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This article is cited in 2 scientific papers (total in 2 papers) ‘Far interaction’ of small spectral perturbations of the Neumann boundary conditions for an elliptic system of differential equations in a three-dimensional domain S. A. Nazarov Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, St. Petersburg, Russia
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     § 1. Introduction1.1. The statement of the problemOn the smooth (C∞-smooth for simplicity: cf. § 4.1) boundary ∂Ω of a domain Ω⊂R3 with compact closure ¯Ω=Ω∪∂Ω we distinguish pairwise different points P1,…,PJ. For j=1,…,J, in a neighbourhood Vj∋Pj we introduce a local Cartesian coordinate system xj=(xj1,xj2,xj3) with origin Pj so that the xj3-axis is co-directed with the outward normal to ∂Ω, and the xj1- and xj2-axes lie in the tangent plane Πj. Also let (sj1,sj2,nj) be curvilinear coordinates in Vj associated with ∂Ω: nj is the oriented distance to ∂Ω such that nj<0 in Ω∩Vj and sjk is the oriented distance to Pj measured along the projection of the xjk-axis onto ∂Ω. We also consider the sets where ε is a small positive parameter, ϖj is a domain in R2 with smooth boundary and compact closure, sj=(sj1,sj2) and ηj′=(ηj1,ηj2). We show these sets and the coordinate system in Figure 1.Consider the formally selfadjoint system of partial differential equations with Neumann boundary conditions and with spectral conditions on the small sets (1.1) which are similar to the classical Steklov conditions: Here A is a symmetric positive definite real N×N matrix and D(∇x) is an N×K matrix of first-order differential operators with constant real coefficients such that D(0)=ON×K is the zero matrix, ∇x=grad and ⊤ denotes transposition. In addition, n(x) is the unit outward normal vector at x∈∂Ω and Q(x) is a nontrivial orthogonal projection in R3 which depends smoothly on x∈∂Ω. Finally, λε is the spectral projector and uε=(uε1,…,uεK)⊤ is a vector-valued eigenfunction.The variational statement of problem (1.2)–(1.4) refers to the integral identity (see [1] and [2]), where (⋅,⋅)Ξ is the natural inner product in the Lebesgue space L2(Ξ), H1(Ω) is a Sobolev space, and the last superscript K in (1.5) indicates the number of components of the vector-valued test function ψ=(ψ1,…,ψK)⊤ (however, this superscript does not participate in our notation for inner products and norms).Assume that we have chosen the domains (1.1) and the projection Q so that Korn’s inequality holds, where the coefficient Cε is independent of uε∈H1(Ω)K. We endow the Sobolev space Hε:=H1(Ω)K with the inner product We define a positive symmetric continuous — and therefore selfadjoint — operator Kε by the identity As the embedding H1(Ω)⊂L2(Ω) is compact, the essential spectrum of Kε consists of the single point κ=0, and its discrete spectrum is a positive infinitesimal sequence (for instance, see [3], Theorems 10.1.5 and 10.2.2).By the definitions (1.7) and (1.8) the integral identity (1.5) is equivalent to the abstract equation with spectral parameterFormula (1.11) transforms the sequence (1.9) into the unbounded monotone sequence of eigenvalues of problem (1.5) (or (1.2)–(1.4)) The first inequality in (1.12) holds because by (1.7) and (1.8) the norm of Kε is at most 1.On vector-valued eigenfunctions uε(1),uε(2),…,uε(m),…∈Hε of problem (1.5) we impose conditions of orthogonality and normalization: where δm,n is the Kronecker delta.The main aim of this paper is to construct an asymptotic formula for the eigenvalues (1.12) of problem (1.5) (or (1.2)–(1.4)); for several reasons, which we explain in what follows, they must be sought in the form 1.2. Assumptions about the problemWe assume that the matrix D is algebraically complete (see [4], Ch. 3), that is, there exists a positive integer ϱD∈N such that for each row p(ζ)=(p1(ζ),…,pK(ζ)) of homogeneous polynomials in ζ=(ζ1,ζ2,ζ3)⊤∈R3 of degree degpk=ϱ⩾ϱD there exists a row of polynomials q(ζ)=(q1(ζ),…,qN(ζ)) such that Note that, necessarily, N⩾K.If (1.15) is satisfied, then the operator L(∇x) in system (1.2) is said to be formally positive, and in each bounded domain Ξ with Lipschitz boundary ∂Ξ we have Korn’s inequality with coefficient CΞ which is independent of the vector-valued function u (see [4], § 3.7.4).In addition, the bilinear form E in (1.5) enjoys the polynomial property (see [5] and [6]), namely, where P is a finite-dimensional subspace of vector-valued polynomials. It is not difficult to see that and degpk⩽ϱD−1.From the polynomial property we can extract various useful information about the solvability of the problems below and the behaviour of their solutions. (The reader can find some details in [6].) In particular, it is clear that the first d:=dimP elements of the sequence (1.12) are zeros and, as appropriate vector-valued eigenfunctions, we can take some basis p1,…,pd of the subspace (1.18). Since the constant columns belong to P, we have d⩾K. We investigate the spectrum (1.12) under an additional restriction, which is the key condition in this work: Lemma 1. Under assumptions (1.17) and (1.19) Korn’s inequality holds, where the coefficient CΩQ is independent of uε∈H1(Ω)K and ε∈(0,ε0] for some ε0>0. Proof. We represent uε∈H1(Ω)K in the following form: where p(x)=(p1(x),…,pd(x)) is a K×d matrix composed of the elements of a basis of P, and P is the d×d Gram matrix constructed from the linearly independent vector-valued functions p1,…,pd by use of the inner product in L2(Ω)K: The matrix (1.23) is symmetric and positive definite. Finally, the orthogonality conditions are satisfied: and since, by the polynomial property (1.17), the quadratic form E vanishes only on the subspace (1.18), relations (1.24) and the lemma on equivalent norms show that Korn’s inequality (1.16) takes the following form in Ξ=Ω:
