrightarrow{w})_{\mathbb{C}^J}\,{=}\,0. \end{aligned}
\end{equation*}
\notag
$$
It remains to mention that the right-hand side of (3.29) is real, because it concides with its complex conjugate. Thus, $\mu\in\mathbb{R}$, the result required. 3.5. Spectrum of the pencilAccording to Propositions 3, 4 and the analytic Fredholm alternative [32], Theorem 1.5.1, the spectrum $\sigma^\mathfrak{z}_\beta$ consists of normal eigenvalues in the real axis, and, for any $R>0$, the interval $[-R,-1/R]\subset\mathbb{R}_-$ is free of the eigenvalues for large $\mathfrak{z}$. Remark 2 will explain that each eigenvalue is algebraically simple, that is, the eigenvectors have no associated vectors. We formulate the final result. Theorem 2. For any $R$, there exists $\mathfrak{z}^0_R>0$ such that, for $\mathfrak{z}\geqslant\mathfrak{z}^0_R$ the spectrum of the pencil (3.15) inside the disc $\mathbb{B}_\circledR\subset\mathbb{C}$ contains only eigenvalues from the non-positive monotone sequence arranged with due account of the geometric multiplicity. All the eigenvalues are algebraically simple. The null eigenvalue corresponds to the constant vector function $\overrightarrow{w}^{\,\mathfrak z}_1= (1,1,\dots,1)$ with the projections $\pi^-_\Omega\overrightarrow{w}^{\,\mathfrak z}_1= \pi^-_\omega\overrightarrow{w}^{\,\mathfrak z}_1=0\in \mathbb{C}^J$ and $\pi^+_\Omega \overrightarrow{w}^{\,\mathfrak z}_1= \pi^+_\omega\overrightarrow{w}^{\,\mathfrak z}_1=\mathbf{e}$. The first (strict) inequality in (3.30) is secured by Proposition 5 and Theorem 8. § 4. The operator model4.1. The limit passage and a remark on analyticitySetting $\varepsilon=0$ and $\mathfrak{z}=+\infty$ in formula (3.15), we get the operator pencil
$$
\begin{equation}
\mathfrak{A}^{\infty}_\beta(\mu):=(\mathfrak{B}_\beta-\mu\mathfrak{M}_\beta, \pi^-_\Omega- \pi^-_\omega ,\pi^-_\Omega)\colon \mathfrak{V}_\beta\to \mathfrak{R}_\beta\times \mathbb{C}^J\times \mathbb{C}^J .
\end{equation}
\tag{4.1}
$$
The structure of the algebraical components of this pencil demonstrates that projections (3.11) of the vector $\overrightarrow{w}^{\,\infty}\in\mathfrak{V}_\beta$ satisfy
$$
\begin{equation}
\pi^-_\Omega\overrightarrow{w}^{\,\infty}=\pi^-_\omega\overrightarrow{w}^{\,\infty}=0\in\mathbb{C}^J.
\end{equation}
\tag{4.2}
$$
Hence, the eigenvectors lie in the subspace
$$
\begin{equation*}
\mathfrak{V}^+_\beta=\{ \overrightarrow{w}^{\,\infty}\in \mathfrak{V}_\beta\colon \pi^-_\Omega\overrightarrow{w}^{\,\infty}=\pi^-_\omega\overrightarrow{w}^{\,\infty}=0\in \mathbb{C}^J\},
\end{equation*}
\notag
$$
where decompositions (2.23) and (2.32) of components of vector functions do not contain logarithms, so that
$$
\begin{equation}
\mathfrak{V}^+_\beta\subset \mathfrak{H}:=H^1(\Omega)\times\prod_{j=1}^J\mathcal{H}_j.
\end{equation}
\tag{4.3}
$$
Thus, according to (4.2), Propositions 1 and 2 mean that the restriction $\mathfrak{A}^+_\beta(\mu)$ of operator (4.1) onto the subspace $\mathfrak{V}^+_\beta$ inherits all general properties of the operator
$$
\begin{equation*}
\bigl(\mathcal{B}^+_0(\mu),\mathcal{B}^+_1(\mu),\dots, \mathcal{B}^+_J(\mu)\bigr)\colon \mathfrak{H}\to\mathfrak{H}^\ast
\end{equation*}
\notag
$$
of the limit problems (3.17) and (3.18), $j=1,\dots,J$, in particular, the spectrum (1.9) is the union of sequences (2.6), $j=1,\dots,J$, as well as (2.8) in case $\gamma=2$. Besides, in view of (2.10) and (2.7), the eigenvectors $\overrightarrow{w}^{\,\infty}_{(p)}=(w_{0(p)}^\infty, w_{1(p)}^\infty,\dots,w_{J(p)}^\infty)$ of the pencil satisfy the orthogonality and normalization conditions
$$
\begin{equation}
I_\gamma(\overrightarrow{w}^{\,\infty}_{(p)},\overrightarrow{w}^{\,\infty}_{(q)} ):=\delta_{\gamma,2}\bigl(w^\infty_{0(p)},w^\infty_{0(q)}\bigr)_\Omega+ \sum_{j=1}^J\bigl(w^\infty_{j(p)},w^\infty_{j(q)}\bigr)_{\omega_j}=\delta_{p,q}, \quad p,q\in \mathbb{N}.
\end{equation}
\tag{4.4}
$$
Remark 2. Since the pencils (3.15) and (4.1), which differ from each other by a finite-dimensional operator with the norm $O(\mathfrak{z}^{-1})$, the relations (4.4) for the eigenvectors $\overrightarrow{w}^{\,\mathfrak{z}}_{(p)}\in \mathfrak{V}_\beta$, $p\in\mathbb{N}$, of the pencil $\mu\mapsto\mathfrak{A}^\mathfrak{z}_\beta(\mu)$ also have infinitesimal perturbations as $\mathfrak{z} \to+\infty$. Thus, there are no vectors associated with the eigenvector $\overrightarrow{w}^{\mathfrak{z},1}_{(p)}\in \mathfrak{V}_\beta$, because the compatibility conditions in the equations for them Now the next result is a consequence of classical results of the perturbation theory of linear operators (see, for example, [25], Ch. 7, and [32], Ch. 1). Theorem 3. The entries of sequences (3.30) and (1.9) are related by In particular, for any $k\in\mathbb{N}$, for $\mathfrak{z}\in[\mathfrak{z}^0_k,+\infty)$ with some $\mathfrak{z}^0_k>0$ and $c^0_k>0$. We also mention that pencil (3.15) depends linearly on the small parameter $1/\mathfrak{z}$, and, by Theorem 7.1.8 in [25], in the case of a simple limit eigenvalue $\mu_k$ the function
$$
\begin{equation}
\biggl[0,\frac1{\mathfrak{z}^0_k}\biggr]\ni\mathfrak{z}\mapsto\mu^\mathfrak{z}_k
\end{equation}
\tag{4.6}
$$
is analytic for a certain $\mathfrak{z}^0_k>0$. However, this theorem suggest that the multiple eigenvalues $\mu_k^\mathfrak{z},\dots, \mu^\mathfrak{z}_{k+\varkappa_k-1}$ may have much more involved behaviour as $\mathfrak{z}\to+\infty$; that is, for $\varkappa_k>1$ and
$$
\begin{equation}
\mu_{k-1}<\mu_k=\dots=\mu_{k+\varkappa_k-1}<\mu_{k+\varkappa_k},
\end{equation}
\tag{4.7}
$$
We will discuss the analyticity problem also in § 4.5. 4.2. Asymptotics in inverse powers of logarithms; case (1.17)We construct an asymptotics of the eigenvalues $\mu^\mathfrak{z}_k$ of pencil (3.15) in inverse powers of the large parameter (3.8). In what follows, objects of the asymptotic analysis are real. First of all, we consider the case $\gamma=2$, where all limit problems are spectral. Let first $\mu^\infty_k=\mu_k$ be a simple eigenvalue of the pencil $\mathfrak{A}^\infty_\beta$, and the corresponding eigenvector $\overrightarrow{w}^{\,\infty}_{(k)}=\overrightarrow{w}_{(k)}\in\mathfrak{V}_\beta$ be subject to the normalization condition (4.4) for $p=q=k$. We accept the simplest asymptotic ansätze
$$
\begin{equation}
\mu^\mathfrak{z}_p=\mu^\infty_p+\mathfrak{z}^{-1}\mu'_p+ \widetilde{\mu}^{\,\mathfrak z}_p
\end{equation}
\tag{4.8}
$$
and
$$
\begin{equation}
\overrightarrow{w}^{\,\mathfrak{z}}_{(p)}=\overrightarrow{w}^{\,\infty}_{(p)}+\mathfrak{z}^{-1} \overrightarrow{w}^{\,\prime}_{(p)}+\widetilde{\overrightarrow{w}}^{\,\mathfrak z}_{(p)}
\end{equation}
\tag{4.9}
$$
with index $p=k$ and small remainders, which were estimated in Theorem 4. Inserting ansätze (4.9) and (4.8) into the abstract equation
$$
\begin{equation*}
\mathfrak{A}_\beta^\mathfrak{z}(\mu^\mathfrak{z}) \overrightarrow{w}^{\,\mathfrak{z}}=0\in \mathfrak{R}_\beta\times\mathbb{C}^J\times\mathbb{C}^J,
\end{equation*}
\notag
$$
and observing that that terms of order $1$ cancel each othe, we collect the multipliers of $\mathfrak{z}^{-1}$. As a result, for the the asymptotic correction terms, we have the differential problems
$$
\begin{equation}
\begin{gathered} \, -\Delta_xw'_{0(p)}(x)-\delta_{\gamma,2}\mu^\infty_pw'_{0(p)}(x)= \delta_{\gamma,2}\mu'_pw^\infty_{0(p)}(x),\qquad x\in\Omega, \\ \partial_nw'_{0(p)}(x)=0,\qquad x\in\partial\Omega, \end{gathered}
\end{equation}
\tag{4.10}
$$
$$
\begin{equation}
-\Delta_\xi w'_{j(p)}(\xi)-\mu^\infty_pX_j(\xi) w'_{j(p)}(\xi) =\mu'_pX_j(\xi)w^\infty_{j(p)}(\xi),\qquad \xi\in\mathbb{R}^2,\quad j=1,\dots,J,
\end{equation}
\tag{4.11}
$$
and the algebraic equations
$$
\begin{equation}
\pi^-_\Omega\overrightarrow{w}^{\,\prime}_{(p)}-\pi^-_\omega\overrightarrow{w}^{\,\prime}_{(p)} =0\in\mathbb{C}^J,\qquad \pi^-_\Omega\overrightarrow{w}^{\,\prime}_{(p)} =\pi^+_\omega\overrightarrow{w}^{\,\infty}_{(p)}-\pi^+_\Omega\overrightarrow{w}^{\,\infty}_{(p)},
\end{equation}
\tag{4.12}
$$
where $p=k$ and the Kronecker delta $\delta_{\gamma,2}$ was introduced in order to apply formulas also in case (1.18). The right-hand side of the second algebraic transmission condition (4.12) has appeared because of the presence of the small parameter $\mathfrak{z}^{-1}$ in the original conditions (3.13). Since the eigenvalue $\mu^\infty_k$ is simple, there is only one compatibility condition for problem (4.10)–(4.12), which we find by inserting the vector functions $\overrightarrow{w}^{\,\prime}_{(k)}$ and $\overrightarrow{w}^{\,\infty}_{(k)}$ into the generalized Green formula (3.27), thereby fixing the correction term $\mu_p'$. Taking into account the normalization condition (4.4) for $\overrightarrow{w}^{\,\infty}_{(k)}$, we get
$$
\begin{equation}
\begin{aligned} \, \mu'_p &=\mu'_p\biggl( \delta_{\gamma,2}(w_{0(k)},w_{0(k)})_\Omega+ \sum_{j=1}^J(w_{j(k)},w_{j(k)})_{\omega_j}\biggr) \nonumber \\ &=-\bigl(\Delta_x w'_{0(k)}+\delta_{\gamma,2}\mu^\infty_kw'_{0(k)},w^\infty_{0(k)}\bigr)_\Omega- \sum_{j=1}^J\bigl(\Delta_\xi w_{j(k)}+\mu^\infty_kX_jw'_{0(k)},w^\infty_{j(k)}\bigr)_{\mathbb{R}^2} \nonumber \\ &=-\bigl(\pi^+_\Omega\overrightarrow{w}^{\,\infty}_{(k)}\bigr)^\top \pi^-_\Omega\overrightarrow{w}^{\,\prime}_{(k)}+ \bigl(\pi^+_\omega\overrightarrow{w}^{\,\infty}_{(k)}\bigr)^\top \pi^-_\omega\overrightarrow{w}^{\,\prime}_{(k)} \nonumber \\ &=\bigl(\pi^+_\omega\overrightarrow{w}^{\,\infty}_{(k)}-\pi^+_\Omega\overrightarrow{w}^{\,\infty}_{(k)} \bigr)^\top \pi^-_\Omega\overrightarrow{w}^{\,\prime}_{(k)}=\bigl| \pi^+_\Omega\overrightarrow{w}^{\,\infty}_{(k)} -\pi^+_\omega\overrightarrow{w}^{\,\infty}_{(k)} \bigr|^2. \end{aligned}
\end{equation}
\tag{4.13}
$$
Here, we took into account the integral identities (2.8) for $w^\infty_{0(k)}\in H^1(\Omega)$, and (2.2), for $w^\infty_{j(k)}\in \mathcal{H}_j$; these inclusions are accompanied by relations (4.2). The last couple of equalities in (4.13) follow from algebraic conditions (4.12). Finally formula (4.13) (without its middle part) gives the correction term in the asymptotic expansion (4.8) of the eigenvalue $\mu^\mathfrak{z}_k$, which is simple by Theorem 3 and our assumption on the limit eigenvalue $\mu^\infty_k$. Furthermore, from problem (4.10)–(4.12), which is now solvable, we get the correction term in the asymptotic expansion (4.9) of the eigenvector $\overrightarrow{w}_{(k)}^\mathfrak{z}$ (up to the term $c'_p\overrightarrow{w}^{\,\infty}_{(k)}$). Similar asymptotic procedure also apply to the multiple eigenvalue $\mu_k$ from (4.7). Ansaätze (4.8) and (4.9) are accepted for the eigenpairs
$$
\begin{equation*}
\bigl\{\mu^\mathfrak{z}_k;\overrightarrow{w}_{(k)}^\mathfrak{z}\bigr\}, \dots, \bigl\{\mu^\mathfrak{z}_{k+\varkappa_k-1};\overrightarrow{w}_{(k+\varkappa_k-1)}^\mathfrak{z} \bigr\}
\end{equation*}
\notag
$$
while $\mu^\infty_k=\dots=\mu^\infty_{k+\varkappa_k-1}=\mu_k$ and
$$
\begin{equation}
\overrightarrow{w}^{\,\infty}_{(p)}=c^p_k\overrightarrow{w}_{(k)}+\dots+ c^p_{k+\varkappa_k-1}\overrightarrow{w}_{(k+\varkappa_k-1)},\qquad p=k,\dots,k+\varkappa_k-1,
\end{equation}
\tag{4.14}
$$
where $\overrightarrow{w}_{(k)},\dots,\overrightarrow{w}_{(k+\varkappa_k-1)}\in \ker \mathfrak{A}^\infty_\beta(\mu_k)$ are eigenvectors of the pencil $\mathfrak{A}^\infty_\beta$, which correspond to its eigenvalue $\mu_k$ and are subject to the orthogonality and normalization conditions (4.4), and the unknown columns $c^p=(c^p_k,\dots,c^p_{k+\varkappa_k-1})^\top$ in the linear combinations (4.14) satisfy
$$
\begin{equation}
(c^p)^\top c^q=\delta_{p,q},\qquad p,q=k,\dots,k+\varkappa_k-1.
\end{equation}
\tag{4.15}
$$
The correction terms $\mu'_k$ and $\overrightarrow{w}^{\,\prime}_{(p)}$ in ansätze (4.8) and (4.9) can be found from problems (4.10)–(4.12), which, according to assumption (4.7), have $\varkappa_k$ compatibility conditions. Repeating calculation (3.28) several times, we find that these conditions take form of the system of linear algebraic equations
$$
\begin{equation}
\mu'_p c^p_q= \sum_{m=k}^{k+\varkappa_k-1} \mathbf{T}^p_{qm}c^p_{m},\qquad q=k,\dots,k+\varkappa_k-1.
\end{equation}
\tag{4.16}
$$
Here, the entries of the symmetric ($\varkappa_k\times\varkappa_k$)-matrix $\mathbf{T}^p$ are as follows:
$$
\begin{equation}
\mathbf{T}^p_{qm}=\bigl(\pi^+_\Omega\overrightarrow{w}^{\,\infty}_{(m)}- \pi^+_\omega\overrightarrow{w}^{\,\infty}_{(m)} \bigr)^\top \bigl(\pi^+_\Omega\overrightarrow{w}^{\,\infty}_{(q)}-\pi^+_\omega\overrightarrow{w}^{\,\infty}_{(q)} \bigr).
\end{equation}
\tag{4.17}
$$
It is clear that $\mathbf{T}^p_{(2)}:=\mathbf{T}^p$ is a Gram matrix, which is symmetric and positive, but is not necessarily positive definite (see examples in § 5). Hence, system (4.16) has non-negative real eigenvalues
$$
\begin{equation}
0\leqslant\mu'_k\leqslant\dots\leqslant \mu'_{k+\varkappa_k-1},
\end{equation}
\tag{4.18}
$$
which specify the correction terms in expansions (4.17) of the eigenvalues of the operator pencil $\mathfrak{A}^\mathfrak{z}_\beta$. From the corresponding eigenvectors, that is, from the columns $c^k,\dots,c^{k+\varkappa_k-1} \in\mathbb{R}^J$ satisfying (4.15), one can find both the main terms (4.14) in expansions (4.9) of the eigenvectors, and their correction terms (from problem (4.10)–(4.12), in which the compatibility conditions (4.16) are fulfilled, and, of course, the terms $\overrightarrow{w}^{\,\prime}_{(p)}$ are determined up to a linear combination of eigenvectors $\overrightarrow{w}_{(k)},\dots,\overrightarrow{w}_{(k+\varkappa_k-1)}$). This completes the construction of the initial terms in asymptotics of eigenpairs of pencil (3.15) is completed in case (1.17). 4.3. Asymptotics in inverse powers of logarithm; case (1.18)We consider a multiple eigenvalue $\mu_k$ from formula (4.7): for a simple eigenvalue it is necessary to set $\varkappa_k=1$ in further calculations. The limit problem in the punctured domain $\Omega\setminus\mathcal{P}$ has no spectral parameter, and its solution takes the form (3.25). Moreover, the orthogonality and normalization conditions (4.4) lose the first inner product in $L^2(\Omega)$, so that
$$
\begin{equation}
\pi^+_\Omega\overrightarrow{w}^{\,\infty}_{(p)}=C_p\mathbf{e},\qquad \pi^+_\Omega\overrightarrow{w}^{\,\infty}_{(p)}=0,\qquad \pi^-_\Omega\overrightarrow{w}^{\,\prime}_{(p)}=\pi^-_\omega\overrightarrow{w}^{\,\prime}_{(p)}= \mathbb{R}^J_\bot:=\mathbb{P}\mathbb{R}^J.
\end{equation}
\tag{4.19}
$$
The factors $C_k,\dots,C_{k+\varkappa_k-1}$ do not have a direct relation to the eigenvectors $\overrightarrow{w}_{(k)},\dots, \overrightarrow{w}_{(k+\varkappa_k-1)}$ but they and coefficients of linear combinations (4.14) are to be found mutually. We apply the projectors $\mathbb{P}$ and $\mathbb{I}_J-\mathbb{P}$ to the second line in (4.12), and derive from (4.19) the formulas
$$
\begin{equation}
\pi^-_\Omega\overrightarrow{w}^{\,\prime}_{(p)}=\pi^-_\omega\overrightarrow{w}^{\,\prime}_{(p)}= \mathbb{P}\pi^+_\omega\overrightarrow{w}^{\,\infty}_{(p)},
\end{equation}
\tag{4.20}
$$
$$
\begin{equation}
C_p=J^{-1}\mathbf{e}^\top\pi^+_\omega\overrightarrow{w}^{\,\infty}_{(p)}.
\end{equation}
\tag{4.21}
$$
Expression (4.21) specifies the first column in (4.19), and expressions (4.20) indicate singular components of the solutions $\overrightarrow{w}^{\,\prime}_{(k)},\dots, \overrightarrow{w}^{\,\prime}_{(k+\varkappa_k-1)}$ of equations (4.11) in the plane. We emphasize that the Neumann problem (4.10) is homogeneous, because the parameters $\mu_p^\infty$ and $\mu_p'$ are absent due to the factor $\delta_{\gamma,2}$, so that there is no need to verify the compatibility conditions, and the solution takes the form $w_{(p)}'(x)=a^{0\,\prime}_0+G(x)b^{0\,\prime}$ with coefficients $a^{0\,\prime}_0\in\mathbb{R}$ and $b^{0\,\prime}\in\mathbb{R}^J_\bot$, which can be found at the next steps of the asymptotic procedure. The compatibility conditions in the obtained problem (4.11), (4.20) are found via (4.13), which still involves the vectors $\overrightarrow{w}^{\,\prime}_{(p)}$ and $\overrightarrow{w}_{(m)}$, but requires a modification of the last transition only. As a result, according to relations (4.20) and (4.21), we derive the following system of algebraic equations:
$$
\begin{equation}
\begin{aligned} \, \mu'_p c^p_m &= -\bigl(\pi^+_\Omega\overrightarrow{w}_{(m)}\bigr)^\top \pi^-_\Omega\overrightarrow{w}^{\,\prime}_{(p)} +\bigl(\pi^+_\omega\overrightarrow{w}_{(m)}\bigr)^\top \pi^-_\omega\overrightarrow{w}^{\,\prime}_{(p)} \nonumber \\ &=-0+\sum_{q=k}^{k+\varkappa_k-1} c^p_q \bigl(\pi^+_\omega\overrightarrow{w}_{(m)}\bigr)^\top \mathbb{P}\pi^+_\omega\overrightarrow{w}_{(q)},\qquad m=k,\dots,k+\varkappa_k-1. \end{aligned}
\end{equation}
\tag{4.22}
$$
In the matrix form, this system reads as
$$
\begin{equation}
\mu'_p c^p= \mathbf{T}^kc^p\ \text{ in subspace }\mathbb{R}^J_\bot.
\end{equation}
\tag{4.23}
$$
$\mathbf{T}^k_{(\gamma)}:=\mathbf{T}^k= (\mathbf{T}^k_{mq})_{m,q=k}^{k+\varkappa_k-1}$ is a symmetric and positive ($\varkappa_k\times \varkappa_k$)-matrix with entries
$$
\begin{equation}
\mathbf{T}^k_{mq}=\bigl(\mathbb{P}\pi^+_\omega\overrightarrow{w}_{(m)}\bigr)^\top \mathbb{P}\pi^+_\omega\overrightarrow{w}_{(q)}.
\end{equation}
\tag{4.24}
$$
System (4.23) possesses the eigenvalue family (4.18). Thus, in case (1.18), we have found the correction term $\mathfrak{z}^{-1}\mu'_p$ for ansatz (4.8), and obtained the terms $\mathfrak{z}^{-1}\overrightarrow{w}^{\,\prime}_{(p)}$ for ansatz (4.9) (with arbitrariness specified above). 4.4. Theorem on logarithmic asymptoticsWe formulate error estimates for asymptotics of eigenvalues, which are secured by the classical results of the perturbation theory of linear operators (see, for example, [25], [32], [33]). Theorem 4. For any $m\in\mathbb{N}\setminus \{1\}$, there exist positive numbers $\mathfrak{z}^0_m$ and $c^0_m$ such that, for $\mathfrak{z}\geqslant\mathfrak{z}^0_m$ (that is, for $\varepsilon\in(0,\varepsilon^0_m]$ with some $\varepsilon^0_m>0$), the first $m$ positive terms in sequence (3.30) of eigenvalues of pencil (3.15) admit the asymptotic formulas Furthermore, the eigenvectors $\overrightarrow{w}^{\,\mathfrak{z}}_k$ and $\overrightarrow{w}_k$ of pencils (3.15) and (4.1), which correspond to the simple ($\varkappa=1$) eigenvalue $\mu_k\in M$ and meet the orthogonality and normalization conditions (4.4), satisfy the inequality
$$
\begin{equation}
\|\overrightarrow{w}^{\,\mathfrak{z}}_k-\overrightarrow{w}^{\,\infty}_k;\mathfrak{H}\|\leqslant c^0_k\,\mathfrak{z}^{-1},
\end{equation}
\tag{4.26}
$$
where $\overrightarrow{w}^{\,\infty}_k=\overrightarrow{w}_k$. In case of a multiple eigenvalue (see (4.7)) $\overrightarrow{w}^{\,\infty}_k$ become linear combinations (4.14) with certain coefficient columns $c^p(\mathfrak{z})$ which depend on $\mathfrak{z}$ and satisfy the orthogonality and normalization conditions (4.15). We mention that correction terms $\overrightarrow{w}^{\,\prime}_k$ are found from problems (4.10)–(4.12) with some natural arbitrariness, that is, they are not defined yet, and, therefore, are excluded from the left-hand sides of estimate (4.26). The asymptotic procedure, of course, can be continued so that infinite formal asymptotic series can be obtained (we discuss their convergence in the nest section). If the eigenvalue $\overrightarrow{w}^{\,\prime}_k$ is simple, the correction term can be computed at the next step of the iterative process. 4.5. Remark on analyticityIn both cases (1.17) and (1.18), the matrix $\mathbf{T}^k_{(\gamma)}$ can be degenerate, and, in particular, may have multiple null eigenvalue. Family (4.18) can also contain multiple positive eigenvalues. At the same time, if the eigenvalue $\mu'_q$ from (4.18) is simple, then Theorem 4 assures that the eigenvalue $\mu^\mathfrak{z}_q$ of pencil (3.15) is also simple; by applying Theorem 7.1.8 in [25] to the pencil $\mu\mapsto\mathfrak{A}_\beta^\mathfrak{z}(\mu-\mu_k- \mathfrak{z}^{-1}\mu'_q)$ we guarantee analyticity of function (4.6) with index $k=q$. If $\mu'_q$ is an eigenvalue of the matrix $\mathbf{T}^k_{(\gamma)}$ of multiplicity $\varkappa_{kq}>1$, then classical results are not capable of directly specifying the quality of the dependence of the eigenvalues $\mu^\mathfrak{z}_q,\dots, \mu^\mathfrak{z}_{q+\varkappa_{kq}-1}$ on the small parameter $1/\mathfrak{z}$. In this way, one has to construct higher-order asymptotic terms in (4.8) for $p=q,\dots,q+\varkappa_{kq}-1$. Thanks to a primitive dependence of the pencil $\mathfrak{A}_\beta^\mathfrak{z}$ on $1/\mathfrak{z}$, the next steps of the asymptotic procedure do not differ from the executed steps. However, two situations may occur. First of all, at the level $\mathfrak{z}^{-N_q}$ one may meet the asymptotic splitting of eigenvalues, that is, all the numbers $\mathfrak{z}^{-N_q}\mu^{N_q}_q,\dots, \mathfrak{z}^{-N_q}\mu^{N_q}_{q+\varkappa_{kq}-1}$ become different and, hence, functions (4.6) with indexes $k=q,\dots,q+\varkappa_{kq}-1$ are analytic by the above. If the splitting does not appear after a finite number of steps of the process, then again two possibilities may occur. First, the eigenvalue
$$
\begin{equation}
\mu^\mathfrak{z}_q=\dots=\mu^\mathfrak{z}_{q+\varkappa_{kq}-1}
\end{equation}
\tag{4.27}
$$
of the perturbed pencil (3.15) is still multiple, for instance, by the geometric symmetry of the original compound domain $\Omega^\varepsilon\,\cup\,\omega^\varepsilon$ (cf. Fig. 1, (b)). Next, the theorem from [25] just cited implies analyticity of quantities (4.27). Second, splitting of eigenvalues may occur at super-power-law level with respect to the parameter $1/\mathfrak{z}$, so that the analyticity, which is now an open problem, cannot be verified by a direct reference. We also mention that, for several ($J>1$) congruent inclusions (1.1) (Fig. 1, (a)), all (the case $\gamma>2$) or many (the case $\gamma=2$) eigenvalues in the sequence $M$ are multiple. By using the geometric symmetry in Fig. 1, (b), the original problem can be reduced to a part of the domain $\Omega$ by imposing artificial boundary conditions, and so, in the majority of cases, here one has to deal only with simple eigenvalues. We formulate the corresponding result in a reduced form and give some comments. Theorem 5. Let an eigenvalue $\mu_k$ of pencil (3.15) be simple or let an eigenvalue $\mu_q'$ of the matrix $\mathbf{T}^k_{(\gamma)}$ be simple in the case of multiple $\mu_k$ (see (4.6) and (4.18)). Then function (4.6) with index $k$ or $q$ is analytic, and the corresponding eigenvector $\overrightarrow{w}^{\,\mathfrak{z}}_k$ or $\overrightarrow{w}^{\,\mathfrak{z}}_k$ normalized by (4.4) depends analytically on the small parameter $1/\mathfrak{z}$. We emphasize that Theorem 5 deals only with model (3.15), rather than with problem (1.2), (1.3) in the compound domain $\Omega^\varepsilon\cup\omega^\varepsilon$. However, the present paper does not give an answer about analyticity of eigenvalues (1.6) with respect to the couple of small parameters $\varepsilon$ and $1/|{\ln\varepsilon}|$. It is quite possible that open questions can be solved with the help of the machinery developed in [19], [20], [34], etc., but at present, there are no available results on eigenvalues for problems with concentrated masses. § 5. Perturbation of null eigenvalue5.1. The case $\gamma=2$The eigenvalue $\mu=0$ of the limit pencil $\mathfrak{A}^\infty_\beta$ corresponds to the eigenvectors
$$
\begin{equation}
\overrightarrow{w}^{\,\infty}_{(j)}=(0,|\omega_1|^{-1/2}\delta_{1,j}, \dots,|\omega_J|^{-1/2}\delta_{J,j}), \qquad j=1,\dots,J,
\end{equation}
\tag{5.1}
$$
$$
\begin{equation}
\overrightarrow{w}^{\,\infty}_{(J+1)}=(|\Omega|^{-1/2},0,\dots,0).
\end{equation}
\tag{5.2}
$$
The matrix $\mathbf{T}^1$ of size $(J+1)\times(J+1)$ with entries (4.17) takes form
$$
\begin{equation}
\mathbf{T}^1= \begin{pmatrix} \mathbb{T}_\omega & -|\Omega|^{-1/2}t_\omega \\ -|\Omega|^{-1/2} t_\omega^\top & J|\Omega|^{-1} \end{pmatrix},
\end{equation}
\tag{5.3}
$$
where $\mathbb{T}_\omega$ is the diagonal positive definite ($J\times J$)-matrix $\operatorname{diag}\{|\omega_1|^{-1}, \dots,|\omega_J|^{-1}\}$, and $t_\omega$ is the column $(|\omega_1|^{-1/2},\dots,|\omega_J|^{-1/2})^\top$. The numeration of eigenvectors (5.1), (5.2) is now different from that of their components aimed at arranging the of eigenvalues of matrix (5.3) in ascending order. Proposition 5. The symmetric positive definite matrix (5.3) has the eigenvalues Proof. Let $\bf D$ be the ($(J+1)\times (J+1)$)-matrix $\operatorname{diag}\{|\omega_1|^{1/2},\dots,|\omega_J|^{1/2},|\Omega|^{1/2}\}$. The matrix $\mathbf{D} T^1 \mathbf{D}$ has the form
$$
\begin{equation}
\begin{pmatrix} \mathbb{I}_J & -\mathbf{e} \\ -\mathbf{e}^\top & J \end{pmatrix}.
\end{equation}
\tag{5.9}
$$
The eigenpairs of matrix (5.9) are given by (5.6)–(5.8), where one has to put $|\Omega|=|\omega|=1$. Since $\bf D$ is positive definite, relation (5.4) for eigenvalues of matrix (5.3) is established. In case (5.5), verification of (5.6)–(5.8) can be made by a direct calculation. The proposition is proved. Proposition 5 and Theorem 4 demonstrate that in the particular case (5.5) of equal areas of the inclusions, the initial (small, but positive) terms of sequence (3.15) of eigenvalues of pencil (3.15) can be written as
$$
\begin{equation}
\mu^\mathfrak{z}_k=0+\mathfrak{z}^{-1}|\omega|^{-1}+\mathfrak{z}^{-2}\mu''_k +O(\mathfrak{z}^{-3}),\qquad k=2,\dots,J,
\end{equation}
\tag{5.10}
$$
$$
\begin{equation}
\mu^\mathfrak{z}_{J+1}=0+\mathfrak{z}^{-1}(|\omega|^{-1}+J|\Omega|^{-1}) +\mathfrak{z}^{-2}\mu''_{J+1}+O(\mathfrak{z}^{-3}).
\end{equation}
\tag{5.11}
$$
It has become clear that, at the level $|{\ln \varepsilon}|^{-1}=(2\pi\mathfrak{z})^{-1}$, no interaction of inclusions (1.1) occurs (cf. § 5.3 and § 7.3). We construct terms of order $\mathfrak{z}^{-2}$ in asymptotics (5.10) and (5.11). Simultaneously, we improve the asymptotic ansätze (4.9) for the eigenvectors
$$
\begin{equation}
\overrightarrow{w}^{\,\mathfrak{z}}_{(p)}=\overrightarrow{w}^{\,\infty}_{(p)}+\mathfrak{z}^{-1} \overrightarrow{w}^{\,\prime}_{(p)}+\mathfrak{z}^{-2} \overrightarrow{w}^{\,\prime\prime}_{(p)}+\cdots,\qquad p=2,\dots,J+1.
\end{equation}
\tag{5.12}
$$
The coefficients of linear combination (4.14), that is, the main term of expansion (5.12), are now determined in the two cases $p=1$ and $p=J+1$, that is, for the simple eigenvalues $\tau_1=0$ and $\tau_{J+1}$. Note that $\tau_2=\dots=\tau_J$ is a multiple eigenvalue for $J>2$ (the particular case $J=2$ will be discussed below in § 5.3). Pencil (3.15) has null eigenvalue, which is associated with the constant (non-normalized) eigenvector
$$
\begin{equation}
\overrightarrow{w}^{\,\mathfrak{z}}_1= \{1,1,\dots,1\},
\end{equation}
\tag{5.13}
$$
that is, there is no need to treat eigenpair (5.6). We consider the first correction term in the asymptotics of the another simple eigenvalue. It is associated with the constant vector function from ansatz (5.12) given by (5.1), (5.2) and (5.8), namely
$$
\begin{equation}
\overrightarrow{w}^{\,\infty}_{J+1}=\alpha_{J+1}^{-1/2}(-J|\Omega|^{-1}|\omega|^{1/2}, |\omega|^{-1/2},\dots,|\omega|^{-1/2})
\end{equation}
\tag{5.15}
$$
with the following normalization factor
$$
\begin{equation}
\alpha_{J+1}=J|\omega|^{-1}(1+J|\Omega|^{-2}|\omega|^2)
\end{equation}
\tag{5.16}
$$
and the attributes
$$
\begin{equation*}
\begin{gathered} \, \pi^-_\Omega\overrightarrow{w}^{\,\infty}_{J+1} =\pi^-_\omega\overrightarrow{w}^{\,\infty}_{J+1}=0\in\mathbb{R}^J, \\ \pi^+_\Omega\overrightarrow{w}^{\,\infty}_{J+1}=- \alpha_{J+1}^{-1/2} J|\Omega|^{-1}|\omega|^{1/2}\mathbf{e},\qquad \pi^+_\omega\overrightarrow{w}^{\,\infty}_{J+1}=\alpha_{J+1}^{-1/2} |\omega|^{-1/2}\mathbf{e}. \end{gathered}
\end{equation*}
\notag
$$
Moreover, in view of the algebraic relations (4.12) involved into the model under consideration, we have
$$
\begin{equation}
\begin{gathered} \, b'=\pi^-_\Omega\overrightarrow{w}^{\,\prime}_{J+1} =\pi^-_\omega\overrightarrow{w}^{\,\prime}_{J+1}= \pi^+_\omega\overrightarrow{w}^{\,\infty}_{J+1}-\pi^+_\Omega\overrightarrow{w}^{\,\infty}_{J+1} =A_{J+1}\mathbf{ e}, \\ A_{J+1}=\alpha_{J+1}^{-1/2}(|\omega|^{-1/2}+J|\Omega|^{-1}|\omega|^{1/2})= \alpha_{J+1}^{-1/2}|\omega|^{1/2}\mu'_{J+1}. \end{gathered}
\end{equation}
\tag{5.17}
$$
Hence, the components of the correction term $\overrightarrow{w}^{\,\prime}_{J+1}$ in expansion (5.12) are solutions of the problems
$$
\begin{equation}
\begin{gathered} \, -\Delta_xw'_{(J+1)0}(x)=A_{J+1}\sum_{j=1}^J\delta(x-P^j)- \mu'_{J+1}\alpha_{J+1}^{-1/2}J|\Omega|^{-1}|\omega|^{1/2},\qquad x\in\Omega, \\ \partial_nw'_{(J+1)0}(x)=0,\qquad x\in\partial\Omega, \end{gathered}
\end{equation}
\tag{5.18}
$$
and
$$
\begin{equation}
\begin{gathered} \, -\Delta_\xi w'_{(J+1)j}(\xi^j)=\mu'_{J+1}\alpha_{J+1}^{-1/2}|\omega|^{-1/2} X_j(\xi^j),\qquad \xi^j\in\mathbb{R}^2, \\ w'_{(J+1)j}(\xi^j)=\alpha_{J+1}^{-1/2}|\omega|^{-1/2} \Phi(\xi^j)+O(1),\qquad |\xi^j|\to+\infty, \quad j=1,\dots,J. \end{gathered}
\end{equation}
\tag{5.19}
$$
The compatibility condition in the Neumann problem (5.18) is satisfied, because “integration” in the domain $\Omega$ (based on the formula $\int_\Omega\delta(x-P^j)\, dx=1$ in the framework of the distribution theory) gives, in view of (5.17),
$$
\begin{equation*}
A_{J+1}\sum_{j=1}^J1-|\Omega| \mu'_{J+1}\alpha_{J+1}^{-1/2}J|\Omega|^{-1}|\omega|^{1/2}=0.