Thus, that is, it only remains to deal with the column (1.22). First using coordinate dilation we verify the elementary trace inequalities Furthermore, where we have Iε⊥⩽cεE(uε,uε;Ω) and where |ωεj|=ε2|ϖj|+O(ε3) and |ϖj| are the areas of the two-dimensional shapes ωεj⊂∂Ω and ϖj⊂R2, respectively. As a result, we obtainNow we verify that where the coefficient c is the same for all columns aε. This will show that which implies the required bound (1.20). If (1.28) fails for some nontrivial column aε, then we have Hence the polynomial p(x)=p(x)aε∈P satisfies Q(Pj)p(Pj)=0, j=1,…,J, and therefore p=0 in view of (1.19). This contradiction completes the proof of the lemma.The above assumptions ensure, in particular, Korn’s inequality (1.6) (see (1.20)), and thus justify the operator setting (1.10) of problem (1.2)–(1.4). Remark 1. Representation (1.21) and estimates (1.26) and (1.29) for its ingredients reflect more accurately the dependence of uε on the small parameter ε than (1.20) does. This is why in § 3, in dealing with discrepancies, we have to appeal to intermediate steps of the proof of Lemma 1. 1.3. Examples1∘. Surface waves. Let A=I3 be the 3×3 identity matrix, D(∇x)=∇x be a column of height 3, and let Q=1. The scalar problem involving Laplace’s equation (1.2) with Neumann (1.3) and Steklov (1.4) boundary conditions has a direct relation to the linear theory of waves on the surface of a weighty fluid (see [7], [8] and other works). In a physically meaningful setting Ω is bounded by a connected compact part Γ of the plane {x:x3=0} and a smooth surface Σ⊂R3−={x:x3<0}, which meet each other along a smooth one-dimensional ‘edge’ Υ=Γ∩¯Σ (Figure 2, a). The small sets (1.1) lie inside the domain Γ∖Υ (Figure 2, b). Assumption (1.19) holds for any set of points P1,…,PJ∈Γ∖Υ. Problem (1.2)–(1.4) specialized in this way, involves the velocity potential uε and the spectral parameter λε=g−1ς2ε, where ςε>0 is the oscillation frequency and g>0 is the gravitational acceleration. Problem (1.30)–(1.32) describes surface waves in the ice holes ωε1,…,ωεJ made in the ice cover Γ of the basin Ω with bottom Σ. As the asymptotic formula (1.14), which we establish in what follows for the eigenvalues (1.12) of problem (1.30)–(1.32), shows, the wave processes in several ice holes are linked in leading order independently of the distances between the holes.2∘. Problems in elasticity. Let K=3, N=6 and Then (1.2) should be interpreted as the equilibrium equations for an elastic body Ω in the absence of mass forces, where u=(u1,u2,u3)⊤ is the displacement vector and uk is its projection onto the xk-axis. The 6×6 matrix A is the matrix of elastic constants of the material. In the isotropic case it has the form
\begin{equation}
\begin{gathered} \, {\mathcal A}=\begin{pmatrix} {\mathcal A}_{11}&{\mathbb O}_3 \\ {\mathbb O}_3&{\mathcal A}_{22} \end{pmatrix}, \\ {\mathcal A}_{11}= \begin{pmatrix} \boldsymbol\lambda+2\boldsymbol\mu&\boldsymbol\lambda &\boldsymbol\lambda \\ \boldsymbol\lambda&\boldsymbol\lambda+2\boldsymbol\mu &\boldsymbol\lambda \\ \boldsymbol\lambda&\boldsymbol\lambda& \boldsymbol\lambda+2\boldsymbol\mu \end{pmatrix}\quad\text{and} \quad {\mathcal A}_{22}=\begin{pmatrix} \boldsymbol\mu&0&0 \\ 0&\boldsymbol\mu&0 \\ 0&0&\boldsymbol\mu \end{pmatrix}, \end{gathered}
\end{equation}
\tag{1.34}
where \boldsymbol \lambda\geqslant0 and \boldsymbol\mu>0 are the Lamé constants and {\mathbb O}_3 is the 3\times3 zero matrix. The description of other special cases, namely, orthotropy and transverse isotropy, can be bound, for instance, in [9], Ch. 2, § 2, and for the reduction of some anisotropic cases to orthotropy, see [10]. Boundary conditions (1.3) mean that the surface {\partial\Omega\setminus{\overline{\omega^\varepsilon}}} is free from external efforts; they involve the normal stress vector {\mathcal N}(x,\nabla_x)u(x) calculated from the stress column
\begin{equation}
{\mathcal A}{\mathcal D}(\nabla_x)u=\bigl(\sigma_{11}(u),\sigma_{22}(u),\sigma_{33}(u), \sqrt{2}\sigma_{23}(u),\sqrt{2}\sigma_{31}(u),\sqrt{2}\sigma_{12}(u)\bigr)^\top,
\end{equation}
\tag{1.35}
where the \sigma_{pq}(u) are the Cartesian components of the stress tensor. We introduce the coefficients 2^{\pm1/2} into the definitions (1.33) and (1.35) to equate the natural norms of a second-rank tensor and the column of height 6 representing it. Boundary conditions (1.4) are conditions of elastic pinching, which are also called Winkler-Steklov conditions (cf. [11]), and two settings are physically meaningful here. In the first the body is attached to an absolutely rigid profile by means of a dense family of tiny springs, which only react to tension in the normal direction; then the projection has the form In the second setting the body of fixed by spots of glue, which resist displacements of \Omega in any direction, so that
\begin{equation}
{\mathcal Q}(x)u(x)=u(x), \qquad {\mathcal Q}(x)={\mathbb I}_3.