\end{equation*}
\notag
$$
Hence, in view of (2.11) and the comments to Proposition 1,
$$
\begin{equation}
w'_{(J+1)0}(x)=A_{J+1}\sum_{j=1}^JG_j(x) +|\Omega|^{-1/2}a'_0,
\end{equation}
\tag{5.20}
$$
where $a'_0$ is a number. To specify the functions $w'_{(J+1)j}$ satisfying (5.19), we need a new object. Lemma 4. The equation Proof. It suffices to verify that
$$
\begin{equation*}
\begin{aligned} \, 1 &=-\lim_{R\to+\infty}\int_{\mathbb{B}_R} \Delta_\xi\mathbf{w}_j(\xi)\, d\xi =-\lim_{R\to+\infty}\int_{\partial\mathbb{B}_R} \partial_\rho\mathbf{w}_j(\xi)\, ds_\xi \\ &=-\lim_{R\to+\infty} R\int_0^{2\pi} \frac{1}{2\pi}\, \frac{\partial}{\partial\rho}\ln\frac{1}{\rho}\, d\varphi=1 \end{aligned}
\end{equation*}
\notag
$$
and then employ the maximum principle, because solution (5.22) is negative in a neighbourhood of infinity and the right-hand side of the Poisson equation (5.21) is non-negative. This makes formula (5.23) evident. Lemma 4 is proved. Thus,
$$
\begin{equation}
w'_{(J+1)j}(\xi^j)=\mu'_{J+1}\alpha_{J+1}^{-1/2}|\omega|^{-1/2}\mathbf{w}_j(\xi^j) +|\omega|^{-1/2}a'_j.
\end{equation}
\tag{5.24}
$$
Here, $\mathbf{w}_j$ are the functions mentioned in Lemma 4 (note that the equalities $|\omega|=|\omega_j|$ are possible for different shapes of $\omega_1,\dots,\omega_J$) and the column $a'=(a'_1,\dots, a'_J)^\top\in\mathbb{R}^J$ has to be determined. Moreover, the coefficient column $(a'_1,\dots,a'_J,a'_0)^\top$ from representations (5.24) and (5.20) can be fixed orthogonal to the eigenvector indicated in (5.8) and involved into the main term of ansatz (5.15), namely
$$
\begin{equation}
\mathbf{e}^\top a'-J|\Omega|^{-1/2}|\omega|^{1/2}a'_0=0.
\end{equation}
\tag{5.25}
$$
We now compose problems to find the second correction terms in ansätze (5.11) and (5.12). According to (5.20), (2.13), (2.14) and (5.24), (5.22), we have
$$
\begin{equation*}
\begin{aligned} \, \pi^+_\Omega\overrightarrow{w}^{\,\prime}_{(J+1)} &=A_{J+1}\mathbf{G}^0\mathbf{e}+|\Omega|^{-1/2}a'_0\mathbf{ e} =|\Omega|^{-1/2}a'_0\mathbf{e}, \\ \pi^+_\omega\overrightarrow{w}^{\,\prime}_{(J+1)} &=|\omega|^{-1/2}a':=(|\omega|^{-1/2}a'_1, \dots,|\omega|^{-1/2}a'_J)^\top=|\omega|^{-1/2}(a')^\top. \end{aligned}
\end{equation*}
\notag
$$
Now from (3.12) and (3.13) we find that
$$
\begin{equation}
\pi^-_\Omega\overrightarrow{w}^{\,\prime\prime}_{(J+1)}= \pi^+_\omega\overrightarrow{w}^{\,\prime\prime}_{(J+1)}=b'':= |\omega|^{-1/2}a'-|\Omega|^{-1/2}a'_0\mathbf{e}.
\end{equation}
\tag{5.26}
$$
Furthermore, inserting the asymptotic ansaätze into problem (1.2), (1.3) and collecting the coefficients of $\mathfrak{z}^{-2}$, which are written in the slow $x$ and fast $\xi^j$ variables, respectively, we arrive at the Neumann problem
$$
\begin{equation}
\begin{gathered} \, \begin{aligned} \, &-\Delta_xw''_{(J+1)0}(x) =f''_{(J+1)0}(x):= \sum_{j=1}^J(|\omega|^{-1/2}a'_j- |\Omega|^{-1/2}a'_0)\delta(x-P^j) \\ &\qquad-\mu''_{J+1}\alpha_{J+1}^{-1/2}J|\Omega|^{-1}|\omega|^{1/2}+\mu'_{J+1} w'_{(J+1)0}(x),\qquad x\in\Omega, \end{aligned} \\ \partial_nw''_{(J+1)0}(x)=0,\qquad x\in\partial\Omega, \end{gathered}
\end{equation}
\tag{5.27}
$$
and the differential equations with indexes $j=1,\dots,J$
$$
\begin{equation}
\begin{aligned} \, -\Delta_\xi w''_{(J+1)j}(\xi^j) &= X_j(\xi^j)\bigl(\mu''_{J+1}\alpha_{J+1}^{-1/2} |\omega|^{-1/2}+\mu'_{J+1}w'_{(J+1)j}(\xi^j)\bigr),\qquad \xi^j\in\mathbb{R}^2, \\ w'_{(J+1)j}(\xi^j) &=(|\omega|^{-1/2}a'_j-|\Omega|^{-1/2}a'_0) \Phi(\xi^j)+O(1),\qquad |\xi^j|\to+\infty. \end{aligned}
\end{equation}
\tag{5.28}
$$
The compatibility condition $\int_\Omega f''_{(J+1)0}(x)\, dx=0$ in problem (5.27) takes the form
$$
\begin{equation}
\underbrace{|\omega|^{-1/2}\mathbf{e}^\top a'-J|\Omega|^{-1/2}a'_0} -\mu''_{J+1}\alpha_{J+1}^{-1/2}J|\omega|^{1/2}+\mu'_{J+1}|\Omega|^{1/2}a'_0=0.
\end{equation}
\tag{5.29}
$$
The underbraced difference vanishes due to requirement (5.25) so that (5.29) converts into
$$
\begin{equation}
-\mu''_{J+1}\alpha_{J+1}^{-1/2}J|\Omega|^{-1/2}=h''_{J+1}:= \overbrace{-\mu'_{J+1}|\omega|^{-1/2}a'_0}.
\end{equation}
\tag{5.30}
$$
We now consider equations (5.28). Fulfilling the growth conditions (5.26) at infinity, we represent their solutions as follows:
$$
\begin{equation*}
w''_{(J+1)j}=(|\omega|^{-1/2}a'_j-|\Omega|^{-1/2}a'_0) \mathbf{w}_j+\widehat{w}^{\,\prime\prime}_{(J+1)j},\qquad \widehat{w}^{\,\prime\prime}_{(J+1)j}\in\mathcal{H}_j.
\end{equation*}
\notag
$$
Taking relations (5.21), (5.22) and (5.24) into account, we achieve the following equations for “energy” remainders:
$$
\begin{equation}
\begin{aligned} \, &-\Delta_\xi \widehat{w}^{\,\prime\prime}_{(J+1)j}(\xi^j)=f''_{(J+1)j}(\xi^j):= X_j(\xi^j)\bigl(\mu''_{J+1}\alpha_{J+1}^{-1/2} |\omega|^{-1/2} \nonumber \\ &\qquad+(\mu'_{J+1})^2\alpha_{J+1}^{-1/2} |\omega|^{-1/2}\mathbf{w}_j(\xi^j) +\mu'_{J+1}|\omega|^{-1/2}a'_j \nonumber \\ &\qquad-|\omega|^{-1}(|\omega|^{-1/2}a'_j-|\Omega|^{-1/2}a'_0)\bigr),\qquad \xi^j\in\mathbb{R}^2. \end{aligned}
\end{equation}
\tag{5.31}
$$
The compatibility conditions $\int_{\mathbb{R}^2}f''_{(J+1)j}(\xi^j)\, d\xi^j$ in problems (5.31) take the form
$$
\begin{equation*}
\begin{aligned} \, &\mu''_{J+1}\alpha_{J+1}^{-1/2}|\omega|^{1/2}+(\mu'_{J+1})^2\alpha_{J+1}^{-1/2} |\omega|^{1/2}\mathbf{m}_j+\mu'_{J+1}|\omega|^{1/2}a'_j \\ &\qquad-|\omega|^{-1/2}a'_j+|\Omega|^{-1/2}a'_0=0,\qquad j=1,\dots,J, \end{aligned}
\end{equation*}
\notag
$$
and can be converted into
$$
\begin{equation}
\begin{aligned} \, \mu''_{J+1}\alpha_{J+1}^{-1/2} |\omega|^{-1/2} &=h''_j:=\overbrace{-\mu'_{J+1}|\omega|^{-1/2}a'_j- |\omega|^{-1}(|\omega|^{-1/2}a'_j-|\Omega|^{-1/2}a'_0)} \nonumber \\ &\qquad-(\mu'_{J+1})^2\alpha_{J+1}^{-1/2}|\omega|^{-1/2}\mathbf{m}_j,\qquad j=1,\dots,J. \end{aligned}
\end{equation}
\tag{5.32}
$$
Considering the system of algebraic equations (5.32) and (5.30), we find., in the left-hand sides, components of the normalized eigenvector of matrix (5.3)
$$
\begin{equation*}
s_{(J+1)}= \alpha_{J+1}^{-1/2}(|\omega|^{-1/2},\dots,|\omega|^{-1/2},-J|\Omega|^{-1/2})^\top.
\end{equation*}
\notag
$$
We multiply the system by this vector and observe that the overbraced terms on the right of (5.32) and (5.30) cancel due to the orthogonality condition (5.25). As a result, we have
$$
\begin{equation}
\mu''_{J+1}=(s_{(J+1)})^\top h''_{(J+1)}=-(\mu'_{J+1})^2\alpha_{J+1}^{-1} |\omega|^{-1}\sum_{j=1}^J\mathbf{m}_j>0.
\end{equation}
\tag{5.33}
$$
This completes the construction of first asymptotic correction terms in ansatz (5.11). We now find the three-term asymptotics (5.10) and (5.12) of eigenpairs of pencil (3.15) corresponding to eigenpairs (5.7) of matrix (5.3). As the main term of ansatz (5.12), we choose the vector
$$
\begin{equation}
\overrightarrow{w}^{\,\infty}_{(p)}=\bigl(0,|\omega|^{-1/2}a^\infty_{(p)1},\dots, |\omega|^{-1/2}a^\infty_{(p)J}\bigr)
\end{equation}
\tag{5.34}
$$
with the coefficient column $a_{(p)}^\infty=(a^\infty_{(p)1},\dots,a^\infty_{(p)J})^\top$ in the root subspace $\mathbb{R}^J_\bot$ of the truncated (without the lower row and the right column) matrix $\mathbb{T}^1-|\omega|^{-1}\mathbb{I}_{J+1}$ (see (5.7)). We normalize this vector; namely, we set $|a^\infty_{(p)}|=1$, and note that
$$
\begin{equation}
\begin{gathered} \, \pi^-_\Omega\overrightarrow{w}^{\,\infty}_p =\pi^-_\omega\overrightarrow{w}^{\,\infty}_{(p)}=0\in\mathbb{R}^J, \\ \pi^+_\Omega\overrightarrow{w}_{(p)}^{\,\infty}=0,\qquad \pi^+_\omega\overrightarrow{w}_{(p)}^{\,\infty}=|\omega|^{-1/2}a^\infty_{(p)}. \end{gathered}
\end{equation}
\tag{5.35}
$$
Thus, taking into account the transmission conditions (3.12) and (3.13), we find that
$$
\begin{equation}
\pi^-_\Omega\overrightarrow{w}^{\,\prime}_{(p)} =\pi^-_\omega\overrightarrow{w}^{\,\prime}_{(p)}= \pi^+_\omega\overrightarrow{w}_{(p)}^{\,\infty}- \pi^+_\Omega\overrightarrow{w}_{(p)}^{\,\infty}=|\omega|^{-1/2}a^\infty_{(p)}
\end{equation}
\tag{5.36}
$$
and
$$
\begin{equation}
\begin{aligned} \, w'_{(p)0}(x) &=|\omega|^{-1/2}\sum_{j=1}^Ja_{(p)j}^\infty G_j(x) +|\Omega|^{-1/2}a'_0, \\ w'_{(p)j}(\xi^j) &=|\omega|^{-1/2}a_{(p)j}^\infty\mathbf{w}_j(\xi^j)+|\omega|^{-1/2}a'_j, \end{aligned}
\end{equation}
\tag{5.37}
$$
and the column $a_{(p)}'=(a_{(p)1}',\dots,a_{(p)J}')^\top$ can be fixed orthogonal to the columns $\mathbf{e}$ and $a_{(p)}^\infty$, which appear in the vector functions (5.13) and (5.34), so that
$$
\begin{equation}
\mathbf{e}^\top a_{(p)}'=0, \qquad (a_{(p)}^\infty)^\top a_{(p)}'=0.
\end{equation}
\tag{5.38}
$$
We mention that the equations
$$
\begin{equation}
-\Delta_\xi w'_{(p)j}(\xi^j)=\mu'_p|\omega|^{-1/2}a^\infty_{(p)}X_j(\xi^j),\qquad \xi^j\in\mathbb{R}^2,
\end{equation}
\tag{5.39}
$$
are valid in view of (5.21) and (5.7). The last relation means that Thus, we derive from expansions (5.37), in addition to projections (5.36), the following formulas for the other projections
$$
\begin{equation}
\begin{aligned} \, \pi^+_\Omega\overrightarrow{w}^{\,\prime}_{(p)} &=|\omega|^{-1/2}\mathbf{G}^0a^\infty_{(p)}+ |\Omega|^{-1/2}a'_{(p)0}\mathbf{e}, \\ \pi^+_\omega\overrightarrow{w}^{\,\prime}_{(p)} &=|\omega|^{-1/2}a'_{(p)}= |\omega|^{-1/2}(a'_{(p)1},\dots,a'_{(p)J})^\top. \end{aligned}
\end{equation}
\tag{5.41}
$$
Hence, from the algebraic conditions (3.12) and (3.13) we have
$$
\begin{equation}
\pi^-_\Omega\overrightarrow{w}^{\,\prime\prime}_{(p)}= \pi^+_\omega\overrightarrow{w}^{\,\prime\prime}_{(p)}=b''_{(p)}:= |\omega|^{-1/2}a'_{(p)}-|\Omega|^{-1/2}a'_{(p)0}\mathbf{e}- |\omega|^{-1/2}\mathbf{G}^0a^\infty_{(p)}.
\end{equation}
\tag{5.42}
$$
We now compose problems for second correction terms in ansätze (5.10), (5.12). Since the first component of vector (5.34) is null, we do not need the Neumann problem for $w''_{(p)0}$ for constructing the asymptotics of the eigenvalue (5.10). At the same time, the equations and asymptotic conditions for $w''_{(p)1},\dots,w''_{(p)J}$ are as follows:
$$
\begin{equation}
\begin{gathered} \, -\Delta_\xi w''_{(p)j}(\xi^j)\,{=}\, X_j(\xi^j)\bigl(\mu''_p|\omega|^{-1/2}a^\infty_{(p)j} +\mu'_p|\omega|^{-1/2}(a^\infty_{(p)j}\mathbf{w}_j(\xi^j)+a'_j)\bigr),\quad \xi^j\,{\in}\,\mathbb{R}^2, \\ w'_{(p)j}(\xi^j)=b''_j\Phi(\xi^j)+O(1),\quad|\xi^j|\to+\infty. \end{gathered}
\end{equation}
\tag{5.43}
$$
In the same way as for problem (5.28), the compatibility conditions in problems (5.43) convert into the system of algebraic equations
$$
\begin{equation}
\mu''_p |\omega|^{1/2}a^\infty_{(p)j}+\mu'_p|\omega|^{1/2}(a^\infty_{(p)j}\mathbf{m}_j +a'_j)-b''_j=0,\qquad j=1,\dots,J.
\end{equation}
\tag{5.44}
$$
Inserting the expression (5.42) for $b''_j$ into (5.44), and using (5.40), we have
$$
\begin{equation*}
\mu''_p a^\infty_{(p)j}+\mu'_p\mathbf{m}_ja^\infty_{(p)j}+|\omega|^{-1}\sum_{k=1}^J \mathbf{G}^0_{jk}a^\infty_k+|\Omega|^{-1/2}a'_{(p)0}=0,\qquad j=1,\dots,J.
\end{equation*}
\notag
$$
We rewrite this system in the matric form
$$
\begin{equation}
\mu''_pa^\infty_{(p)}+\mu'_p\mathbf{m}a^\infty_{(p)}+|\Omega|^{-1/2} a'_{(p)0}\mathbf{e}+ |\omega|^{-1}\mathbf{G}^0a^\infty_{(p)}=0\in\mathbb{R}^J
\end{equation}
\tag{5.45}
$$
with the diagonal matrix $\mathbf{m}=\operatorname{diag}\{\mathbf{m}_1,\dots,\mathbf{m}_J\}$, and then project it onto the subspace $\mathbb{R}^J_\bot$ (see (2.14) and (2.27)), and, finally, according to the orthogonality conditions (5.38), reduce it to the eigenvalue problem
$$
\begin{equation}
\mathbf{M}\,a^\infty_{(p)}=\mu''_pa^\infty_{(p)}\ \text{ in subspace } \mathbb{R}^J_\bot.
\end{equation}
\tag{5.46}
$$
Here, the symmetric ($J\times J$)-matrix
$$
\begin{equation}
\mathbf{M}=-|\omega|^{-1}(\mathbf{G}^0+\mathbb{P}\mathbf{m}\mathbb{P}),
\end{equation}
\tag{5.47}
$$
annihilates the column $\mathbf{e}$ (see (2.14) and (2.28)). Thus, problem (5.46) has the eigenvalues
$$
\begin{equation}
\mu''_2\leqslant\mu''_3\leqslant\dots\leqslant\mu''_J
\end{equation}
\tag{5.48}
$$
which are the coefficients of $\mathfrak{z}^{-2}$ in expansions (5.10) of the desired terms $\mu^\mathfrak{z}_2,\dots,\mu^\mathfrak{z}_J$ in sequence (3.30) of eigenvalues of pencil (3.15). The eigenvectors $a^\infty_{(2)},\dots,a^\infty_{(J)}$ corresponding to (5.48) are subject to the orthogonality and normalization conditions
$$
\begin{equation*}
(a^\infty_{(q)})^\top a^\infty_{(p)}=\delta_{p,q}, \qquad p,q=2,\dots,J.
\end{equation*}
\notag
$$
At last, multiplying system (5.45) by the column $\mathbf{e}$, we find that
$$
\begin{equation*}
a'_{(p)0}=-\frac{|\Omega|^{1/2}}{J|\omega|}\sum_{j=1}^J \mathbf{m}_ja^\infty_{(p)j}.
\end{equation*}
\notag
$$
The columns $a'_{(p)}$ in (5.41) remain unknown, but they can be found at the next steps of the algorithm. If there are multiple eigenvalues among (5.48), s the corresponding columns $a^\infty_{(p)}$ also are not determined uniquely. This completes the construction of the three-term asymptotics of the first $J+1$ eigenvalues of the pencil $\mathfrak{A}^\mathfrak{z}_\beta$ with $\gamma=2$. The corresponding result is as follows. Theorem 6. In case (1.17), (5.5), the first $J$ positive entries of sequence (3.30) admit expansions (5.10) and (5.11) involving eigenvalues (5.48) of problem (5.46) with matrix (5.47) and the quantity 5.2. Case $\gamma>2$Since the spectral parameter is absent in the limit Neumann problem in the domain $\Omega$, the asymptotic procedure changes a bit. The null eigenvalue is associated with $J$ eigenvectors (5.1) of pencil (4.1) and the ($J\times J$)-matrix $\mathbf{T}^1_{(\gamma)}$ with entries (4.24) has the form
$$
\begin{equation}
\mathbf{T}^1_{(\gamma)}=\mathbb{P}\mathbb{T}_\omega\mathbb{P}
\end{equation}
\tag{5.50}
$$
where $\mathbb{P}$ is the orthogonal projector (2.28) and $\mathbb{T}_\omega= \operatorname{diag}\{|\omega_1|^{-1},\dots,|\omega_J|^{-1}\}$ is the same diagonal matrix as in (5.3) for the matrix $\mathbf{T}^1_{(2)}:=\mathbf{T}^1$. The following result is clear. Proposition 6. The symmetric positive matrix (5.50) has the eigenvalues We adopt the simplifying assumption (5.5). The construction of the asymptotics (5.10) and (5.12) for eigenpairs of pencil (3.15) repeats entirely the procedure in § 5.1. The main term of expansion (5.12) of eigenvalue is given by (5.34) with the unknown column $a_{(p)}^\infty\in\mathbb{R}^J_\bot$ and projections (5.35). The components of the correction term $\overrightarrow{w}^{\,\prime}_{(p)}$ satisfy relations (5.36) and (5.37), and the new unknown coefficients $a'_0$ and $a'_1,\dots,a'_J$ are subject to equalities (5.38). Equations (5.39) and (5.43) are intact, and from their compatibility conditions we get (5.40) and (5.45). This, in this way, we have found both asymptotic terms in expansions (5.10) of eigenvalues of tje pencil $\mathfrak{A}^\mathfrak{z}_\beta$. The corresponding result is as follows. Theorem 7. In situation (1.18), (5.5), the first $J-1$ positive terms in (3.30) admit expansions (5.10), where the second correction terms are eigenvalues (5.48) of problem (5.46) with matrix (5.47). 5.3. Explicit formulas in the particular case $J=2$Theorems 6 and 7 give the second correction terms only in case (5.5), where all inclusions (1.1) have the same area (and mass). This restriction was introduced, because the author could not find a common formula for eigenvalues of matrices (5.3) and (5.50) in the general situation $J>2$. If $J=2$, the first of these matrix takes the form
$$
\begin{equation*}
\begin{pmatrix} |\omega_1|^{-1} &0 &-|\omega_1|^{-1/2}|\Omega|^{-1/2} \\ 0 &|\omega_2|^{-1} &-|\omega_2|^{-1/2}|\Omega|^{-1/2} \\ -|\omega_1|^{-1/2}|\Omega|^{-1/2} &-|\omega_2|^{-1/2}|\Omega|^{-1/2} &2|\Omega|^{-1} \end{pmatrix}
\end{equation*}
\notag
$$
and has the eigenvalues
$$
\begin{equation}
\tau_0= 0,\qquad \tau_\pm= \frac{1}{|\Omega|}+\frac{1}{2}\biggl(\frac{1}{|\omega_1|}+\frac{1}{|\omega_2|}\biggr)\pm \sqrt{\frac{1}{|\Omega|^2}+\biggl(\frac{1}{|\omega_1|^2}-\frac{1}{|\omega_2|^2}\biggr)^2}.
\end{equation}
\tag{5.52}
$$
The quantities $\mu'_2=\tau_-$ and $\mu'_3=\tau_+$ appear in asymptotics (5.10) and (5.11) of eigenvalues of pencil (3.15) for $\gamma=2$. The second matrix is quite clear
$$
\begin{equation*}
\begin{pmatrix} m &-m \\ -m &m \end{pmatrix}, \quad \text{where}\quad m=\frac{1}{4} \biggl(\frac{1}{|\omega_1|}+\frac{1}{|\omega_2|}\biggr).
\end{equation*}
\notag
$$
It has the eigenvalues
$$
\begin{equation}
\tau_0= 0,\qquad \tau_1= \frac{1}{2}\biggl(\frac{1}{|\omega_1|}+\frac{1}{|\omega_2|}\biggr).
\end{equation}
\tag{5.53}
$$
It is the quantity $\mu'_2=\tau_1>0$ which appears in the asymptotic expansion (5.10). Formulas (5.52) and (5.53) demonstrate interaction of inclusions at the level $\mathfrak{z}^{-1}$. § 6. Justification of asymptotics6.1. Abstract formulation of the original problemWe endow the Sobolev space $\mathcal{H}^\varepsilon:=H^1(\Omega)$ with the inner product
$$
\begin{equation}
\langle u,\psi\rangle_\varepsilon=(\nabla_xu,\nabla_x\psi)_\Omega+ \varepsilon^{\gamma-2}\bigl((u,\psi)_\Omega +\varepsilon^{-\gamma}(u,\psi)_{\omega^\varepsilon}\bigr)
\end{equation}
\tag{6.1}
$$
which depends on the small parameter $\varepsilon\in(0,\varepsilon_0]$ and which is in accord with the integral identity (1.5). We also introduce the positive symmetric continuous (and, therefore, self-adjoint) operator $\mathcal{K}^\varepsilon$,
$$
\begin{equation}
\langle \mathcal{K}^\varepsilon u,\psi\rangle_\varepsilon= \varepsilon^{\gamma-2}\bigl((u,\psi)_\Omega+\varepsilon^{-\gamma}(u,\psi)_{\omega^\varepsilon} \bigr), \qquad u,\psi\in \mathcal{H}^\varepsilon.
\end{equation}
\tag{6.2}
$$
This operator is compact, and so, according to Theorems 10.1.5 and 10.2.2 in [35], its essential spectrum $\wp^\varepsilon_e$ consists of the only point $\kappa=0$ , and the discrete spectrum $\wp^\varepsilon_d$ is a positive monotone decreasing null sequence
$$
\begin{equation}
1=\kappa^\varepsilon_1>\kappa^\varepsilon_2\geqslant\kappa^\varepsilon_3\geqslant\dots \geqslant\kappa^\varepsilon_k\geqslant\ \to +0.
\end{equation}
\tag{6.3}
$$
Comparing (6.1), (6.2) and (1.5), we see that the variational statement of spectral problem (1.2), (1.3) is equivalent to the abstract equation
$$
\begin{equation*}
\mathcal{K}^\varepsilon u^\varepsilon=\kappa^\varepsilon u^\varepsilon \text{ in } \mathcal{H}^\varepsilon
\end{equation*}
\notag
$$
with the new spectral parameter
$$
\begin{equation}
\kappa^\varepsilon=\varepsilon^{\gamma-2}(\lambda^\varepsilon+\varepsilon^{\gamma-2})^{-1}.
\end{equation}
\tag{6.4}
$$
Relation (6.4) transforms sequence (1.6) into sequence (6.3). The next assertion, which is known as the lemma on “almost eigenvalues and eigenvectors” (cf. [36]), follows from the spectral decomposition of the resolvent (see, for example, [35], Ch. 6). Lemma 5. Let $\mathbf{k}^\varepsilon\in\mathbb{R}_+$ and $\mathbf{u}^\varepsilon\in\mathcal{H}^\varepsilon$ be such that 6.2. Almost eigenvalues and eigenvectorsLet $\mu^\mathfrak{z}_k$ be an eigenvalue of pencil (3.15) of multiplicity $\varkappa^{\,\mathfrak{z}}_k\geqslant1$. According to Remark 2, the matrix of size $\varkappa^{\,\mathfrak{z}}_k\times\varkappa^{\,\mathfrak{z}}_k$ with entries $I_\gamma(\overrightarrow{w}^{\,\mathfrak{z}}_{(p)},\overrightarrow{w}^{\,\mathfrak{z}}_{(q)})$ (see (4.4)) is not degenerate, that is, the diagonalization process provides the orthogonality and normalization conditions
$$
\begin{equation}
I_\gamma\bigl(\overrightarrow{w}^{\,\mathfrak{z}}_{(p)},\overrightarrow{w}^{\,\mathfrak{z}}_{(q)} \bigr):= \bigl(\mathfrak{M}_\beta\overrightarrow{w}^{\,\mathfrak{z}}_{(p)}, \overrightarrow{w}^{\,\mathfrak{z}}_{(q)}\bigr)_{\Omega\times\omega_1\times\dots\times\omega_J}= \delta_{p,q},\qquad p,q=k,\dots,k+\varkappa^{\,\mathfrak{z}}_k-1.
\end{equation}
\tag{6.8}
$$
The eigenvectors $\overrightarrow{w}^{\,\mathfrak{z}}_{(k)},\dots, \overrightarrow{w}^{\,\mathfrak{z}}_{(k+\varkappa_k^\mathfrak{z}-1)}$ are solutions of the equations
$$
\begin{equation}
\mathfrak{A}^\mathfrak{z}_\beta(-1)\overrightarrow{w}^{\,\mathfrak{z}}_{(p)}= (1+\mu_k^\mathfrak{z})(\mathfrak{M}_\beta,0,0) \overrightarrow{w}^{\,\mathfrak{z}}_{(p)},
\end{equation}
\tag{6.9}
$$
and, since the invertible operators $\mathfrak{A}^\mathfrak{z}_\beta(-1)$ and $\mathfrak{A}^\infty_\beta(-1)=(\mathfrak{B}_\beta+\mathfrak{M}_\beta, \pi^-_\Omega-\pi^-_\omega,\pi^-_\Omega)$ differ by the small term $\mathfrak{z}^{-1}(0,0,\pi^+_\Omega-\pi^+_\omega)$, the solution of equation (6.9) satisfies
$$
\begin{equation}
\begin{aligned} \, &\mathfrak{z}\bigl(\bigl|\pi^-_\Omega\overrightarrow{w}^{\,\mathfrak{z}}_{(p)}\bigr| +\bigl|\pi^-_\omega\overrightarrow{w}^{\,\mathfrak{z}}_{(p)}\bigr|\bigr)+ \bigl\|\overrightarrow{w}^{\,\mathfrak{z}}_{(p)};\mathfrak{V}_\beta\bigr\|\leqslant c(1+\mu_k^\mathfrak{z})\bigl\|\mathfrak{M}_\beta\overrightarrow{w}^{\,\mathfrak{z}}_{(p)}; \mathfrak{R}_\beta\bigr\| \nonumber \\ &\qquad\leqslant c(1+c_k)= C_k, \qquad p=k,\dots,k+\varkappa^{\,\mathfrak{z}}_k-1. \end{aligned}
\end{equation}
\tag{6.10}
$$
Here, we take into account relation (6.8) for a large $\mathfrak{z}$ and inequality (4.5). In other words, the projections $\pi^-_\Omega\overrightarrow{w}^{\,\mathfrak{z}}_{(p)}= \pi^-_\omega\overrightarrow{w}^{\,\mathfrak{z}}_{(p)}$ become of order $\mathfrak{z}^{-1}$. The almost eigenvalues and eigenvectors are searched in the form
$$
\begin{equation}
\mathbf{k}^\varepsilon_p=(\mu^\mathfrak{z}_k+1)^{-1},\qquad p=k,\dots,k+\varkappa^{\,\mathfrak{z}}_k-1,
\end{equation}
\tag{6.11}
$$
and
$$
\begin{equation}
\mathbf{u}^\varepsilon_p=\|\mathcal{W}^\varepsilon_p;\mathcal{H}^\varepsilon\|^{-1} \mathcal{W}^\varepsilon_p,\qquad p=k,\dots,k+\varkappa^{\,\mathfrak{z}}_k-1,
\end{equation}
\tag{6.12}
$$
where
$$
\begin{equation}
\begin{aligned} \, \mathcal{W}^\varepsilon_p(x) &=\biggl(1-\sum_{j=1}^J\chi_\omega^j \bigl(\varepsilon^{-1}(x-P^j)\bigr)\biggr) w^{\,\mathfrak{z}}_{(p)0}(x)+ \sum_{j=1}^J\chi_j(x) w^{\,\mathfrak{z}}_{(p)j}\bigl(\varepsilon^{-1}(x-P^j)\bigr) \nonumber \\ &\qquad-\sum_{j=1}^J\chi_j(x)\bigl(1-\chi_\omega^j\bigl(\varepsilon^{-1}(x-P^j)\bigr)\bigr) \bigl(a^{0\mathfrak{z}}_{(p)j}+b^{0\mathfrak{z}}_{(p)j}\Phi(x-P^j)\bigr). \end{aligned}
\end{equation}
\tag{6.13}
$$
Let us clarify on the last formula. First, we mention that the eigenvectors $\overrightarrow{w}^{\,\mathfrak z}_{(p)}= \bigl(w^{\,\mathfrak{z}}_{(p)0},w^{\,\mathfrak{z}}_{(p)1},\dots, w^{\,\mathfrak{z}}_{(p)J}\bigr)\in\mathfrak{V}_\beta$ satisfy the orthogonality and normalization conditions (6.8) and inequalities (6.10). Here, we apply an asymptotic construction using cut-off functions with “overlapping supports” (see the papers [37], [23], and the book [16]), the cut-off functions $\chi_j$ and $\chi^j_\omega$ are taken from (2.24) and (2.33). The terms matched in § 3.2 are taken into account twice, namely in the first addendum and in the first sum on the right-hand side of (6.13), but the last subtrahend compensates for such duplication. In view of transformation (3.9), using the asymptotic transmission conditions (3.6), (3.7) (or, what is the same, (3.12), (3.13)) we can rewrite expression (6.13) in the following two ways:
$$
\begin{equation}
\mathcal{W}^\varepsilon_p(x) =\mathcal{X}_\varepsilon(x)\widetilde{w}^{\,\mathfrak z}_{(p)0}(x)+ \sum_{j=1}^J\chi_j(x) w^{\,\mathfrak{z}}_{(p)j}\bigl(\varepsilon^{-1}(x-P^j)\bigr),
\end{equation}
\tag{6.14}
$$
$$
\begin{equation}
\mathcal{W}^\varepsilon_p(x) =\mathcal{X}_\varepsilon(x)w^{\,\mathfrak{z}}_{(p)0}(x)+ \sum_{j=1}^J\chi_j(x) \widetilde{w}^{\,\mathfrak z}_{(p)j}\bigl(\varepsilon^{-1}(x-P^j)\bigr).
\end{equation}
\tag{6.15}
$$
The remainders in representations (2.23) and (2.32) of the components of the vector $\overrightarrow{w}^{\,\mathfrak z}_{(p)}$ decay exponentially, and the cut-off function $\mathcal{X}_\varepsilon$ is a multiplier of $\overrightarrow{w}^{\,\mathfrak z}_{0(p)}$ in construction (6.13), that is,
$$
\begin{equation}
\mathcal{X}_\varepsilon(x)=1- \sum_{j=1}^J\chi_\omega^j\bigl(\varepsilon^{-1}(x-P^j)\bigr).
\end{equation}
\tag{6.16}
$$
6.3. Auxiliary inequalitiesIn next two lemmas, we will show how representations (6.13)–(6.15) are used differently on an example of calculation of the norm $\|\mathcal{W}^\varepsilon_p;\mathcal{H}^\varepsilon\|$. It is worth mentioning that further computations can be derived on the basis of smoothness of components of the eigenvectors $\overrightarrow{w}^{\,\mathfrak{z}}_k$ and pointwise estimates of the remainders in their expansions (see § 2.6). However, with the aim at staying in the framework of the developed model, we will use only weighted integral estimates of the above remainders. Since the weight exponent $\beta\in(0,1)$ is arbitrary, the accuracy of error estimates does is not spoiled (cf. the comments to Theorem 8). Lemma 6. The inequalities hold with a multiplier $C_k$ independent of $\varepsilon\in(0,\varepsilon_k]$ for some $\varepsilon_k>0$. Proof. Applying (6.15) and taking into account inequality (6.10), which means that
$$
\begin{equation}
\bigl|\pi^-_\Omega\overrightarrow{w}^{\,\mathfrak{z}}_{(p)}\bigr| +\bigl|\pi^-_\omega\overrightarrow{w}^{\,\mathfrak{z}}_{(p)}\bigr| \leqslant c_q\mathfrak{z}^{-1},
\end{equation}
\tag{6.18}
$$
we obtain
$$
\begin{equation*}
\begin{aligned} \, &\|\nabla_x\mathcal{W}^\varepsilon_q,L^2(\Omega)\|^2 \leqslant c\biggl(\bigl\|\nabla_x \widetilde{w}^{\,\mathfrak z}_{(q)0};L^2(\Omega)\bigr\|^2 +\sum_{j=1}^J \biggl(\bigl\|\nabla_x\bigl(\chi_j \widetilde{w}^{\,\mathfrak z}_{j(q)}\bigr);L^2(\Omega)\bigr\|^2 \\ &\qquad\qquad\qquad +\bigl|b^{0\mathfrak{z}}_{(q)j}\bigr|^2\int_{\varepsilon R^j_\omega}^R \frac{dr_j}{r_j}+ \frac{1}{\varepsilon^2} \int_{\varepsilon R^j_\omega}^{2\varepsilon R^j_\omega} \bigl(\bigl|b^{0\mathfrak{z}}_{(q)j}\bigr|^2|{\ln r_j}|^2 +\bigl|a^{0\mathfrak{z}}_{(q)j}\bigr|^2\bigr) r_j\, dr_j \biggr)\biggr) \\ &\qquad\leqslant c_q\biggl(\bigl\| \widetilde{w}^{\,\mathfrak z}_{(q)0};V_{-\beta}^2(\Omega;\mathcal{P})\bigr\|^2 \\ &\qquad\qquad\qquad+\sum_{j=1}^J \bigl(\bigl\| \widetilde{w}^{\,\mathfrak z}_{(q)j};V^1_\beta(\mathbb{R}^2)\bigr\|^2+ \bigl|b^{0\mathfrak{z}}_{(q)j}\bigr|^2(1+|{\ln \varepsilon}|^2)+\bigl|a^{0\mathfrak{z}}_{(q)j}\bigr|^2 \bigr)\biggr)\leqslant C_q. \end{aligned}
\end{equation*}
\notag
$$
The norm $\|\mathcal{W}^\varepsilon_q,L^2(\Omega)\|$ is estimated in the same way. This proves inequality (6.17), and, therefore, Lemma 6. Lemma 7. Functions (6.13) satisfy the inequalities Proof. We first process the last inner products in definition (6.1). Thanks to the presence of the cut-off function (6.16) (cf. (1.1) and (2.33)), the first and third terms on the right of (6.13) vanish on the inclusions $\omega^\varepsilon_j$ and, hence,
$$
\begin{equation}
\begin{gathered} \, \mathcal{W}^\varepsilon_m(x)=w^{\,\mathfrak{z}}_{(m)j}\bigl(\varepsilon^{-1}(x-P^j)\bigr),\qquad x\in \omega^\varepsilon_j,\quad m=p,q, \\ \varepsilon^{-2}(\mathcal{W}^\varepsilon_p,\mathcal{W}^\varepsilon_q)_{\omega^\varepsilon}= \varepsilon^{-2}\sum_{j=1}^J\bigl(\widetilde{w}^{\,\mathfrak{z}}_{(p)j}, \widetilde{w}^{\,\mathfrak{z}}_{(q)j}\bigr)_{\omega_j}= \sum_{j=1}^J\bigl(w^{\,\mathfrak{z}}_{(p)j},w^{\,\mathfrak{z}}_{(q)j}\bigr)_{\omega_j}. \end{gathered}
\end{equation}
\tag{6.22}
$$
Moreover, using (6.15), we write
$$
\begin{equation}
\begin{aligned} \, &\varepsilon^{\gamma-2}(\mathcal{W}^\varepsilon_p,\mathcal{W}^\varepsilon_q)_\Omega= \varepsilon^{\gamma-2}\bigl(w^{\,\mathfrak{z}}_{(p)0},w^{\,\mathfrak{z}}_{(q)0}\bigr)_\Omega +\varepsilon^{\gamma-2}\bigl((\mathcal{X}_\varepsilon^2-1)w^{\,\mathfrak{z}}_{(p)0}, w^{\,\mathfrak{z}}_{(q)0}\bigr)_\Omega \nonumber \\ &\ \quad +\varepsilon^{\gamma-2}\sum_{j=1}^J\bigl(\bigl(\mathcal{X}_\varepsilon w^{\,\mathfrak{z}}_{(p)0},\chi_j\widetilde{w}^{\,\mathfrak z}_{(q)j}\bigr)_\Omega +\bigl(\chi_j\widetilde{w}^{\,\mathfrak z}_{(p)j},\mathcal{X}_\varepsilon w^{\,\mathfrak{z}}_{(q)0}\bigr)_\Omega+ \bigl(\chi_j\widetilde{w}^{\,\mathfrak z}_{(p)j},\chi_j\widetilde{w}^{\,\mathfrak z}_{(q)j}\bigr)_\Omega\bigr) \nonumber \\ &\ =:\varepsilon^{\gamma-2}\biggl(\bigl(w^{\,\mathfrak{z}}_{(p)0},w^{\,\mathfrak{z}}_{(q)0}\bigr)_\Omega +J^\varepsilon_{pq}+\sum_{j=1}^J\bigl(\widetilde{J}^{\,j\varepsilon}_{pq} +\widetilde{J}^{\,j\varepsilon}_{qp}+{\widehat{J}}^{\,j\varepsilon}_{pq}\bigr)\biggr). \end{aligned}
\end{equation}
\tag{6.23}
$$
We then process the terms from the end of (6.23). We have
$$
\begin{equation}
\begin{aligned} \, |J^\varepsilon_{pq}| &\leqslant c\varepsilon^{\gamma-2} \bigl(|\omega^\varepsilon|^{1/2}\bigl(\bigl|b^{0\mathfrak{z}}_{(p)}\bigr| +\bigl|a^{0\mathfrak{z}}_{(p)}\bigr|\bigr) +\varepsilon^{1+\beta}\bigl\|\widetilde{w}^{\,\mathfrak z}_{(p)0}; V^1_{-\beta}(\Omega;\mathcal{P})\bigr\|\bigr) \nonumber \\ &\qquad \times\bigl(|\omega^\varepsilon|^{1/2}\bigl(\bigl|b^{0\mathfrak{z}}_{(q)}\bigr| +\bigl|a^{0\mathfrak{z}}_{(p)}\bigr|\bigr) +\varepsilon^{1+\beta} \bigl\|\widetilde{w}^{\,\mathfrak z}_{(q)0};V^1_{-\beta}(\Omega;\mathcal{P})\bigr\|\bigr) \leqslant c\varepsilon^\gamma. \end{aligned}
\end{equation}
\tag{6.24}
$$
The multipliers in the middle part of (6.24) appeared in the following way: the norms of $\chi_j(b^{0\mathfrak{z}}_{(m)j} \Phi(x^j)+a^{0\mathfrak{z}}_{(m)j})$ in $L^2(\mathbb{B}_{2\varepsilon R^j_\omega})$ with $m\,{=}\,p$ or $m\,{=}\,q$ are summed over $j=1,\dots,J$ , the remainders $\widetilde{w}^{\,\mathfrak{z}}_{(m)0}\in V^1_{-\beta}(\Omega;\mathcal{P}) \subset L^2_{-\beta-1} (\Omega;\mathcal{P})$ in representations (2.23) are estimated via $\mathbf{r}(x)\leqslant 2\varepsilon\max\{R^1_\omega,\dots,R^J_\omega\}$ on the support of the cut-off function (6.16), and inequality (6.10) is taken into account in (6.24).