\end{equation}
\tag{1.37}
In either case the external effects have a constant strength \gamma>0, so the spectral parameter has the form \lambda^\varepsilon=\gamma\varsigma^2_\varepsilon, where \varsigma_\varepsilon>0 is the frequency of oscillations. We stress that \Omega is made of ‘weightless’ material and equations (1.2) do not contain the spectral parameter (cf. § 4.4). The elastic energy functional \frac{1}{2}{\mathcal E}(u,u;\Omega) has the polynomial property (1.17) (see [6], Example 1.12), in which {\mathcal P}=\{p(x)=d(x)c \mid c\in{\mathbb R}^6\} is the six-dimensional linear space of rigid motions; here
\begin{equation}
d(x)=(d^t,d^r(x))= \begin{pmatrix} 1&0&0&0&2^{-1/2}x_3&-2^{-1/2}x_2 \\ 0&1&0&-2^{-1/2}x_3&0&2^{-1/2}x_1 \\ 0&0&1&2^{-1/2}x_2&-2^{-1/2}x_1&0 \end{pmatrix},
\end{equation}
\tag{1.38}
and d^t={\mathbb I}_3 (the 3\times3 identity matrix) and d^r are the 3\times3 translation and rotation blocks in the 3\times6 matrix of rigid motions. As shown in [12], § 2.2, in the case (1.37), to meet the restriction (1.19) is its sufficient that the set P^1,\dots,P^J contains three vertices of a nondegenerate triangle; however, in the case (1.36) the linear span of the columns must be equal to {\mathbb R}^6; in particular, we must have J\geqslant6. One example of a suitable fixing is shown in Figure 3. Figure 3.The sets \omega^\varepsilon_1,\dots,\omega^\varepsilon_6 and the normals to the surface of a parallelepiped that ensure the geometric condition on the linear span of the vectors (1.39). 1.4. The contents of the paperIn § 2 we construct the formal asymptotics, as \varepsilon\to+0, of the spectral pairs \{\lambda^\varepsilon_m,u^\varepsilon_{(m)}\} of problem (1.5) (or (1.2)–(1.4)) and deduce the limiting problem, which produces the (nonzero) limits \mu_m=\mu^0_{m-\mathbf d} involved in (1.14) and contains J boundary value problems in the half-space {\mathbb R}^3_-. It is a quite unexpected fact that each of these problems involves algebraic components, the mean values of vector-valued eigenfunctions over the sets \varpi_j\subset\partial {\mathbb R}^3_- for j=1,\dots,J, so that the limiting problems in half-spaces are interconnected and make up a single problem. In other words, we discover the interaction of the spectral boundary conditions (1.4), prescribed on small distant pieces (1.1) of the boundary. We stress that if we set Dirichlet conditions on a set \overline{\gamma}\subset \partial\Omega\setminus{\overline{\omega^\varepsilon}} of positive area, then the interaction phenomenon does not appear (see § 4.3) because the limiting (\varepsilon=0) mixed boundary value problem (2.18), (4.13), (4.14) in \Omega is uniquely solvable. This suggests that the underlying cause of the above phenomenon is the interaction of the root subspaces corresponding to the eigenvalue \lambda=0 of the limiting Neumann problem in the bounded domain \Omega and to the small eigenvalues \varepsilon\mu of the family (j=1,\dots,J) of limiting problems in the half-space {\mathbb R}^3_- which describe the boundary layer phenomenon. It is important here that the point \mu=0 lies away from the spectra of the latter boundary problems. The reader interested in the results of the asymptotic analysis in the simplest case of the (scalar) Steklov-Neumann problem for the Laplace operator, which we mentioned in § 1.3, 1^\circ, can turn to § 2.5, 1^\circ, where we present the system of interconnected limiting problems and discuss the relationships between its spectrum and the spectra of separate problems in the half-space. In particular, we show that when there is symmetry, the interaction phenomenon is damped. A more complicated example is given by the (vector) elasticity problem in §§ 1.3, 2^\circ and 2.5, 2^\circ. The property of leading asymptotic terms that we have discovered distinguishes the problem under consideration from a number of investigations of singularly perturbed scalar problems with Steklov spectral conditions and from vector elasticity problems with Winkler-Steklov spectral conditions (see [13]–[27] and [11], [28], respectively). Most investigations in the above — certainly, incomplete — list concern problems involving dense periodic families of small sets, on which Steklov spectral conditions are set: in this series of papers their authors use the so-called techniques of boundary homogenization, which is very different from the techniques used in our paper. On the other hand, in several of these papers their authors consider ‘isolated’ perturbations, related to those described in § 1.1: Steklov conditions are prescribed on the boundaries of small interior holes (see [17] and [26]) on a thin toroidal subset (see [25] and cf. Figure 2, c). In [17] and [26] the authors find asymptotic formulae1 for the eigenvalues and eigenfunctions of mixed boundary value problems for the Laplace operator, for various arrangements of Dirichlet, Neumann and Steklov conditions. A distinctive feature of the asymptotic constructions in [25] is that an additional limiting problem arises, which is an integral equation on the axis of the toroidal set. We also point out the papers [18] and [24], where the authors showed that the eigenvalues of a problem with Steklov conditions on a small set depend analytically on the small parameter \varepsilon. Note that the phenomenon of far interaction of singular perturbations, similar to what we consider here, was originally discovered in [29] (also see [30] and [31]) for the Neumann scalar problem in a multidimensional domain with concentrated masses; on the formal level various asymptotic constructions for the frequencies of eigenoscillations of an elastic body with small