Changing $x\mapsto\xi^j$ and observing that $\widetilde{w}^{\,\mathfrak{z}}_{(q)j}\in L^2_{\beta-1}(\mathbb{R}^2)$ and $(1+|\xi^j|)^{1-\beta}= (1+\varepsilon^{-1}|x^j|)^{1-\beta}\leqslant c_j\varepsilon^{\beta-1}$ on t $\operatorname{supp}\chi_j$ (cf. (2.24) and the restriction $\beta\in(0,1)$), we get the inequalities
$$
\begin{equation}
|\widehat{J}^{\,j\varepsilon}_{pq}|\leqslant c\varepsilon\varepsilon^{\beta-1} \|\widetilde{w}_{(p)j};L^2_{\beta-1}(\mathbb{R}^2)\|\,\varepsilon\varepsilon^{\beta-1} \|\widetilde{w}_{(q)j};L^2_{\beta-1}(\mathbb{R}^2)\|\leqslant c\varepsilon^{2\beta}.
\end{equation}
\tag{6.25}
$$
Finally, evaluating the $L^2(\operatorname{supp}\mathcal{X}_\varepsilon)$-norms of $\chi_j(x)(b^{0\mathfrak{z}}_{(p)j}\Phi(x^j)+a^{0\mathfrak{z}}_{(p)j})$, taking into account (6.18), and applying the estimate of the product $\chi_j\widetilde{w}^{\,\mathfrak{z}}_{(p)j}$ used in (6.25), we find that
$$
\begin{equation}
|\widetilde{J}^{\,j\varepsilon}_{pq}|\leqslant c \bigl(\bigl|b^{0\mathfrak{z}}_{(p)j}\bigr| |{\ln\varepsilon}|+ \bigl|a^{0\mathfrak{z}}_{(p)j}\bigr|+ \bigl\|\widetilde{w}^{\,\mathfrak{z}}_{(p)0};L^2_{-\beta-1}(\Omega)\bigr\|\bigr) \varepsilon^\beta \bigl\|\widetilde{w}^{\,\mathfrak{z}}_{(q)j};L^2_{-\beta-1}(\mathbb{R}^2) \bigr\| \leqslant c_{pq}\varepsilon^\beta.
\end{equation}
\tag{6.26}
$$
Now, consider the inner product $(\nabla_x\mathcal{W}^\varepsilon_p, \nabla_x\mathcal{W}^\varepsilon_q)_\Omega$. We transfer the cut-off functions $\mathcal{X}_\varepsilon$, $\chi_j$ and $\mathcal{X}_\varepsilon\chi_j$ from (6.13) into the function $\mathcal{W}^\varepsilon_q$ on the second position in the inner product. Commutation of the cut-off functions and the gradient-operator $\nabla_x$ results in the operators of multiplication by vector functions with localized supports, namely, we get
$$
\begin{equation}
\begin{aligned} \, &|\nabla_x\mathcal{X}_\varepsilon(x)|\leqslant c\varepsilon^{-1}\ \text{ and }\ \varepsilon^{-1}\leqslant c\mathbf{r}(x)^{-1}\ \text{ in the set} \\ &\qquad\operatorname{supp} |\nabla_x\mathcal{X}_\varepsilon|\subset \bigcup_{j=1}^J\{x:\,\varepsilon R^j_\omega \leqslant r_j\leqslant2\varepsilon R^j_\omega\}, \\ &(\varepsilon+|x^j|)^{-1}\leqslant c\mathbf{r}(x)^{-1}\ \text{ in the set } \ \operatorname{supp}\mathcal{X}_\varepsilon, \\ &1+|\xi^j|\leqslant c\varepsilon^{-1}\ \text{ in the set} \\ &\qquad\{\xi^j\colon P^j+\varepsilon\xi^j\in\operatorname{supp}|\nabla_x\chi_j|\}\subset \biggl\{x\colon \frac{R}2\leqslant r_j\leqslant R\biggr\}. \end{aligned}
\end{equation}
\tag{6.27}
$$
Moreover, the equality
$$
\begin{equation*}
\nabla_x\bigl(\mathcal{X}_\varepsilon(x)\chi_j(x)\bigr)= \varepsilon^{-1}\nabla_\xi\chi_\omega^j\bigl(\varepsilon^{-1}(x-P^j)\bigr)+ \nabla_x\chi_j(x),
\end{equation*}
\notag
$$
which follows from definitions (6.16) and (2.33) of cut-off functions, is important for redistribution of the commutator terms.
Following this scenario, after redistribution of the terms in analogy with (6.15) and (6.14), we find that
$$
\begin{equation*}
\begin{aligned} \, &(\nabla_x\mathcal{W}^\varepsilon_p, \nabla_x\mathcal{W}^\varepsilon_q)_\Omega =\bigl(\nabla_x\widetilde{w}^{\,\mathfrak z}_{(p)0}, \nabla_x(\mathcal{X}_\varepsilon\mathcal{W}^\varepsilon_q)\bigr)_\Omega \\ &\qquad+\sum_{j=1}^J\bigl(\bigl(\nabla_x w^{\,\mathfrak z}_{(p)j}, \nabla_x(\chi_j\mathcal{W}^\varepsilon_q)\bigr)_\Omega-\bigl(\nabla_x\bigl(b^\mathfrak{z}_{(p)j} \Phi(x^j)+a^\mathfrak{z}_{(p)j}\bigr),\nabla_x(\chi_j\mathcal{W}^\varepsilon_q)\bigr)_\Omega \bigr) \\ &\qquad+\mathcal{J}_{pq}^{\nabla 0\varepsilon}+ \sum_{j=1}^J\mathcal{J}_{pq}^{\nabla j\varepsilon}, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation}
\begin{aligned} \, \mathcal{J}_{pq}^{\nabla 0\varepsilon} &= \bigl(\widetilde{w}^{\,\mathfrak{z}}_{(p)0} \nabla_x\mathcal{X}_\varepsilon,\nabla_x\mathcal{W}^\varepsilon_q\bigr)_\Omega- \bigl(\nabla_x\widetilde{w}^{\,\mathfrak{z}}_{(p)0},\mathcal{W}^\varepsilon_q \nabla_x\mathcal{X}_\varepsilon\bigr)_\Omega, \\ \mathcal{J}_{pq}^{\nabla j\varepsilon} &= \bigl(\widetilde{w}^{\,\mathfrak{z}}_{(p)j} \nabla_x\chi_j,\nabla_x\mathcal{W}^\varepsilon_q\bigr)_\Omega- \bigl(\nabla_x\widetilde{w}^{\,\mathfrak{z}}_{(p)j},\mathcal{W}^\varepsilon_q \nabla_x\chi_j\bigr)_\Omega. \end{aligned}
\end{equation}
\tag{6.28}
$$
We emphasize that the above commutation with the cut-off function $\mathcal{X}_\varepsilon$ is based on representation (6.14), and the commutation with the cut-off function $\chi_j$ depends on (6.15).
Since the products $\mathcal{X}_\varepsilon\mathcal{W}^\varepsilon_q$ vanish near the points $P^1,\dots,P^J$, we have $\mathcal{X}_\varepsilon\mathcal{W}^\varepsilon_q\in V^1_{-\beta}(\Omega;\mathcal{P})$, and, therefore, using the integral identity (2.20) (with zero right-hand side $f_0$) for the eigenpair $\{\delta_{\gamma,2}\mu^\mathfrak{z}_k,w^{\,\mathfrak{z}}_{(p)0}\}$, we get the equality
$$
\begin{equation}
\bigl(\nabla_xw^\mathfrak{z}_{(p)0}, \nabla_x(\mathcal{X}_\varepsilon\mathcal{W}^\varepsilon_q)\bigr)_\Omega= \delta_{\gamma,2}\mu^\mathfrak{z}_k\bigl(w^{\,\mathfrak{z}}_{(p)0}, \mathcal{X}_\varepsilon\mathcal{W}^\varepsilon_q\bigr)_\Omega.
\end{equation}
\tag{6.29}
$$
Here, according to the above calculations,
$$
\begin{equation}
\delta_{\gamma,2}\mu^\mathfrak{z}_k\big|\bigl(w^{\,\mathfrak{z}}_{(p)0}, \mathcal{X}_\varepsilon\mathcal{W}^\varepsilon_q\bigr)_\Omega- \bigl(w^{\,\mathfrak{z}}_{(p)0},w^{\,\mathfrak{z}}_{(q)0}\bigr)_\Omega \bigr|\leqslant c \varepsilon^\beta.
\end{equation}
\tag{6.30}
$$
The function $\xi^j \mapsto \chi_j(P^j+\varepsilon\xi^j)\mathcal{W}^\varepsilon_q(P^j+\varepsilon\xi^j)$ is compactly supported on the plane, and hence, it can be taken as a test function in the integral identity (2.29) (with zero right-hand side $f_j$) for the eigenpair $\{\mu^\mathfrak{z}_k,w^{\,\mathfrak{z}}_{(p)j}\}$. So, using the first of (6.22), we get
$$
\begin{equation}
\begin{aligned} \, \bigl(\nabla_xw^\mathfrak{z}_{(p)j}, \nabla_x(\chi_j\mathcal{W}^\varepsilon_q)\bigr)_\Omega &=\bigl(\nabla_\xi w^{\,\mathfrak{z}}_{(p)j}, \nabla_\xi(\chi_j\mathcal{W}^\varepsilon_q)\bigr)_{\mathbb{R}^2} \nonumber \\ &= \mu^\mathfrak{z}_k\bigl( w^{\,\mathfrak{z}}_{(p)j}, \mathcal{W}^\varepsilon_q\bigr)_{\omega_j}=\mu^\mathfrak{z}_k\bigl( w^{\,\mathfrak{z}}_{(p)j}, w^{\,\mathfrak{z}}_{(q)j}\bigr)_{\omega_j}. \end{aligned}
\end{equation}
\tag{6.31}
$$
Finally, the support of the product $\mathcal{X}_\varepsilon\chi_j\mathcal{W}^\varepsilon_q$ lies in an annulus around the point $P^j$, and hence, for the function harmonic inside this annulus, we get
$$
\begin{equation}
\bigl(\nabla_x\bigl(b^\mathfrak{z}_{(p)j}\Phi+a^\mathfrak{z}_{(p)j}\bigr), \nabla_x(\mathcal{X}_\varepsilon\chi_j\mathcal{W}^\varepsilon_q)\bigr)_\Omega=0.
\end{equation}
\tag{6.32}
$$
It remains to process the inner product (6.28). In view of (6.27), the supports of the involved functions are localized. Thus, in view of the inclusion $\widetilde{w}^{\,\mathfrak{z}}_{(p)0}\in V^1_{-\beta}(\Omega;\mathcal{P})$ and inequality (2.19) for the function $\mathcal{W}^\varepsilon_q\in H^1(\Omega)$ (see Lemma 6), we find that
$$
\begin{equation}
\begin{aligned} \, |\mathcal{J}_{pq}^{\nabla 0\varepsilon}| &\leqslant c\varepsilon^{-1}\bigl(\varepsilon^{1+\beta}\|\mathcal{W}^\varepsilon_q;H^1(\Omega)\| +\varepsilon^\beta \varepsilon(1+|{\ln \varepsilon}|)\|\mathbf{r}^{-1} (1+|{\ln\mathbf{r}}|)^{-1} \|\mathcal{W}^\varepsilon_q;L^2\Omega\|\bigr) \nonumber \\ &\leqslant c\varepsilon^\beta(1+|{\ln \varepsilon}|). \end{aligned}
\end{equation}
\tag{6.33}
$$
The inner products $\mathcal{J}_{pq}^{\nabla 1\varepsilon},\dots,\mathcal{J}_{pq}^{\nabla J\varepsilon}$ are dealt with similarly. Namely, using the last line and the inclusion $\widetilde{w}^{\,\mathfrak{z}}_{(p)j}\in V^1_\beta(\mathbb{R}^2)$, we have
$$
\begin{equation}
\begin{aligned} \, |\mathcal{J}_{pq}^{\nabla j\varepsilon}| &\leqslant c \bigl(\varepsilon^{\beta-1}\varepsilon\bigl\|\widetilde{w}^{\,\mathfrak{z}}_{(p)j}; L^2_{\beta-1}(\mathbb{R}^2)\bigr\| \|\mathcal{W}^\varepsilon_q;H^1(\Omega)\| \nonumber \\ &\qquad +\varepsilon^\beta\bigl\|\nabla_\xi\widetilde{w}^{\,\mathfrak{z}}_{(p)j}; L^2_\beta(\mathbb{R}^2)\bigr\| (1+|{\ln \varepsilon}|)\|\mathbf{r}^{-1} (1+|{\ln\mathbf{r}}|)^{-1} \|\mathcal{W}^\varepsilon_q;L^2\Omega\|\bigr) \nonumber \\ &\leqslant c\varepsilon^\beta(1+|{\ln \varepsilon}|). \end{aligned}
\end{equation}
\tag{6.34}
$$
Collecting the derived formulas, we find in (6.22), (6.23) and (6.29), (6.31) the quantity $(1+\mu^\mathfrak{z}_k)I_\gamma (\overrightarrow{w}^{\,\mathfrak{z}}_{(p)}, \overrightarrow{w}^{\,\mathfrak{z}}_{(q)})$, as determined by (6.8). The remaining components of the inner product $\langle \overrightarrow{w}^{\,\mathfrak{z}}_{(p)}, \overrightarrow{w}^{\,\mathfrak{z}}_{(q)}\rangle_\varepsilon$ satisfy the inequalities with the bound indicated in (6.22): it is worth to mention that quantity $\varepsilon^\beta(1+|{\ln\varepsilon}|)$ appeared in estimates (6.33) and (6.34). Finally, the differences in (6.20) for the infinitesimal value $\ell_\beta(\varepsilon)$ for $\gamma=2$ and for $\gamma>2$ are explained by the fact that in the case $\gamma=2$ the expression $(w^{\,\mathfrak{z}}_{(p)0}, w^{\,\mathfrak{z}}_{(q)0})_\Omega$ in (6.23) is included into the orthogonality and normalization conditions (6.8) by definition (3.14) of the operator $\mathfrak{M}_\beta$, but in the case $\gamma>2$ this expression is absent. Lemma 7 is proved. 6.4. Processing the discrepanciesLet us perform the next step in the asymptotics justification procedure. Namely, we evaluate the quantities $\alpha^\varepsilon_k,\dots,\alpha^\varepsilon_{k+\varkappa^{\,\mathfrak{z}}_k-1}$ from (6.5) for couples (6.11), (6.12). By definitions (6.1) and (6.2), we have
$$
\begin{equation}
\begin{aligned} \, \alpha^\varepsilon_p &=\sup|\langle \mathcal{K}^\varepsilon \mathbf{u}^\varepsilon_p- \mathbf{k}^\varepsilon_p\mathbf{u}^\varepsilon_p ,\psi\rangle_\varepsilon| \nonumber \\ &=(\mu^\mathfrak{z}_k+1)^{-1}\|\mathcal{W}^\varepsilon_p;\mathcal{H}^\varepsilon\|^{-1} \sup|\langle (\mu^\mathfrak{z}_k+1)\mathcal{K}^\varepsilon \mathcal{W}^\varepsilon_p-\mathcal{W}^\varepsilon_p,\psi\rangle_\varepsilon| \nonumber \\ &=(\mu^\mathfrak{z}_k+1)^{-1}\|\mathcal{W}^\varepsilon_p;\mathcal{H}^\varepsilon\|^{-1} \nonumber \\ &\qquad\qquad\times\sup\bigl|(\nabla_x\mathcal{W}^\varepsilon_p,\nabla_x\psi)_\Omega -\varepsilon^{\gamma-2}\mu^\mathfrak{z}_k \bigl((\mathcal{W}^\varepsilon_p,\psi)_\Omega+\varepsilon^{-\gamma} (\mathcal{W}^\varepsilon_p,\psi)_{\omega^\varepsilon}\bigr)\bigr|. \end{aligned}
\end{equation}
\tag{6.35}
$$
Here, the supremum is taken over the unit sphere of $\mathcal{H}^\varepsilon$, that is, $\|\psi;\mathcal{H}^\varepsilon\|= 1$. In the case $\gamma=2$, we obtain $\|\psi;\mathcal{H}^\varepsilon\|\geqslant \|\psi;H^1(\Omega)\|$ by (6.1), and, therefore, from inequality (2.19) we have
$$
\begin{equation}
\|\nabla_x\psi;L^2(\Omega)\|+ \|\mathbf{r}^{-1}(1+|{\ln\mathbf{r}}|)^{-1}\psi;L^2(\Omega)\|\leqslant C\|\psi;\mathcal{H}^\varepsilon\|=C.