heavy inclusions and boundary free from external effects were described in [32], but justifying asymptotic expansions in such problems in elasticity theory is still an open question. In § 3 we present estimates for asymptotic remainders, which are obtained in a standard way: establishing Theorem 3 on convergence, dealing with the discrepancies of the asymptotic solution constructed and using Lemma 4 on ‘almost eigenvalues’ and ‘almost eigenvectors’. Note, however, that some peculiarities of our calculations, stemming from our general setting, are based on the geometric restriction (1.19) and the polynomial property (1.17), which, in particular, yields all necessary information about the solutions of limiting problems (see [6] and our §§ 2.1 and 2.2). Many of the assumptions in § 1.1 on the boundaries of the domains \Omega and \varpi_j and the coefficients of \mathcal L and \mathcal N are only needed for a clearer presentation, and in § 4.1 we discuss questions related to discarding these assumptions. On the other hand some generalizations, namely, setting Dirichlet conditions on subsets of \partial\Omega, introducing the spectral parameter into system (1.2), considering the special case J=1, or increasing the orders of differential operators make a considerable impact on both the asymptotic procedure and the structure of the limiting spectral problem itself. Our comments on the corresponding modifications, based on concrete examples, are presented in § 4. On the way we also state a number of open questions. Our presentation in § 4 makes no pretension on a complete analysis; its only aim is to attract the reader’s attention to the wide range of problems in which one does not even knows the asymptotic pattern when the geometric restriction (4.36) is not met. It is always satisfied only in the simplest Steklov scalar problem (see §§ 1.3, 1^\circ and 4.5); however, for spatial problems in elasticity or planar problems in the theory of plates and shells which have a clear applied interest, dropping this restrictions provokes significant changes in the final asymptotic formulae (cf. §§ 4.5 and 4.6). § 2. Deducing the limiting problem2.1. The Neumann problem in a half-spaceWe dilate local coordinates:2
\begin{equation}
x\mapsto \xi^j=\varepsilon^{-1}x^j=\varepsilon^{-1}\Theta^j(x-P^j),
\end{equation}
\tag{2.1}
where we need the 3\times3 orthogonal matrix \Theta^j in order to turn the x^j_3-axis in the direction of the outward normal n(P^j) (see § 1.1). The change of variables (2.1) and the formal transition to \varepsilon=0 straighten the surface {\partial\Omega\cap{\mathcal V}^j}, thus transforming \Omega into the half-space {\mathbb R}^3_-=\{\xi^j\colon \xi^j_3<0\} and \omega^\varepsilon_j into the set \varpi_j on its boundary {\mathbb R}^2=\partial {\mathbb R}^3_- (cf. Definition (1.1)). Moreover, since \partial\Omega is smooth, we have
\begin{equation}
\nabla_{(s^j_1,s^j_2,n^j)}=({\mathbb I}_3 +\phi(x))\nabla_{x^j}, \quad |\nabla_x\phi(x)|+|x-P^j|^{-1}|\phi(x)|\leqslant c_j, \qquad x\in{\mathcal V}^j,
\end{equation}
\tag{2.2}
so that the differential operators \varepsilon^2{\mathcal L}(\nabla_x) and \varepsilon{\mathcal N}(x,\nabla_x) take the following form after these changes:
\begin{equation}
\begin{gathered} \, {\mathcal L}^j(\nabla_{\xi^j})={\mathcal D}^j(-\nabla_{\xi^j})^\top {\mathcal A}{\mathcal D}^j(\nabla_{\xi^j}), \\ {\mathcal N}^j(\nabla_{\xi^j})={\mathcal D}^j(e_{(3)})^\top {\mathcal A}{\mathcal D}^j(\nabla_{\xi^j}), \end{gathered}
\end{equation}
\tag{2.3}
where {\mathcal D}^j(\nabla_{\xi^j})={\mathcal D}(\Theta^j\nabla_{\xi^j}) and e_{(3)}=(0,0,1)^\top. We forget for a while about the spectral parameter and boundary condition (1.4). After scaling the variables, straightening the boundary and freezing the coefficients, from (1.2)–(1.4) we obtain the problem
\begin{equation}
{\mathcal L}^j(\nabla_{\xi^j})w^j(\xi^j)=0, \qquad \xi^j\in{\mathbb R}^3_-,
\end{equation}
\tag{2.4}
\begin{equation}
{\mathcal N}^j(\nabla_{\xi^j})w^j(\xi^{j\prime},0)=g^j(\xi^{j\prime}), \qquad \xi^{j\prime}=(\xi^j_1,\xi^j_2)\in{\mathbb R}^2.
\end{equation}
\tag{2.5}
Here g^j\in L^2({\mathbb R}^2)^K is a vector-valued function with compact support (we need only such functions in § 2.3). Because the Neumann data are not sufficiently smooth, we must use a weak setting of problem (2.4), (2.5); as the domain is unbounded, weighted Sobolev spaces are required. We define the widely used Kondrat’ev space V^1_\beta({\mathbb R}^3_-) with weight exponent \beta\in {\mathbb R} (see [33] and, for instance, [34]) to be the completion of the linear space C^\infty_c({\overline{{\mathbb R}^3_-}}) (of infinitely differentiable functions with compact support) with respect to the norm
\begin{equation}
\|w;V^1_\beta({\mathbb R}^3_-)\|=\biggl(\sum_{p=0}^\ell\|(1+\rho_j)^{\beta -\ell+p}\nabla_{\xi^j}^pw;L^2({\mathbb R}^3_-)\|^2\biggr)^{1/2}.
\end{equation}
\tag{2.6}
We stress that V^1_\beta({\mathbb R}^3_-) consists of the functions w\!\in\! H^1_{\mathrm{loc}}({\overline{{\mathbb R}^3_-}}) with finite norm (2.6). The weighted Lebesgue space V^0_\gamma({\mathbb R}^3_-)=L^2_\gamma({\mathbb R}^3_-) is equipped with the norm \|(1+\rho_j)^\gamma\,\cdot\,; L^2({\mathbb R}^3_-)\|. In this section we need the case \ell=1, but we use the space V^2_0({\mathbb R}^3_-) in § 4.6. By a weak solution of (2.4), (2.5) we mean a vector function w^j\in V^1_\beta({\mathbb R}^3_-)^K satisfying the integral identity
\begin{equation}
{\mathcal E}(w^j,\psi^j):=({\mathcal A}{\mathcal D}^j(\nabla_{\xi^j})w^j, {\mathcal D}^j(\nabla_{\xi^j})\psi^j)_{{\mathbb R}^3_-}=(g^j,\psi^j)_{{\mathbb R}^2} \quad \forall\, \psi^j\in V^1_{-\beta}({\mathbb R}^3_-)^K.