\end{equation}
\tag{6.36}
$$
For $\gamma>2$, a substituter for inequality (6.36) is given in the next lemma. Lemma 8. In the case $\gamma>2$, the function $\psi\in \mathcal{H}^\varepsilon$ satisfies Proof. We represent the function $\psi$ as the sum $\psi={\overline{\psi}}+\psi_\bot$, where
$$
\begin{equation*}
\overline{\psi}=\frac{1}{|\Omega|}\int_\Omega\psi(x)\, dx,\qquad \int_\Omega\psi_\bot(x)\, dx=0.
\end{equation*}
\notag
$$
The last orthogonality condition implies the Poincaré inequality
$$
\begin{equation*}
\int_\Omega |\psi_\bot(x)|^2\, dx\leqslant c \int_\Omega |\nabla_x\psi_\bot(x)|^2\, dx=c \int_\Omega |\nabla_x\psi(x)|^2\, dx.
\end{equation*}
\notag
$$
Moreover,
$$
\begin{equation*}
\begin{aligned} \, &\varepsilon^{-2}\int_{\omega^\varepsilon_1}|\psi(x)|^2\, dx\geqslant\frac{1}{2}\, |\omega_1|\, |\overline{\psi}|^2-\int_{\omega^\varepsilon_1} |\psi_\bot(x)|^2\, dx \\ &\qquad \begin{aligned} \, \Longrightarrow\quad |\overline{\psi}|^2 &\leqslant c\bigl(\varepsilon^{-2}\|\psi;L^2(\omega^\varepsilon)\|^2+ (1+|{\ln\varepsilon}|)^2\|\mathbf{r}(1+|{\ln\mathbf{r}}|)^{-1};L^2(\Omega)\|^2\bigr) \\ &\leqslant C(1+|{\ln\varepsilon}|)^2\|\psi;\mathcal{H}^\varepsilon\|^2. \end{aligned} \end{aligned}
\end{equation*}
\notag
$$
These inequalities together with the Hardy inequality (2.19) give the desired estimate of the second term on the left-hand side of (6.37). Estimates of terms in the sum over $j=1,\dots,J$ are derived via the Poincaré–Friedrichs inequality in the stretched coordinates
$$
\begin{equation*}
\|w;L^2(\mathbb{B}_{R^j_\omega})\|^2\leqslant c_j\bigl(\|\nabla_\xi w;L^2(\mathbb{B}_{R^j_\omega})\|^2+\|w;L^2(\omega_j)\|^2\bigr).
\end{equation*}
\notag
$$
Lemma 8 is proved. Let us transform the expression $\mathcal{J}^\varepsilon_p(\psi)$ between the last modulus sign in (6.35). To this aim, we transfer the cut-off functions (6.16) and (2.24), as introduced in construction (6.13), to the test function $\psi$, and treat the ensuing commutators of $\mathcal{X}_\varepsilon$ and $\nabla_x$ via (6.27) and (6.14) and with the help of inequality (6.36):
$$
\begin{equation}
\begin{aligned} \, &\bigl|\bigl(\widetilde{w}^{\,\mathfrak z}_{(p)0}\nabla_x\mathcal{X}_\varepsilon, \nabla_x\psi\bigr)_\Omega-\bigl(\nabla_x\widetilde{w}^{\,\mathfrak z}_{(p)0},\psi \nabla_x\mathcal{X}_\varepsilon\bigr)_\Omega\bigr| \nonumber \\ &\qquad\leqslant c\varepsilon^{-1}\bigl(\varepsilon^{\beta+1}\|\nabla_x\psi;L^2(\Omega)\| +\varepsilon^\beta\varepsilon(1+|{\ln\varepsilon}|) \|\mathbf{r}(1+|{\ln\mathbf{r}}|)^{-1}\psi;L^2(\Omega)\|\bigr) \nonumber \\ &\qquad\leqslant c\varepsilon^\beta(1+|{\ln\varepsilon}|). \end{aligned}
\end{equation}
\tag{6.38}
$$
In the case of the cut-off functions $\chi_j$, the analogous estimates are as follows:
$$
\begin{equation}
\begin{aligned} \, &\bigl|\bigl(\widetilde{w}^{\,\mathfrak z}_{(p)j}\nabla_x\chi_j, \nabla_x\psi\bigr)_\Omega-\bigl(\nabla_x\widetilde{w}^{\,\mathfrak z}_{(p)j},\psi \nabla_x\chi_j\bigr)_\Omega\bigr| \nonumber \\ &\qquad\leqslant c\bigl(\varepsilon \varepsilon^{\beta-1}\bigl\|\widetilde{w}^{\,\mathfrak z}_{(p)j};L^2_{\beta-1}(\mathbb{R}^2)\bigr\| \|\nabla_x\psi;L^2(\Omega)\| \nonumber \\ &\qquad\qquad+\varepsilon^\beta\bigl\|\nabla_\xi\widetilde{w}^{\,\mathfrak z}_{(p)j};L^2_\beta(\mathbb{R}^2)\bigr\| \varepsilon^{-1} \|\psi;L^2(\Omega)\|\bigr)\leqslant c\varepsilon^\beta(1+|{\ln\varepsilon}|). \end{aligned}
\end{equation}
\tag{6.39}
$$
We emphasize that in view of (6.37) with $\gamma>2$, the bound in (6.38) loses the multiplier $1+|{\ln\varepsilon}|$, and the same multiplier does not appear for $\gamma=2$ in view of (6.36). Thus,
$$
\begin{equation*}
\biggl|\mathcal{J}^\varepsilon_p(\psi)-\mathcal{J}^{0\varepsilon}_p(\psi)\sum_{j=1}^J \bigl(\mathcal{J}^{j\varepsilon}_p(\psi)+ \mathcal{J}^{j\varepsilon}_{\mu p}(\psi)\bigr) \biggr| \leqslant c\varepsilon^\beta(1+|{\ln\varepsilon}|),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{aligned} \, \mathcal{J}^{0\varepsilon}_p(\psi) &= \bigl(\nabla_x w^{\,\mathfrak{z}}_{(p)0}, \nabla_x(\mathcal{X}_\varepsilon\psi)\bigr)_\Omega -\varepsilon^{\gamma-2}\mu_k^\mathfrak{z} \bigl(w^{\,\mathfrak{z}}_{(p)0},\mathcal{X}_\varepsilon\psi\bigr)_\Omega , \\ \mathcal{J}^{j\varepsilon}_p(\psi) &= \bigl(\nabla_x w^{\,\mathfrak{z}}_{(p)j}, \nabla_x(\chi_j\psi)\bigr)_{\mathbb{R}^2}-\varepsilon^{-2} \mu^\mathfrak{z}_k\bigl(w^{\,\mathfrak{z}}_{(p)j},\psi\bigr)_{\omega_j}, \\ \mathcal{J}^{j\varepsilon}_{\mu p}(\psi) &=-\varepsilon^{\gamma-2}\mu_k^\mathfrak{z} \bigl(\widetilde{w}^{\,\mathfrak z}_{(p)j}, \chi_j\psi\bigr)_{\mathbb{R}^2}. \end{aligned}
\end{equation*}
\notag
$$
The product $\mathcal{X}_\varepsilon\psi$ vanishes near the points $P^1,\dots,P^J$, and, therefore, $\mathcal{X}_\varepsilon\psi\in V^1_{-\beta}(\Omega;\mathcal{P})$, which implies
$$
\begin{equation*}
\mathcal{J}^{0\varepsilon}_p(\psi)=0\quad \text{for}\quad \gamma=2\qquad {\rm or}\quad |\mathcal{J}^{0\varepsilon}_p(\psi)|\leqslant c\varepsilon^{\gamma-2}\quad \text{for }\quad \gamma>2.
\end{equation*}
\notag
$$
Since the function $\xi^j\mapsto \chi_j(P^j+\varepsilon\xi^j)\psi(P^j+\varepsilon\xi^j)$ has compact support, and $\{\mu^\mathfrak{z}_k,w^{\,\mathfrak{z}}_{(p)j}\}$ is an eigenpair of the limit equation (1.10), we have
$$
\begin{equation*}
\mathcal{J}^{j\varepsilon}_p(\psi)=0,\qquad j=1,\dots,J.
\end{equation*}
\notag
$$
Finally, we have
$$
\begin{equation*}
\begin{aligned} \, |\mathcal{J}^{j\varepsilon}_{\mu p}(\psi)| &\leqslant c\varepsilon^{\gamma-2} \bigl\|(\varepsilon+r_j)^{-1}(1+|\ln(\varepsilon+r_j)|)^{-1}\widetilde{w}^{\,\mathfrak z}_{(p)j};L^2(\mathbb{R}^2)\bigr\| \\ &\qquad\times\bigl\|(\varepsilon+r_j)\bigl(1+|\ln(\varepsilon+r_j)\chi_j\psi|\bigr);L^2(\Omega)\bigr\| \\ &\leqslant c\varepsilon^{\gamma-2}\varepsilon(1+|{\ln\varepsilon}|)\bigl\|\widetilde{w}^{\,\mathfrak z}_{(p)j};L^2_{\beta+1}(\mathbb{R}^2)\bigr\| (1+|{\ln\varepsilon}|)\|\psi;\mathcal{H}^\varepsilon\| \\ &\leqslant c\varepsilon^{\gamma-1}(1+|{\ln\varepsilon}|)^2\leqslant c\varepsilon^\beta(1+|{\ln\varepsilon}|). \end{aligned}
\end{equation*}
\notag
$$
To summarize. In view of (6.21) and (4.5), using the first two multipliers on the right-hand side of (6.35), we have the desired estimates
$$
\begin{equation}
\alpha^\varepsilon_p\leqslant c_k\ell_{\gamma,\beta}(\varepsilon),\qquad p=k,\dots,k+\varkappa^{\,\mathfrak{z}}_k-1.
\end{equation}
\tag{6.40}
$$
6.5. On asymptotic accuracy of the model concerning approximation of eigenvaluesNow we have all the ingredients for proving the main theorem of this section. Theorem 8. For any $p\in\mathbb{N}$, there exist positive $\mathfrak{z}_p$ and $c_p$ such that the terms of the sequences (1.6) and (3.30) satisfy The proof follows the traditional scheme. According to Lemma 5 and estimates (4.5), taking an eigenvalue $\mu^\mathfrak{z}_k$ of pencil (3.15) of multiplicity $\varkappa^{\,\mathfrak{z}}_k$, we find the eigenvalues $\kappa^\varepsilon_{N^\varepsilon(p)}$, $p=k,\dots,k+\varkappa^{\,\mathfrak{z}}_k-1$, of the operator $\mathcal{K}^\varepsilon$ which meet inequalities
$$
\begin{equation}
|\kappa^\varepsilon_{N^\varepsilon(p)}-(\mu^\mathfrak{z}_k+1)^{-1}| \leqslant c_k\ell_\gamma(\varepsilon),\qquad p=k,\dots,k+\varkappa^{\,\mathfrak{z}}_k-1.
\end{equation}
\tag{6.42}
$$
Since the spectral parameters are related by (6.4), we have from (6.42)
$$
\begin{equation}
\begin{aligned} \, &|(\lambda^\varepsilon_{N^\varepsilon(p)}+\varepsilon^{\gamma-2})-\varepsilon^{\gamma-2} (\mu^\mathfrak{z}_k+1)|\leqslant c_k\ell_\gamma(\varepsilon) (\mu^\mathfrak{z}_k+1)(\lambda^\varepsilon_{N^\varepsilon(p)}+\varepsilon^{\gamma-2}) \nonumber \\ &\Longrightarrow\quad\lambda^\varepsilon_{N^\varepsilon(p)}+\varepsilon^{\gamma-2}\leqslant \mu^\mathfrak{z}_k+1+c_k\ell_\gamma(\varepsilon) (\mu^\mathfrak{z}_k+1)(\lambda^\varepsilon_{N^\varepsilon(p)}+\varepsilon^{\gamma-2}) \nonumber \\ &\Longrightarrow\quad\lambda^\varepsilon_{N^\varepsilon(p)}+\varepsilon^{\gamma-2} \leqslant2(\mu^\mathfrak{z}_k+1)\quad \text{as}\quad c_k\ell_\gamma(\varepsilon) (\mu^\mathfrak{z}_k+1)\leqslant\frac{1}{2} \nonumber \\ &\Longrightarrow\quad |\lambda^\varepsilon_{N^\varepsilon(p)}-\varepsilon^{\gamma-2} \mu^\mathfrak{z}_k|\leqslant 2c_k\ell_\gamma(\varepsilon) (\mu^\mathfrak{z}_k+1)^2 \quad \text{as} \quad c_k\ell_\gamma(\varepsilon) (\mu^\mathfrak{z}_k+1)\leqslant\frac{1}{2}. \end{aligned}
\end{equation}
\tag{6.43}
$$
Thus, we have found the eigenvalues $\lambda^\varepsilon_{N^\varepsilon(k)}, \dots,\lambda^\varepsilon_{N^\varepsilon(k+\varkappa^{\,\mathfrak{z}}_k-1)}$ satisfying inequalities (6.41), however, no information on their indexes $N^\varepsilon(p)$ in the ordered sequence (1.6) is still available. Let us first verify that these indexes differ from each other, that is, let us show that it can be assumed that
$$
\begin{equation*}
N^\varepsilon(p)=N^\varepsilon(k)+p-k\quad\text{for}\quad p=k,\dots,k+\varkappa^{\,\mathfrak{z}}_k-1.
\end{equation*}
\notag
$$
To this end, we apply the second part of Lemma 5 and denote by $\mathbf{s}^\varepsilon_{(p)}$ and $\mathbf{c}^\varepsilon_{(p)}$, respectively, the linear combination $\sum c^\varepsilon_{(p)q} \mathcal{U}^\varepsilon_q$ and the column of its coefficients (normalized in $\mathbb{R}^{\mathbf{X}_\varepsilon}$), which appear in (6.6) for the almost eigenvector (6.12). Besides,
$$
\begin{equation}
\alpha^\varepsilon=\max\{\alpha^\varepsilon_k,\dots, \alpha^\varepsilon_{k+\varkappa^{\,\mathfrak{z}}_k-1}\}, \qquad \alpha^\varepsilon_\bullet= \frac{\alpha^\varepsilon}{t}\quad\text{with}\quad t>1
\end{equation}
\tag{6.44}
$$
(see (6.40)). The columns $\mathbf{c}^\varepsilon_{(k)},\dots, \mathbf{c}^\varepsilon_{(k+\varkappa^{\,\mathfrak{z}}_k-1)}$ belong to the space $\mathbb{R}^{\mathbf{X}_\varepsilon}$, whose dimension $\mathbf{X}_\varepsilon$ is the total multiplicity of the spectrum of the operator $\mathcal{K}^\varepsilon$ in the interval
$$
\begin{equation}
[(1+\mu^\mathfrak{z}_k)^{-1}-t\alpha^\varepsilon,\, (1+\mu^\mathfrak{z}_k)^{-1}+t\alpha^\varepsilon]\subset(0,1).
\end{equation}
\tag{6.45}
$$
By means of simple transformations and using the orthogonality and normalization conditions (6.7), we find that
$$
\begin{equation*}
\begin{aligned} \, &|(\mathbf{c}^\varepsilon_{(q)})^\top\mathbf{c}^\varepsilon_{(p)}-\delta_{p,q}| =|\langle\mathbf{s}^\varepsilon_{(p)},\mathbf{s}^\varepsilon_{(q)} \rangle_\varepsilon-\delta_{p,q}| \\ &\ \ =|\langle\mathbf{s}^\varepsilon_{(p)},\mathbf{s}^\varepsilon_{(q)} -\mathbf{u}^\varepsilon_{(q)}\rangle_\varepsilon+\langle\mathbf{s}^\varepsilon_{(p)} -\mathbf{u}^\varepsilon_{(p)},\mathbf{u}^\varepsilon_{(q)} \rangle_\varepsilon+\langle\mathbf{u}^\varepsilon_{(p)},\mathbf{u}^\varepsilon_{(q)} \rangle_\varepsilon -\delta_{p,q}| \\ &\ \ \leqslant (\|\mathbf{u}^\varepsilon_{(p)};\mathcal{H}^\varepsilon\|+\|\mathbf{s}^\varepsilon_{(p)} -\mathbf{u}^\varepsilon_{(p)};\mathcal{H}^\varepsilon\|) \|\mathbf{s}^\varepsilon_{(q)} -\mathbf{u}^\varepsilon_{(q)};\mathcal{H}^\varepsilon\|+\|\mathbf{s}^\varepsilon_{(p)} -\mathbf{u}^\varepsilon_{(p)};\mathcal{H}^\varepsilon\|\,\|\mathbf{u}^\varepsilon_{(q)}; \mathcal{H}^\varepsilon\| \\ &\ \ \qquad+\|\mathcal{W}^\varepsilon_p;\mathcal{H}^\varepsilon\|^{-1} \|\mathcal{W}^\varepsilon_q;\mathcal{H}^\varepsilon\|^{-1}\bigl| \langle\mathcal{W}^\varepsilon_p,\mathcal{W}^\varepsilon_q \rangle_\varepsilon -\delta_{p,q}\|\mathcal{W}^\varepsilon_p;\mathcal{H}^\varepsilon\|\, \|\mathcal{W}^\varepsilon_q;\mathcal{H}^\varepsilon\|\bigr|. \end{aligned}
\end{equation*}
\notag
$$
Now according to definitions (6.12), estimates (6.19), and inequality (6.6), where the bound is $2/t$ by (6.44), we derive that
$$
\begin{equation}
|(\mathbf{c}^\varepsilon_{(q)})^\top\mathbf{c}^\varepsilon_{(p)}-\delta_{p,q}| \leqslant\biggl(\biggl(1+\frac{2}{t}\biggr)\,\frac{2}{t}+\frac{2}{t}+ \frac{1}{1+\mu_k^\mathfrak{z}}\ell_{\gamma,\beta}(\varepsilon)\biggr).
\end{equation}
\tag{6.46}
$$
Inequality (6.46) means that, for large $t$ and small $\varepsilon$, the columns $\mathbf{c}^\varepsilon_{(k)},\dots, \mathbf{c}^\varepsilon_{(k+\varkappa^{\,\mathfrak{z}}_k-1)}$ are “almost orthonoralized” in $\mathbb{R}^{\mathbf{X}_\varepsilon}$, which is possible only in the case $\varkappa^{\,\mathfrak{z}}_k\leqslant \mathbf{X}_\varepsilon$. Thus, the interval (6.45) contains at least $\varkappa^{\,\mathfrak{z}}_k$ eigenvalues of the operator $\mathcal{K}^\varepsilon$ and they satisfy the estimates
$$
\begin{equation}
|\kappa^\varepsilon_{N^\varepsilon(k)+p-k}-(1+\mu^\mathfrak{z}_k)^{-1}| \leqslant t\alpha^\varepsilon,\qquad p=k,\dots,k+\varkappa^{\,\mathfrak{z}}_k-1.
\end{equation}
\tag{6.47}
$$
Fixing a proper value of $t$, we repeat calculations (6.43) and turn formula (6.47) into ineqality (6.41) with replacements $c_k\mapsto tc_k$ and $\lambda^\varepsilon_p\mapsto\lambda^\varepsilon_{N^\varepsilon(k)+p-k}$. The first replacement does not matter, because $t$ is fixed, and the second one is compensated for by the following observations. First of all, the number $N^\varepsilon(k)$ cannot be strictly smaller than $k$, because we have found $k-1$ eigenvalues $\lambda^\varepsilon_1,\dots,\lambda^\varepsilon_{k-1}\in [0,\varepsilon^{\gamma-2}(\mu_{k-1}+\mu_k)/2]$. Finally, the inequality $N^\varepsilon(k)>k$ is rejected by Theorem 1 as contradicting the convergence (2.17) (see also the comments to Proposition 7). Theorem 8 is proved. Here, an important observation is worth making. Since the weight exponent $\beta\in(0,1)$ can be fixed arbitrarily without affecting the properties of the operator pencil (3.15), the infinitesimal value in the bound $c_p\ell_{\beta,\gamma}(\varepsilon)$ of estimate (6.41) can be replaced by
$$
\begin{equation}
\widehat{\ell}^{\,\vartheta}_\gamma(\varepsilon)= \begin{cases} \varepsilon^\vartheta, &\gamma=2, \\ \varepsilon^{\min\{\gamma-2,\, \vartheta\}}, &\gamma>2. \end{cases}
\end{equation}
\tag{6.48}
$$
Here, $\vartheta$ is any exponent in the interval $(0,1)$. The multiplier $c_p=c_{p,\gamma}(\vartheta)$, of course, grows unboundedly as $\vartheta\to 1-0$. 6.6. On approximation of eigenfunctions of the original Neumann problemThe second part of Lemma 5, which was already used in verification of Theorem 8, can be employed to estimate the accuracy of approximation of the eigenfunctions of problem (1.5) by those of pencil (3.15). Let us find an estimate for the asymptotic remainder for a simple eigenvalue $\mu_k\in M$. By Theorems 8 and 3, the eigenvalues $\lambda^\varepsilon_k$ and $\mu^\mathfrak{z}_k$ from sequences (1.6) and (3.30), respectively, also are simple, and so is the eigenvalue $\kappa^\varepsilon_k=\varepsilon^{\gamma-2}(\lambda^\varepsilon_k+\varepsilon^{\gamma-2})^{-1}$ of the operator $\mathcal{K}^\varepsilon$ (see (6.4) and (6.3)). A similar result can also be obtained for a multiple eigenvalue, but the corresponding assertion becomes much more cumbersome and less informative, and hence omitted. It is worth pointing out that quite involved linear combinations (6.13) (or (6.14), (6.15)) of solutions to limit problems also provide asymptotic approximations of the eigenfunctions $u^\varepsilon_k$ of the original problem on concentrated masses. Theorem 9. Let $k\in\mathbb{N}$ and let $\mu_k\in M$ be a simple eigenvalue of the family of limit problems. Then, for any $\vartheta\in(0,1)$, there exist positive $\mathfrak{z}^\vartheta_k$ and $c^\vartheta_k$ such that, for $\mathfrak{z}\in[\mathfrak{z}^\vartheta_k,+\infty)$, the eigenfunction $u^\varepsilon_k$ of problem (1.5) (or (1.2), (1.3)) normalized according to condition (1.7) satisfies Proof. According to § 6.4 and Theorem 8, the interval with some $\theta>0$, contains a unique eigenvalue $\kappa^\varepsilon_k$ of the operator $\mathcal{K}^\varepsilon$. We apply (6.6) from Lemma 5 with ingredients (6.11), (6.12) for $p=k$ and $\alpha^\varepsilon=\alpha^\varepsilon_k$, $\alpha^\varepsilon_\bullet=\theta/2$ (see (6.40) and (6.50)). This gives us the estimate
$$
\begin{equation}
\|\mathbf{u}_k^\varepsilon-\mathcal{U}^\varepsilon_k;\mathcal{H}^\varepsilon\| \leqslant 4c_k\theta^{-1}\ell_{\beta,\gamma}(\varepsilon).
\end{equation}
\tag{6.51}
$$
To verify inequality (6.49), it remains to recall several formulas. First of all, from the integral identity (1.5), the normalization condition (1.7), and estimate (6.41) we get
$$
\begin{equation*}
\|\nabla_x u^\varepsilon_k;L^2(\Omega)\|=\varepsilon^{2-\gamma}\lambda^\varepsilon_k =\mu^\mathfrak{z}_k+O(\ell_{\beta,\gamma}(\varepsilon))
\end{equation*}
\notag
$$
and, therefore, by (6.1) and (6.7), we have the following relation between the eigenfunction of problem (1.5) and those of the operator $\mathcal{K}^\varepsilon$:
$$
\begin{equation}
\|\nabla_x u^\varepsilon_k-(1+\mu^\mathfrak{z}_k)^{-1}\mathcal{U}^\varepsilon_k; L^2(\Omega)\|\leqslant C_k\ell_{\beta,\gamma}(\varepsilon).
\end{equation}
\tag{6.52}
$$
As a result, in view of estimates (6.21) and (6.52), we can make simultaneously substitutions $\mathbf{u}^\varepsilon_k\mapsto \mathcal{W}^\varepsilon_k$ and $\mathcal{U}^\varepsilon_k \mapsto u^\varepsilon_k$ in (6.51), which convert this formula to (6.49). The change $\ell_{\beta,\gamma}(\varepsilon)\mapsto\widehat{\ell}^{\,\vartheta}_\gamma(\varepsilon)$ was explained in the previous section. Theorem 9 is proved. § 7. The Dirichlet boundary condition7.1. Similarity to the Neumann problemThe study of the two-dimensional Dirichlet problem (1.2), (1.8) on concentrated masses (1.1) differs slightly from the performed study of the Neumann problem.3 We only mention some simple modifications in the arguments. Since the variational statement of the Dirichlet problems, original (1.5) and limit (2.8), is posed in the subspace $H^1_0(\Omega)$ of functions in the Sobolev spaces vanishing at the boundary $\partial\Omega$, the first eigenvalues $\lambda^\varepsilon_1$ and $\mu_{01}$ of these problems are positive (cf. the beginnings of sequences (1.6) and (2.9) in the Neumann case). Spectra (2.6) of the limit equations (1.10) remain the same, and the combined sequence (1.9) is composed by the previous rule, but always involve only $J$ nulls (independently on parameter (1.1); cf. (2.15) and (2.16) for the Neumann problem (1.2), (1.3)). The conclusion of Theorem 1 remains valid (cf. the proof of Proposition 7 in § 8). The operator pencil $\mu\mapsto \mathfrak{A}^\mathfrak{z}_\beta(\mu)$ modeling the variational formulation of problem (1.2), (1.8)
$$
\begin{equation}
(\nabla_xu^\varepsilon,\nabla_x\psi)_\Omega=\lambda^\varepsilon\bigl( (u^\varepsilon,\psi)_{\Omega^\varepsilon}+\varepsilon^{-\gamma} (u^\varepsilon,\psi)_{\omega^\varepsilon}\bigr),\qquad \psi\in H^1_0(\Omega),
\end{equation}
\tag{7.1}
$$
is still given by (3.15) via the operator
$$
\begin{equation*}
V^1_{\beta,0}(\Omega,\mathcal{P})\ni w_0 \mapsto f_0\in V^1_{-\beta,0}(\Omega,\mathcal{P})^\ast
\end{equation*}
\notag
$$
of the problem
$$
\begin{equation*}
(\nabla_x w_0,\nabla_x\psi_0)_\Omega=f_0(\psi)\quad \forall\,\psi\in V^1_{-\beta,0}(\Omega;\mathcal{P}),
\end{equation*}
\notag
$$
as reduced to the subspace $\mathcal{V}^1_{\beta,0} (\Omega;\mathcal{P})$ endowed with norm (2.34) and consisting of the functions $w_0\in V^1_\beta(\Omega;\mathcal{P})$ of the form (2.23), which vanish on the boundary $\partial\Omega$. The study of this pencil follows verbatim the arguments from § 3. The only slight difference is in the construction of asymptotics of the first $J$ entries of the eigenvalue sequence $\{\lambda^\varepsilon_k\}_{k\in\mathbb{N}}$. We present the modified constructions without referring to regularly perturbed pencil but imply the method of matched asymptotic expansions (see [24], [18], [16], Ch. 2, etc.), which is very simple for constructing infinite series in inverse powers of logarithm. 7.2. Small eigenvaluesFirst of all, we mention that the Dirichlet problem in the domain $\Omega$ has the classical Green function $x\mapsto G(x,x')$ (see [27]), positive and harmonic in the punctured domain $\Omega\setminus\{x'\}$. Its particular values (2.12) admit expansions (2.13), and we construct a symmetric ($J\times J$)-matrix $\mathbf{G}^0$ from their coefficients. In contrast to the Neumann case, the matrix $\mathbf{G}^0$ is completely defined, and does not require additional restrictions (cf. conditions (2.14) for the generalized Green function). The asymptotic structures also involve special solutions (5.22) of equations (5.21). The eigenfunctions corresponding to the eigenvalues
$$
\begin{equation}
\lambda^\varepsilon_p=\varepsilon^{\gamma-2}(0+\mathfrak{z}^{-1}\mu'_p +\mathfrak{z}^{-2}\mu''_p+\cdots)+\cdots,\qquad p=1,\dots,J
\end{equation}
\tag{7.2}
$$
(cf. ansatz (4.8)), are supplied with expansions of two types, namely, the outer one
$$
\begin{equation}
u^\varepsilon_p(x)=0+\mathfrak{z}^{-1}\sum_{j=1}^J b_{(p)j}^0G_j(x)+ \mathfrak{z}^{-2}\sum_{j=1}^J b_{(p)j}^{0\prime}G_j(x)+\cdots ,
\end{equation}
\tag{7.3}
$$
which is suitable outside a neighbourhood of the singularity set $\mathcal P$ and the inner ones
$$
\begin{equation}
\begin{aligned} \, u^\varepsilon_p(x) &=a_{(p)j}+\mathfrak{z}^{-1}(a'_{(p)j}+b_{(p)j}\mathbf{w}_j(\xi^j)) \nonumber \\ &\qquad+\mathfrak{z}^{-2f}\bigl(a''_{(p)j}+b'_{(p)j}\mathbf{w}_j(\xi^j) +\widehat{w}^{\,\prime}_{(p)j}(\xi^j)\bigr)+\cdots, \qquad j=1,\dots,J, \end{aligned}
\end{equation}
\tag{7.4}
$$
which are fit in the vicinity of the points $P^j$ and are written in the stretched coordinates $\xi^j=\varepsilon^{-1}(x-P^j)$. Here, $\mathbf{w}_j$ are the special solutions of the differential equations (5.21), and all the coefficients and “energy” components $\widehat{w}^{\,\prime}_{(p)j}\in\mathcal{H}_j$ should be determined. Dots in (7.2)–(7.4) stand for the higher-order asymptotic terms immaterial for our formal analysis. Matching expansions (7.3) and (7.4) at the level $1=\mathfrak{z}^0$ and taking into account the relations $\Phi(x-P^j)=\Phi(\xi^j)+\mathfrak{z}$ (cf. Remark 1), we get and by matching the expansions at the level $\mathfrak{z}^{-1}$, we have
$$
\begin{equation}
a'_{(p)j}=b'_{(p)j}+\sum_{k=1}^J \mathbf{G}^0_{jk}b_{(p)k}, \qquad j=1,\dots,J.
\end{equation}
\tag{7.7}
$$
Moreover, inserting ansätze (1.11) and (7.4) into equation (1.2), changing to the stretched coordinates, and collecting the coefficients of $\varepsilon^{-2}\mathfrak{z}^{-1}$, we get
$$
\begin{equation*}
-\Delta_\xi(b_{(p)j}\mathbf{w}_j(\xi^j))=\mu'_pX_j(\xi^j)a_{(p)j},\qquad \xi^j\in\mathbb{R}^2, \quad j=1,\dots,J,
\end{equation*}
\notag
$$
and so, employing to (5.21), we also have
$$
\begin{equation}
|\omega_j|^{-1}b_{(p)j}=\mu'_p a_{(p)j},\qquad j=1,\dots,J.
\end{equation}
\tag{7.8}
$$
From system (7.5), (7.6) and (7.8) we determine the main terms of asymptotics (7.2) of eigenvalues and the columns in decompositions (7.3) and (7.4) of the eigenfunctions:
$$
\begin{equation}
\mu' _p=|\omega_j|^{-1}\quad\text{and}\quad a_{(p)}=b^0_{(p)}= b_{(p)}=e_{(p)}:=(\delta_{p,1},\dots,\delta_{p,J})^\top,\qquad j=1,\dots,J.
\end{equation}
\tag{7.9}
$$
If the areas of inclusions (1.1) are pairwise different, the choice of the columns in (7.9) is unique, and the construction of the second correction terms in the asymptotic formulas (7.2) is simple. Namely, continuing the matching procedure, we have, in addition to equalities (7.7), the relations and the differential equations
$$
\begin{equation}
\begin{aligned} \, -\Delta_\xi \widehat{w}^{\,\prime}_{(p)j}(\xi^j) &=\mu''_pX_j(\xi^j)a_{(p)j} +\mu'_pX_j(\xi^j)(a'_{(p)j}+b_{(p)j}\mathbf{w}_j(\xi^j)) \nonumber \\ &\qquad-X_j(\xi^j)b'_{(p)j}|\omega_j|^{-1},\qquad \xi^j\in\mathbb{R}^2, \quad j=1,\dots,J. \end{aligned}
\end{equation}
\tag{7.11}
$$
In view of (7.9) and (7.7), (7.10), the compatibility condition in equation (7.11) with $j=p$ gives
$$
\begin{equation}
\begin{aligned} \, &\mu''_p a_{(p)p}+\mu'_p\biggl(b'_{(p)p}+\sum_{k=1}^J \mathbf{G}^0_{pk}a_{(p)k}+b_{(p)p}\mathbf{m}_p\biggr) -\frac{b'_{j(p)}}{|\omega_p|}=0 \nonumber \\ &\qquad \Longrightarrow\quad \mu''_p=-\frac{1}{|\omega_p|}(\mathbf{G}^0_{pp}+\mathbf{m}_p). \end{aligned}
\end{equation}
\tag{7.12}
$$
This relation involves the integral characteristics (5.23) of the solution $\mathbf{w}_p$. The quantities $b'_{(p)p}$ and $a'_{(p)p}$ remain unknown, but they can be computed at the next steps of the algorithm. At the same time, for $j\ne p$, the compatibility condition in (7.7) does not contain the parameter $\mu''_p$, but permits us to find the quantities $b'_{(p)j}$ and $a'_{(p)j}$ via
$$
\begin{equation*}
\begin{aligned} \, &\mu'_p(b'_{(p)j}+\mathbf{G}_{jp}a_{(p)k})-b'_{j(p)}|\omega_j|^{-1}=0 \\ &\qquad \Longrightarrow\quad b'_{j(p)}=\frac{|\omega_j|\mathbf{G}^0_{jp}}{|\omega_p|-|\omega_j|},\quad a'_{j(p)}=\frac{|\omega_p|\mathbf{G}^0_{jp}}{|\omega_p|-|\omega_j|}. \end{aligned}
\end{equation*}
\notag
$$
This completes construction of the three-term asymptotics (7.7) of small eigenvalues of problem (1.2), (1.8), provided that the areas of the inclusions are pairwise different. Let
$$
\begin{equation}
|\omega_1|=\dots=|\omega_K|\quad\text{for}\quad K\leqslant J
\end{equation}
\tag{7.13}
$$
and let $|\omega_j|\ne |\omega_1|$ for $j>K$. Relabeling, if necessary, the inclusions, we assume, for simplicity, that
$$
\begin{equation}
|\omega_j|<|\omega|:=|\omega_1|\quad\text{for}\quad j>K.
\end{equation}
\tag{7.14}
$$
This restriction does not affect the asymptotic procedure, but allows us to deal with the ordered first $K$ eigenvalues $\lambda^\varepsilon_1\leqslant\dots\leqslant\lambda^\varepsilon_K$, for which we accept ansätze (7.2) with Realizing the procedure, we observe that the last $J-K$ components of the columns $a_{(p)}$, $b_{(p)}$ in the outer (7.3) and inner (7.4) expansions vanish. From the first $K$ components of the columns $a'_{(p)}$, $b'_{(p)}$, etc., we form the $K$-dimensional columns $a^{\prime\, \sharp}_{(p)}$, $b^{\prime\,\sharp}_{(p)}$, etc. The remaining components are combined into the columns $a^{\prime\,\flat}_{(p)}, b^{\prime\,\flat}_{(p)}\in \mathbb{R}^{J-K}$. In particular, We apply the matching procedure in the same way as above. Formulas (7.5)–(7.9) remain intact. Furthermore, we have in addition to (7.15); however, it is not possible yet to determine the shortened columns
$$
\begin{equation*}
a^\sharp_{(p)}=b^\sharp_{(p)}=b^{0\,\sharp}_{(p)},\qquad p=1,\dots,K.
\end{equation*}
\notag
$$
We continue the matching procedure and obtain the family of equations (7.11) with $p=1,\dots,K$. Their compatibility conditions take the form
$$
\begin{equation*}
\mu''_pa_{(p)j}+\frac{1}{|\omega|}\biggl( \sum_{k=1}^K\mathbf{G}^0_{pk}a_{(p)k}+\mathbf{m}_ja_{(p)j}\biggr)=0,\qquad j=1,\dots,K.
\end{equation*}
\notag
$$
Thus, the quantities composed of ansätze (7.2) with $p=1,\dots,K$ are the ordered eigenvalues of the symmetric ($K\times K$)-matrix
$$
\begin{equation}
\mathbf{M}^\sharp=(\mathbf{M}_{pj})_{j,p=1}^K=-\frac{1}{|\omega|} (\mathbf{G}^0_{pj}+\delta_{p,j}\mathbf{m}_j)_{j,p=1}^K.