\end{equation}
\tag{2.7}
By the definition (2.6), in the middle of (2.7) we have an extension of the inner product in L^2({\mathbb R}^3_-) to a duality between suitable weighted Lebesgue classes. In Remark 2 we explain the following simple result. and assume that the vector-valued function g^j\in L^2({\mathbb R}^2)^K has a compact support. Then problem (2.7) has a unique solution w^j\in V^1_\beta({\mathbb R}^3_-)^K. Because the data in (2.7) have compact supports, the representation
\begin{equation}
w^j(\xi^j)=\chi^\infty(\xi^j)\Phi^j(\xi^j)b^j+{\widetilde{w}}^{j}(\xi^j)
\end{equation}
\tag{2.9}
holds, where \chi^\infty is a smooth cut-off function vanishing in the ball {\mathbb B}_R=\{\xi^j\colon \rho_j:=|\xi^j|<R\}, which contains \varpi_j (see (1.1)), and equal to 1 for \rho_j>2R; b^j\in{\mathbb R}^K is a constant column, the remainder {\widetilde{w}}^{j} belongs to V^1_{1+\beta}({\mathbb R}^3_-)^K, and \Phi^j is the Green’s matrix of the Neumann problem (2.4), (2.5) on the half-space with singularity at the origin. This matrix satisfies
\begin{equation}
\Phi^j(\xi^j)=\rho_j^{-1}\Phi^j(\rho_j^{-1}\xi^j), \qquad \xi^j\in {\mathbb R}^3,
\end{equation}
\tag{2.10}
\begin{equation}
-\int_{{\mathbb S}^-_R}{\mathcal N}_\cup(\xi^j,\nabla_{\xi^j})\Phi^j(\xi^j)\,ds_{\xi^j} ={\mathbb I}_K,
\end{equation}
\tag{2.11}
where {\mathbb I}_K is the K\times K identity matrix, {\mathbb S}^-_R=\{\xi^j\colon \rho_j=R,\,\xi^j_3<0\} is the lower half-sphere and, in accordance with (2.3), the Neumann boundary operator on the sphere has the form
\begin{equation}
{\mathcal N}_\cup(\xi^j,\nabla_{\xi^j})={\mathcal D}(\rho_j^{-1}\xi^j)^\top {\mathcal A}{\mathcal D}(\nabla_{\xi^j}).
\end{equation}
\tag{2.12}
Remark 2. 1) By Korn’s inequality (1.16) and the one-dimensional Hardy inequality
\begin{equation}
\int_0^{+\infty} |W(\rho)|^2\,d\rho \leqslant 4\int_0^{+\infty}\biggl|\frac{dW}{d\rho}(\rho)\biggr|^2 \rho^2\,d\rho \quad \forall\, W\in C^1[0,+\infty),
\end{equation}
\tag{2.13}
which, after integration with respect to the angle variables, ensures that
\begin{equation}
\|\rho_j^{-1}w^j;L^2({\mathbb R}^3_-)\| \leqslant2\biggl\|\frac{\partial w^j}{\partial \rho_j};L^2({\mathbb R}^3_-)\biggr\| \leqslant2\|\nabla_{\xi^j} w^j;L^2({\mathbb R}^3_-)\|,
\end{equation}
\tag{2.14}
the norm on V^1_0({\mathbb R}^3_-)^K is equivalent to the ‘energy’ norm
\begin{equation*}
\bigl({\mathcal E}(w,w;{\mathbb R}^3_-)+\|w;L^2(\varpi_j)\|^2\bigr)^{1/2}.
\end{equation*}
\notag
Thus, for \beta=0 Proposition 1 follows from Riesz’s theorem on the representation of a continuous functional in a Hilbert space. 2) The condition \beta>-1/2 in (2.8) ensures that the constant vectors do not belong to V^1_\beta({\mathbb R}^3_-)^K. The other condition \beta<1/2 shows that the term \chi^\infty\Phi^jb^j distinguished in (2.9) lies in this space (but does not belong to V^1_{1+\beta}({\mathbb R}^3_-)^K). These two observations, in combination with Kondrat’ev’s theorem on asymptotic behaviour ensure Proposition 1 for all \beta\in(-1/2,1/2) (cf. [6], § 2.1). Proposition 2. In the hypotheses of Proposition 1 representation (2.9) holds for the coefficient column calculated by the formula Proof. Representation (2.9) follows from Kondrat’ev’s theorem on asymptotic behaviour (see [33] and [34], Ch. 3, § 5), and the verification of (2.15) is based on relations (2.11) (cf. [35] and [34], Ch. 4, § 3):
\begin{equation}
\begin{aligned} \, \notag \int_{{\mathbb R}^2} g^j(\xi^{j\prime})\,d\xi^{j\prime} &=\int_{{\mathbb R}^2} {\mathcal N}^j(\nabla_{\xi^j})w^j(\xi^{j\prime},0)\,d\xi^{j\prime} \\ \notag &=-\lim_{R\to+\infty}\int_{{\mathbb S}^-_R} {\mathcal N}^j_\cup(\xi^j,\nabla_{\xi^j})w^j(\xi^j)\,ds_{\xi^j} \\ &=-\lim_{R\to+\infty}\int_{{\mathbb S}^-_R} {\mathcal N}^j_\cup(\xi^j,\nabla_{\xi^j})\Phi^j(\xi^j)\,ds_{\xi^j}b^j=b^j. \end{aligned}
\end{equation}
\tag{2.16}
We stress that Kondrat’ev’s theorem on increasing the smoothness of solutions of elliptic boundary value problems yields the inclusion \nabla_{\xi^j}{\widetilde{w}}^{j}\in V^1_{2+\beta}({\mathbb R}^3_-)^K and paves the way for the limit transition in (2.16). In other words, the remainder {\widetilde{w}}^{j} is rapidly decaying and infinitely differentiable outside the ball {\mathbb B}_{R}, which contains the support of the right-hand side of the boundary condition (2.5).