\end{equation}
\tag{7.17}
$$
It would be also possible to find the columns $a^{\prime\,\flat}_{(p)}$ and $a^{\prime\,\flat}_{(p)}$, but they will not be required in the following theorem on asymptotics. The remaining ($p\,{=}\,K+1,\dots,J$) eigenvalues (7.2) for $p=K+ 1,\dots,J$ are constructed by the above procedure; again, among them it is necessary to single out the multiple ones, and to introduce the blocks of the ($J\times J$)-matrix $\mathbf M$ (as in (7.17)). We also mention that this matrix differs from that (5.47), since no projector $\mathbb P$ is involved. The formally obtained related on asymptotics of eigenvalues is formulate below. Error estimates can be obtained as in § 6 for the Neumann boundary condition. Theorem 10. 1) If the areas of inclusions (1.1) are pairwise different, then, for $\varepsilon\in(0,\varepsilon^0_J]$, the first $J$ eigenvalues of the variational problem (7.1) (or of the boundary value problem (1.2), (1.8) in the case of smooth boundary), the asymptotic formulas 2) Under assumptions (7.13) and (7.14), for $\varepsilon\in(0,\varepsilon^0_K]$, the first $K$ eigenvalues of the variational problem (7.1) (or the boundary value problem (1.2), (1.8) in the case of smooth boundary), the asymptotic formulas
$$
\begin{equation}
|\lambda^\varepsilon_p-\varepsilon^{\gamma-2}\mathfrak{z}^{-1} (|\omega_1|^{-1}+\mathfrak{z}^{-1}\mu''_p)| \leqslant c^0_K\varepsilon^{\gamma-2}\mathfrak{z}^{-3},\qquad j=1,\dots,K,
\end{equation}
\tag{7.18}
$$
hold with some positive $c^0_K$ and $\varepsilon^0_K$. The coefficients $\mu''_1,\dots,\mu''_J$ are the eigenvalues (7.16) of matrix (7.17) of size $K\times K$. 7.3. On “far-interaction” of inclusionsWe discuss the obtained asymptotic expansions of small eigenvalues of the Dirichlet and Neumann problems. Under requirements (5.5) (inclusions have the same area), Theorem 10, 2) with $K=J$ shows that the asymptotic correction terms $O(\varepsilon^{\gamma-2}|{\ln\varepsilon}|^{-2})$ in ansätze (7.2) are found via the eigenvalues of the ($J\times J$)-matrix (7.17) including the matrix $\mathbf{G}^0$ of coefficients in decompositions (2.13) of the Green functions (2.12). This matrix depends not only on the shape of the domain, but also on position of the points $P^1,\dots,P^J\in\mathcal{P}$ inside it. In other words, there is an interaction of the concentrated masses at the level $\mathfrak{z}^{-2}=O(|{\ln\varepsilon}|^{-2})$. The same can be said about the eigenvalues $\lambda^\varepsilon_2,\dots, \lambda^\varepsilon_J$ of the Neumann problem (1.2), (1.3) in cases (1.17) and (1.18), because by Theorems 6 and 7 (see also by Theorem 8), the above correction terms are the eigenvalues (5.48) of matrix (5.47). At the same time, the eigenvalue $\lambda^\varepsilon_{J+1}$ is also small in the case $\gamma=2$, for which the correction term is of the form (5.49) and depends on the areas $|\Omega|$ and $|\omega|$, but is independent of the set $\mathcal P$. It can be verified that the next asymptotic term depends on the matrix $\mathbf{G}^0$. Under the assumption that the areas of the inclusions are pairwise different, assertion 1 of Theorem 10 demonstrates that the main terms of expansions (7.2) are equal to $\varepsilon^{\gamma-2} (\mathfrak{z} |\omega_p|)^{-1}$, and the next ones are given by (7.12), which involves the quantities $\mathbf{G}^0_{pp}$ and $\mathbf{m}_p$ depending on shape of the domains $\Omega$, $\omega_p$ and the position of the point $P_p\in\Omega$, but which is independent of the other inclusions and their positions. In other words, an interaction of the concentrated masses at the level $\varepsilon^{\gamma-2}|{\ln\varepsilon}|^{-2}$ is not observed. At the same time, under the same conditions, the main terms of the asymptotics of the eigenvalues $\lambda^\varepsilon_2,\dots, \lambda^\varepsilon_J$ of the Neumann problem are now eigenvalues (5.51) of matrix (5.50), which is “fringed” by the projector (2.28) that “mixes” its diagonal entries $|\omega_1|^{-1},\dots,|\omega_J|^{-1}$ (see examples in § 5.3). Thus, the case under discussion is exceptional, namely the interaction of planar concentrated masses occurs in the main asymptotic term, and the position of their “centres” $P^1,\dots,P^J\in\Omega$ is immaterial. It is worth pointing out that the same effect was also observed in the multidimensional ($d\geqslant3$) Neumann problem (see [11], [13]). 7.4. An important difference between the Neumann and Dirichlet problemsIn the particular case $\gamma>2$ and $J=1$ (the only superheavy inclusion $\omega^\varepsilon=\omega^\varepsilon_1$), the two-dimensional problem (1.2), (1.3) possesses the following remarkable property: for the eigenfunctions of equation (1.10), which admit the representation with a decaying remainder $\widetilde{w}_{1k}(\xi^1)=O(|\xi^1|^{-1})$ (see § 2.6), the residuals have power-law behaviour $O(\varepsilon^{\gamma-2})$ in equation (1.2) and $O(\varepsilon)$ with the boundary condition (1.3). Thus, for all $k\in\mathbb{N}$, the absolute value of the difference $\lambda^\varepsilon_k-\varepsilon^{\gamma-2}\mu^\mathfrak{z}_k$ of the eigenvalues of the original singularly perturbed problem and of its model admits a power-law majorant $c_k\varepsilon^{\min\{\gamma-2,1\}}$.The described algorithm for construction of the asymptotics of the eigenvalues of the Dirichlet problem (1.2), (1.8) demonstrates that the power-law proximity of the numbers $\lambda^\varepsilon_k$ and $\varepsilon^{\gamma-2}\mu^\mathfrak{z}_k$ occurs only under the condition $a_{1k}=0$, where eigenfunction (7.19) itself decays at infinity. This happens, for example, for a symmetric domain $\omega_1$, and, in particular, for a disc, for which the Fourier method applies, and equation (1.10) has eigenfunctions decaying at infinity due to their oddness in one of the variables $\xi^j_1$ or $\xi^j_2$. At the same time, in § 7.2, we have already shown that divergence of the normalized eigenvalue $\varepsilon^{2-\gamma}\lambda^\varepsilon_1>0$ of the original problem from the first eigenvalue $\mu^\mathfrak{z}_1=0$ of the pencil $\mathfrak{A}^\mathfrak{z}_\beta$ is $O(\mathfrak{z}^{-1}|\omega_1|)$, that is, it has logarithmic decay order. § 8. Verifying convergence of normalized eigenvalues8.1. Modifying statement of Theorem 1For the reader convenience, we give here a proof of a simplified version of Theorem 1 for the Neumann problem (1.2), (1.3) (for the Dirichlet problem (1.2), (1.8), the modifications in the analysis that follow are inessential). Since the justification scheme used in the present paper differs from that of [7], the result obtained below in Proposition 7 is a bit weaker then that given by Theorem 1, but the fact established below- will be sufficient for completion of the proof of Theorem 8 (see § 8.2). Let $\mu_k\in M$ be an eigenvalue of the family of limit problems of multiplicity $\varkappa_k$ (see sequence (1.9), which is the union of sequences (2.6), $j=1,\dots,J$, and (2.9) in the case $\gamma=2$; see also (4.7)). An intermediate conclusion observed in § 6.5 in the proof of Theorem 8 together with Theorem 3 gives, in a small neighbourhood of $\mu_k$, at least $\varkappa_k$ of normalized eigenvalues $\varepsilon^{2-\gamma}\lambda^\varepsilon_{N(k)+k},\dots, \varepsilon^{2-\gamma}\lambda^\varepsilon_{N(k)+k+\varkappa_k-1}$ of problem (1.2), (1.3). So, we have
$$
\begin{equation}
\varepsilon^{2-\gamma}\lambda^\varepsilon_p\leqslant c_k,\qquad p=k,\dots,k+\varkappa_k-1,
\end{equation}
\tag{8.1}
$$
with a common factor $c_k$. Hence, we have the convergences
$$
\begin{equation}
\varepsilon_m^{2-\gamma}\lambda^{\varepsilon_m}_p\to \widehat{\mu}_p\quad\text{as}\quad m\to+\infty,\quad p=k,\dots,k+\varkappa_k-1,
\end{equation}
\tag{8.2}
$$
along some positive null sequence $\{\varepsilon_m\}_{\mathbb{N}}$. In what follows, we omit the subscript $m$ on $\varepsilon$, and keep the same notation when choosing to a subsequence. We need to verify that $\widehat{\mu}_k$ is an eigenvalue of the family of limit problems. The integral identity (1.5), the normalization condition (1.7), and relation (8.1) imply
$$
\begin{equation*}
\|u^\varepsilon_p;\mathcal{H}^\varepsilon\|\leqslant c_p,\qquad p=k,\dots,k+\varkappa_k-1.
\end{equation*}
\notag
$$
In view of the inequality
$$
\begin{equation*}
\|u^\varepsilon_p;H^1(\Omega)\|\leqslant c_p\|u^\varepsilon_p;\mathcal{H}^\varepsilon\|
\end{equation*}
\notag
$$
(see (1.5) and Lemma 8), we find that
$$
\begin{equation}
w^\varepsilon_p:=u^\varepsilon_p\to\widehat{w}^{\,0}_p\quad \text{weakly in } H^1(\Omega) \text{ and strongly in }L^2(\Omega).
\end{equation}
\tag{8.3}
$$
We also introduce the functions
$$
\begin{equation}
\mathbb{R}^2\ni\xi^j\mapsto \widehat{u}^{\,\varepsilon j}_p(\xi^j)=\langle u^\varepsilon_p\rangle+\chi^j_\omega(\varepsilon^{1/2}\xi^j)u^\varepsilon_{p\bot} (P^j+\varepsilon\xi^j),
\end{equation}
\tag{8.4}
$$
where $\chi^j_\omega$ are the cut-off functions (2.33) and
$$
\begin{equation}
\begin{gathered} \, \langle u^\varepsilon_p\rangle =\frac{1}{3\varepsilon(R^j_\omega)^2}\int_{\mathbb{B}_{2\sqrt{\varepsilon}\, R^j_\omega}(P^j)\setminus\mathbb{B}_{\sqrt{\varepsilon}\, R^j_\omega}(P^j)} u^\varepsilon_p(x)\, dx, \\ u^\varepsilon_{p\bot}(x)=u^\varepsilon_p(x)-\langle u^\varepsilon_p\rangle, \qquad\int_{\mathbb{B}_{2\sqrt{\varepsilon}\, R^j_\omega}(P^j) \setminus\mathbb{B}_{\sqrt{\varepsilon}\, R^j_\omega}(P^j)} u^\varepsilon_{p\bot}(x)\, dx=0. \end{gathered}
\end{equation}
\tag{8.5}
$$
Using the last orthogonality condition, we have the Poincaré inequality
$$
\begin{equation}
\begin{aligned} \, &\bigl\|u^\varepsilon_{p\bot};L^2\bigl(\mathbb{B}_{2\sqrt{\varepsilon}\, R^j_\omega}(P^j) \setminus\mathbb{B}_{\sqrt{\varepsilon}\, R^j_\omega}(P^j)\bigr)\bigr\| \nonumber \\ &\qquad\leqslant c\sqrt{\varepsilon}\, \bigl\|\nabla_xu^\varepsilon_{p\bot};L^2\bigl(\mathbb{B}_{2\sqrt{\varepsilon}\, R^j_\omega}(P^j) \setminus\mathbb{B}_{\sqrt{\varepsilon}\, R^j_\omega}(P^j)\bigr)\bigr\| \nonumber \\ &\qquad=c\sqrt{\varepsilon}\, \bigl\|\nabla_xu^\varepsilon_p;L^2\bigl(\mathbb{B}_{2\sqrt{\varepsilon}\, R^j_\omega}(P^j) \setminus\mathbb{B}_{\sqrt{\varepsilon}\, R^j_\omega}(P^j)\bigr)\bigr\|\leqslant c\sqrt{\varepsilon}\, \|u^\varepsilon_p;\mathcal{H}^\varepsilon\|, \end{aligned}
\end{equation}
\tag{8.6}
$$
where the multiplier $c\sqrt{\varepsilon}$ is found by coordinate stretching: it is proportional to the diameter of the annulus $\mathbb{B}_{2\sqrt{\varepsilon}R^j_\omega}(P^j) \setminus\mathbb{B}_{\sqrt{\varepsilon}R^j_\omega}(P^j)$ (the annulus area $3\pi^2(R^j_\omega)^2$ has appeared as the denominator in the first formula in (8.5)). By s definition (6.1),
$$
\begin{equation*}
\begin{aligned} \, \|\widehat{u}^{\,\varepsilon j}_p;\mathcal{H}_j\|^2 &=\bigl\|\nabla_\xi \widehat{u}^{\,\varepsilon j}_p;L^2\bigl(\mathbb{B}_{2\varepsilon^{-1/2}R^j_\omega}\bigr) \bigr\|^2+ \|\widehat{u}^{\,\varepsilon j}_p;L^2(\omega_j)\|^2 \\ &\leqslant c_j\bigl(\bigl\|\nabla_x u^\varepsilon_p;L^2\bigl(\mathbb{B}_{2\sqrt{\varepsilon}R^j_\omega}\bigr)\bigr\|^2+ \varepsilon^{-1}\bigl\|u^\varepsilon_{p\bot};L^2\bigl(\mathbb{B}_{2\sqrt{\varepsilon}R^j_\omega} \setminus\mathbb{B}_{\sqrt{\varepsilon}R^j_\omega}\bigr)\bigr\|^2 \\ &\qquad +\varepsilon^{-2}\|u^\varepsilon_p;L^2(\omega^\varepsilon_j)\|^2\bigr) \leqslant c_j\|u^\varepsilon_p;\mathcal{H}^\varepsilon\|^2 \leqslant C_j. \end{aligned}
\end{equation*}
\notag
$$
The relation $|\nabla_x\chi^j_\omega(\varepsilon^{-1/2}(x-P^j))|\leqslant c\varepsilon^{-1/2}$ and inequality (8.6) were used when processing the $L^2$-norm of the function $x\mapsto u^\varepsilon_{p\bot}(x) \nabla_x\chi^j_\omega(\varepsilon^{-1/2} (x-P^j))$, which appeared as a result of differentiation of the last term in (8.4). So, along a subsequence $\{\varepsilon_m\}_{m\in\mathbb{N}}$, we have the convergence
$$
\begin{equation}
\widehat{u}^{\,\varepsilon j}_p\to\widehat{w}^{\,j}_p\quad \text{wekly in } \mathcal{H}_j \text{ and strongly in }L^2(\omega_j).
\end{equation}
\tag{8.7}
$$
We now take the sum
$$
\begin{equation}
\psi^\varepsilon(x)=\psi_0(x)+\sum_{j=1}^J\psi_j(\varepsilon^{-1}(x-P^j)),
\end{equation}
\tag{8.8}
$$
as a test function in the integral identity (1.5). Here, $\psi_0\in C^\infty_{\mathrm{c}}(\overline{\Omega}\setminus\mathcal{P})$ and $\psi_j\in C^\infty_{\mathrm{c}}(\mathbb{R}^2)$ and, therefore, the supports of the terms on the right of (8.8) are disjoint for small $\varepsilon$. Furthermore,
$$
\begin{equation*}
\begin{aligned} \, \bigl((1+\varepsilon^{-\gamma}X^\varepsilon) u^\varepsilon_p,\psi_0\bigr)_\Omega &= (u^\varepsilon_p,\psi_0)_\Omega, \\ \bigl((1+\varepsilon^{-\gamma}X^\varepsilon)u^\varepsilon_p,\psi_j\bigr)_\Omega &=\varepsilon^{-\gamma}( \widehat{u}^{\,\varepsilon j}_p,\psi_j)_\Omega, \\ (\nabla_xu^\varepsilon_p,\nabla_x\psi_j)_\Omega &=(\nabla_\xi\widehat{u}^{\,\varepsilon j}_p,\nabla_\xi\psi_j)_{\mathbb{R}^2}. \end{aligned}
\end{equation*}
\notag
$$
Finally, using convergences (8.2), (8.3) and (8.7), and making as $\varepsilon\to+0$ in the integral identity (1.5), this gives
$$
\begin{equation*}
\sum_{j=1}^J(\nabla_\xi\widehat{w}^{\,j}_p,\nabla_\xi\psi_j)_{\mathbb{R}^2}+ (\nabla_x\widehat{w}^{\,0}_p,\nabla_x\psi_0)_\Omega =\widehat{\mu}_p\biggl(\sum_{j=1}^J(\widehat{w}^{\,j}_p,\psi_j)_{\omega_j} +\delta_{\gamma,2}(\widehat{w}^{\,0}_p,\psi_0)_\Omega\biggr).
\end{equation*}
\notag
$$
This relation splits into $J+1$ independent integral identities. After taking the closure (recall that the linear spaces $C^\infty_{\mathrm{c}}(\mathbb{R}^2)$ and $C^\infty_{\mathrm{c}}(\overline{\Omega}\setminus\mathcal{P})$ are dense in $\mathcal{H}_j$ and $H^1(\Omega)$, respectively), these identities become the variational statements (2.2) of the limit equations (1.10), $j=1,\dots,J$, and the variational statement (2.8) of the limit Neumann problem for $\gamma=2$, and, for $\gamma>2$, they change into the trivial identity
$$
\begin{equation*}
(\nabla_x \widehat{w}^{\,0}_p,\nabla_x\psi_0)_\Omega=0,\qquad \psi_0\in H^1(\Omega),
\end{equation*}
\notag
$$
meaning that $\widehat{w}^{\,0}_p={\widehat{c}}_p$ is a constant function. We emphasize that the above strong convergences in the Lebesgue spaces imply that
$$
\begin{equation*}
\begin{aligned} \, &\varepsilon^{\gamma-2}(u^\varepsilon_p,u^\varepsilon_q)_\Omega +\varepsilon^{-2}\sum_{j=1}^J(u^\varepsilon_p,u^\varepsilon_q)_{\omega^\varepsilon_j}= \varepsilon^{\gamma-2}(w^\varepsilon_p,w^\varepsilon_q)_\Omega +\sum_{j=1}^J(\widehat{u}^{\,\varepsilon j}_p,\widehat{u}^{\,\varepsilon j}_q)_{\omega_j} \\ &\qquad \to\delta_{\gamma,2}(\widehat{w}^{\,0}_p,\widehat{w}^{\,0}_q)_\Omega +\sum_{j=1}^J(\widehat{w}^{\,j}_p,\widehat{w}^{\, j}_q)_{\omega_j}. \end{aligned}
\end{equation*}
\notag
$$
Thus, the vector $\overrightarrow{\widehat{w}_p}=(\widehat{w}^{\,0}_p, \widehat{w}^{\,1}_p,\dots,\widehat{w}^{\, J}_p)$ is normalized in the space $\mathfrak{H}$ from (4.3) (see (1.7) and (2.7), (2.10)). In other words, the family of the limits $\widehat{w}^{\,0}_p, \widehat{w}^{\,1}_p,\dots,\widehat{w}^{\, J}_p$ thus obtained is not trivial, and, therefore, $\widehat{\mu}_p$ is an eigenvalue of at least one limit problem. Proposition 7. The limit passages (8.2), (8.3) and (8.7) along a positive null sequence $\{\varepsilon_m\}_{m\in\mathbb{N}}$ produce an eigenvalue $\widehat{\mu}_p$ for at least one limit problem (2.2), $j=1,\dots,J$, or (2.8) (for the last one, only in case $\gamma=2$). 8.2. Modification of reasoningThe equalities
$$
\begin{equation*}
\widehat{\mu}_p=\mu_p\quad\text{for}\quad p=k,\dots, k+\varkappa_k-1,
\end{equation*}
\notag
$$
which appear in Theorem 1, are still unverified in Proposition 7, but they are not required in the justification scheme from § 6. Indeed, the reference to Theorem 1 in the last step of the proof of Theorem 8 can be replaced by the following frequently useful argument. Since we have verified that near the points $\mu_2,\dots,\mu_k,\dots,\mu_{k+\varkappa_k-1}\in M$ there are at least $k+\varkappa_k-2$ eigenvalues (3.15), it is sufficient to verify that the half-open interval $\mathcal{I}_k=[0,(\mu_{k+\varkappa_k-1}+\mu_{k+\varkappa_k})/2)$ contains no “extra” normalized eigenvalues of problem (1.5). Having assumed that, for some positive null sequence $\{\varepsilon_m\}_{m\in\mathbb{N}}$, there is an eigenvalue $\varepsilon^{2-\gamma}\lambda^\varepsilon_{N^{\varepsilon_m}}\in\mathcal{I}_k$ whose eigenfunction $u^\varepsilon_{N^{\varepsilon_m}}$ is orthogonal, in sense of (1.7), to the selected $k+\varkappa_k-1$ eigenfunctions, we use Proposition 7 to perform the required passages to the limit, and, as a result, we find a vector eigenfunction of the family of limit problems corresponding to the eigenvalue $\widehat{\mu}_N<\mu_{k+\varkappa_k}$, but which is orthogonal to all vector eigenfunctions associated with the eigenvalues $\mu_1,\dots,\mu_k,\dots,\mu_{k+\varkappa_k-1}\in M$. But this is not possible. This contradiction completes the proof of Theorem 8.
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\Bibitem{Naz23} |
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