The proof is complete. Remark 3. Hölder estimates for solutions of boundary value problems in domains with singular points on the boundary (see [36] and also [34], Ch. 3, § 6, for example). result in pointwise estimates
\begin{equation}
|\nabla^k_{\xi^j}w^j(\xi^j)|\leqslant c_k\rho_j^{-1-k} \quad\text{for } \xi^j\in{\overline{{\mathbb R}^3_-}}\setminus{\mathbb B}_R, \quad k\in {\mathbb N}_0=\{0,1,2,\dots\}.
\end{equation}
\tag{2.17}
The rate of decay of the solution w^j itself depends on the order of homogeneity of the function (2.40) producing the leading term of the asymptotic expansion (2.9), while for the remainder {\widetilde{w}}^{j} the exponent of the bound in (2.17) reduces by 1. We took this fact into account in our calculations in (2.16). 2.2. The Neumann problem in \OmegaAlthough \partial\Omega is smooth, we also require a weak statement of the problem
\begin{equation}
{\mathcal N}(x,\nabla_x)v(x)=g(x), \qquad x\in\partial\Omega.
\end{equation}
\tag{2.19}
As usual (see [33] and [34], Chs. 2 and 4), the Kondrat’ev space V^1_\beta(\Omega) is the completion of the linear space C^\infty_c({\overline{\Omega}}\setminus\{P^1,\dots,P^J\}) with respect to the weighted norm
\begin{equation}
\|v;V^\ell_\beta(\Omega)\|=\biggl(\sum_{p=0}^\ell \|\nabla_x^p v;L^2_{\beta-\ell+p}(\Omega)\|^2\biggr)^{1/2},
\end{equation}
\tag{2.20}
where L^2_\gamma(\Omega)=V^0_\gamma(\Omega) is the weighted Lebesgue space with norm and r=\min\{r_1,\dots,r_J\}, where r_j=|x-P^j|. The space V^\ell_\beta(\Omega) consists of the functions v\in H^\ell_{\mathrm{loc}}({\overline{\Omega}}\setminus\{P^1,\dots,P^J\}) such that the norm (2.20) is finite, We require the case \ell=1 in what follows. By a weak solution of problem (2.18), (2.19) with right-hand sides
\begin{equation}
f\in L^2_{\beta+1}(\Omega)\quad\text{and} \quad g\in L^2_{\beta+1/2}(\partial\Omega)
\end{equation}
\tag{2.22}
we mean a vector-valued function v \in V^1_\beta(\Omega)^K satisfying the integral identity
\begin{equation}
{\mathcal E}(v,\psi;\Omega)=(f,\psi)_\Omega+(g,\psi)_{\partial\Omega} \quad \forall \,\psi\in V^1_{-\beta}(\Omega)^K,
\end{equation}
\tag{2.23}
where (\,\cdot\,{,}\,\cdot\,)_\Omega and (\,\cdot\,{,}\,\cdot\,)_{\partial\Omega} are the extensions of the inner products in L^2(\Omega) and L^2(\partial\Omega) to dualities between suitable pairs of weighted Lebesgue classes. Note that because of the embeddings (2.22), definitions (2.20), (2.21) and the simple trace inequality (see, for instance, [34], Ch. 2)
\begin{equation}
\|\psi;L^2_{-\beta-1/2}(\partial\Omega)\|\leqslant c_\Omega\|\psi;V^1_{-\beta}(\Omega)\|,
\end{equation}
\tag{2.24}
the right-hand side of (2.23) is a continuous functional on the space V^1_{-\beta}(\Omega)\ni\psi. Proposition 3. Assume that the right-hand sides in (2.22) for the exponent (2.8) satisfy
\begin{equation}
(f,p)_\Omega+(g,p)_{\partial\Omega}=0 \quad \forall \,p\in {\mathcal P}.
\end{equation}
\tag{2.25}
Then problem (2.23) has a solution v\in V^1_\beta(\Omega)^K. It is defined up to a term from the linear space (2.8); however, when the orthogonality condition is imposed, it becomes unique and satisfies Proof. First we stress that, in accordance with Remark 2, the assumption \beta>-1/2 puts the smooth vector functions in \overline\Omega in the space V^1_\beta(\Omega)^K, and the condition \beta<1/2 excludes the singular solutions \Phi^j(x^j)b^j from this space (see (2.10)). Furthermore, the one-dimensional Hardy condition (2.13), used in neighbourhoods of the points P^j after introducing the spherical variables r_j\in{\mathbb R}_+ and \theta^j\in{\mathbb S}^2 ( = the unit sphere), shows that the spaces V^1_0(\Omega) and H^1(\Omega) are algebraically and topologically indistinguishable. Hence for \beta=0 the result is obvious because by the polynomial property (1.17) the positive quadratic form {\mathcal E}(\,\cdot\,{,}\,\cdot\,;\Omega) vanishes only at the vector-valued polynomials p \in{\mathcal P}. Finally, Kondrat’ev’s theorem on asymptotic behaviour (see [33] and [34], Theorem 4.2.1) allows us to vary the weight exponent \beta within the interval indicated in (2.8). The proof is complete. We keep the conditions (2.22) and (2.8), but seek the solution of (2.18), (2.19) in the class V^1_{1+\beta}(\Omega)^K, that is, we make the substitution \beta\mapsto1+\beta in the integral identity (2.23). Theorem 1. Under the assumptions of Proposition 3 each solution v\in V^1_{1+\beta}(\Omega)^K of problem (2.18), (2.19) has the representation
\begin{equation}
v(x)=\sum_{j=1}^J\chi_j(x)\Phi^j(x^j)b^j+{\widehat{v}}(x),
\end{equation}
\tag{2.27}
where {\widehat{v}}\in V^1_\beta(\Omega)^K, the b^j=(b^j_1,\dots,b^j_J)^\top\in {\mathbb R}^J are the coefficient columns, \Phi^j is the Green’s matrix expressed in the local Cartesian coordinates x^j (see §§ 1.1 and 2.1), and \chi_j\in C^\infty({\overline{\Omega}}) is a cut-off function with small support that is equal to 1 in a neighbourhood of P^j and, in addition, \operatorname{supp}\chi_j\cap\operatorname{supp}\chi_k=\varnothing for j\ne k. Moreover, Proof. Note that by (2.3), (2.4) and (2.8) the matrix-valued functions
\begin{equation}
\begin{gathered} \, \Omega\ni x\mapsto F^j(x)=\bigl[{\mathcal L}(\nabla_x),\chi_j(x)\bigr]\Phi^j(x^j), \\ \partial\Omega\ni x\mapsto G^j(x)=\bigl[{\mathcal N}(x,\nabla_x),\chi_j(x)\bigr]\Phi^j(x^j), \end{gathered}
\end{equation}
\tag{2.29}
containing the commutators of the differential operators {\mathcal L} and {\mathcal N} with the cut-off functions \chi_j, belong to the spaces V^0_{\beta+1}(\Omega)^K and V^0_{\beta+1/2}(\partial\Omega)^K, respectively. Thus, for the remainder {\widehat{v}} in the representation (2.27) we obtain the problem
\begin{equation}
\begin{gathered} \, {\mathcal L}(\nabla_x){\widehat{v}}(x)=f(x)-\sum_{j=1}^JF^j(x)b^j, \qquad x\in\Omega, \\ {\mathcal N}(x,\nabla_x){\widehat{v}}(x)=g(x)-\sum_{j=1}^JG^j(x)b^j, \qquad x\in \partial\Omega\setminus\{P^1,\dots,P^J\}, \end{gathered}
\end{equation}
\tag{2.30}
with right-hand sides in the spaces from Proposition 3. The orthogonality conditions (2.25), which are necessary and sufficient for the solvability of this problem in the class V^1_\beta(\Omega)^K, assume the form
\begin{equation}
\begin{aligned} \, \notag &(f,p)_\Omega+(g,p)_{\partial\Omega}+\sum_{j=1}^J\bigl((F^jb^j,p)_\Omega+(G^jb^j,p)_{\partial\Omega}\bigr) \\ \notag &\qquad =\sum_{j=1}^J\lim_{R\to+0}\biggl(\int_{\Omega\setminus{\mathbb B}_R(P^j)} p(x)^\top{\mathcal L}(\nabla_x)\bigl(\chi_j(x)\Phi^j(x^j)\bigr)\,dx \\ \notag &\qquad\qquad +\int_{\partial\Omega\setminus{\mathbb B}_R(P^j)} p(x)^\top{\mathcal N}(x,\nabla_x)\bigl(\chi_j(x)\Phi^j(x^j)\bigr)\,ds_x\biggr)b^j \\ &\qquad =\sum_{j=1}^J \lim_{R\to+0}\int_{\partial{\mathbb B}_R(P^j)\cap\Omega} p(x)^\top{\mathcal N}_\cup(x,\nabla_x)\Phi^j(x^j)\,ds_x\,b^j \notag \\ &\qquad=-\sum_{j=1}^J p(P^j)^\top b^j \end{aligned}
\end{equation}
\tag{2.31}
and transform into (2.28).
We outline a few points. First, by contrast to the calculations in (2.16), in the Green’s formula in (2.31) we take the operator (2.12) with minus sign because the outward normal to the sphere \partial{\mathbb B}_R(P^j) = \{x\colon r_j = R\} changes direction. Second, the differential operators {\mathcal L}(\nabla_x), {\mathcal N}(x,\nabla_x) and {\mathcal N}_\cup(x,\nabla_x) extinguish each vector polynomial p\in{\mathcal P}, and we also use the relation p(x)=p(P^j)+O(r_j) in calculating the last integral. Finally, since the smooth surface \partial\Omega is curved, the contour \partial{\mathbb B}_R(P^j)\cap\partial\Omega is not necessarily an equator of \partial{\mathbb B}_R(P^j), but the sets \partial{\mathbb B}_R(P^j)\cap\partial\Omega and \{x\colon r_j=R,\, x^j_3<0\} are only different inside a narrow — of width O(R^2) — strip, so that an the end of the calculation (2.31), which completes the verification of (2.28), we can use relation (2.11). It remains to note that representation (2.27) itself is ensured by Kondrat’ev’s theorem on the asymptotic behaviour of solutions of elliptic boundary value problems in neighbourhoods of boundary conic points (see [33] and also [34], Theorem 4.2.1). Theorem 1 is proved. 2.3. The limiting spectral problemAssume that vector-valued functions w^1, \dots,w^J\!\!\in\! V^1_\beta({\mathbb R}^3_-)^K have representation (2.9) with coefficient columns {b^1,\dots,b^J\!\!\in\! {\mathbb R}^K} and satisfy the homogeneous (f^j=0) systems of equations (2.4) and the boundary conditions (2.5) with right-hand sides
\begin{equation}
\begin{gathered} \, g^j(\xi^{j\prime})=\mu {\mathcal Q}(P^j) \bigl(w^j(\xi^{j\prime},0)+p(P^j)\bigr), \qquad \xi^{j\prime}\in\varpi_j, \\ g^j(\xi^{j\prime})=0, \qquad \xi^{j\prime}\in{\mathbb R}^2\setminus{\overline{\varpi_j}}, \end{gathered}
\end{equation}
\tag{2.32}
where \mu\in{\mathbb R} and p\in{\mathcal P} are the eigenvalue and vector-valued polynomial to be determined. In other words, we have the integral identity
\begin{equation}
{\mathcal E}^j(w^j,\psi^j)=\mu(w^j+p(P^j),{\mathcal Q}(P^j)\psi^j)_{\varpi_j}\ \quad \forall\, \psi^j\in V^1_{-\beta}({\mathbb R}^3_-)^K.
\end{equation}
\tag{2.33}
Relations (2.32) follow from the asymptotic ansatz
\begin{equation}
u^\varepsilon(x)=\sum_{j=1}^J \chi_j(x)w^j(\xi^j)+p(x)+\varepsilon{\widehat{v}}(x)+\dotsb
\end{equation}
\tag{2.34}
for the (nonnormalized) vector-valued eigenfunction, the change (1.14) of the spectral parameter, the scaling (2.1) and the formal transition to \varepsilon=0. In (2.34) we denote by dots the higher-order asymptotic terms which are not essential for our analysis (cf. § 4.2 as concerns the construction of full asymptotic expansions). Using Proposition 2 we find the column
\begin{equation}
b^j=\mu{\mathcal Q}(P^j)\int_{\varpi_j}\bigl(w^j(\xi^{j\prime},0)+p(P^j)\bigr)\,d\xi^{j\prime}
\end{equation}
\tag{2.35}
for the solution w^j of (2.33). Substituting the ansatz (2.34) into (1.2)–(1.4) and noting that
\begin{equation}
w^j(\xi^j)=\Phi^j(\xi^j) b^j+\dots=\varepsilon\Phi^j(x^j) b^j+\dotsb,
\end{equation}
\tag{2.36}
we collect the coefficients of \varepsilon. As a result, for the correction term {\widehat{v}} we obtain problem (2.30), where f=0 and g=0. It has the following variational statement:
\begin{equation}
{\mathcal E}({\widehat{v}},\psi;\Omega)=\sum_{j=1}^J {\mathcal F}^j(\psi)b^j \quad \forall\, \psi\in H^1(\Omega)^K.
\end{equation}
\tag{2.37}
It involves the K\times K matrices of the functionals
\begin{equation}
\begin{aligned} \, \notag {\mathcal F}^j(\psi) &=(F^j,\psi)_\Omega+(G^j,\psi)_{\partial\Omega} \\ &=\bigl({\mathcal A}({\mathcal D}(\nabla_x)\chi_j)\Phi^j,{\mathcal D}(\nabla_x)\psi\bigr)_\Omega -\bigl({\mathcal A}{\mathcal D}(\nabla_x)\Phi^j,({\mathcal D}(\nabla_x)\chi_j)\psi\bigr)_\Omega, \end{aligned}
\end{equation}
\tag{2.38}
constructed from the expressions (2.29) by means of additional integration by parts. By Theorem 1, in the above problem the compatibility conditions (2.28) assume the following form:
\begin{equation}
\sum_{j=1}^J q(P^j)^\top b^j=0 \quad \forall\, q\in{\mathcal P}.
\end{equation}
\tag{2.39}
We seek the vector polynomial p in (2.34) as a linear combination
\begin{equation}
p(x)=\sum_{k=1}^{\mathbf d}\mathbf p^k(x)\mathbf a_k=:\mathbf p(x)\mathbf a,
\end{equation}
\tag{2.40}
where the k\times\mathbf d matrix \mathbf p(x) =(\mathbf p^1(x),\dots,\mathbf p^{\mathbf d}(x)) is formed by basis elements of the linear subspace \mathcal P of vector polynomials (see § 1.2) and \mathbf a= (\mathbf a_1,\dots,\mathbf a_\mathbf d)^\top is an unknown column. Taking (2.35) into account we can write the conditions (2.39) for q=\mathbf p^\ell as the system of linear algebraic equations
\begin{equation}
\sum_{j=1}^J\mathbf p^\ell(P^j)^\top {\mathcal Q}(P^j)\biggl(\int_{\varpi_j} w^j(\xi^{j\prime},0)\,d\xi^{j\prime}+|\varpi_j|\,\sum_{k=1}^{\mathbf d}\mathbf p^k(x)\mathbf a_k \biggr)=0, \qquad \ell=1,\dots,\mathbf d.
\end{equation}
\tag{2.41}
We set
\begin{equation}
{\overline{w}}^{j} =\frac{1}{|\varpi_j|}\int_{\varpi_j} w^j(\xi^{j\prime},0)\,d\xi^{j\prime}, \qquad \mathbf M^j_{\ell k}=|\varpi_j| \mathbf p^\ell(P^j)^\top{\mathcal Q}(P^j)\mathbf p^k(P^j)
\end{equation}
\tag{2.42}
and
\begin{equation}
\mathbf M_{\ell k}= \mathbf M^1_{\ell k}+\dots+ \mathbf M^J_{\ell k} .
\end{equation}
\tag{2.43}
Lemma 2. By condition (1.19) the matrix \mathbf M=(\mathbf M_{\ell k})_{\ell,\, k=1}^{\mathbf d} with entries (2.43) is symmetric and positive definite. Proof. By the definition (2.42) each matrix \mathbf M^j is obviously symmetric and positive definite. Let the full matrix \mathbf M be singular, so that for some column \mathbf a\in{\mathbb R}^{\mathbf d}\setminus\{0\} and vector polynomial p=\mathbf p\mathbf a we have
\begin{equation*}
\begin{aligned} \, 0=\mathbf a^\top\mathbf M\mathbf a &\quad \Longleftrightarrow\quad 0=\mathbf a^\top\mathbf M^j\mathbf a, \qquad j=1,\dots,J \\ &\quad \Longleftrightarrow\quad 0={\mathcal Q}(P^j)\sum_{k=1}^{\mathbf d}\mathbf a_k\mathbf p(P^j) ={\mathcal Q}(P^j)p(P^j), \qquad j=1,\dots,J. \end{aligned}
\end{equation*}
\notag
Then p=0 because of (1.19). This contradiction completes the verification of the lemma. Using the matrix notation and (2.42), from (2.41) we deduce that
\begin{equation}
\mathbf a=-\mathbf M^{-1}\sum_{n=1}^J|\varpi_n| \mathbf p(P^n)^\top{\mathcal Q}(P^n){\overline{w}}^{n}.
\end{equation}
\tag{2.44}
As a result, after taking the sum we can write the system (j=1,\dots,J) of integral identities (2.6) as the following single spectral problem for the row of vector functions \overrightarrow{w}=(w^1,\dots,w^J):
\begin{equation}
{\mathfrak E}(\overrightarrow{w},\overrightarrow{\psi})= \mu{\mathfrak Q}(\overrightarrow{w},\overrightarrow{\psi}) \quad\forall \, \overrightarrow{\psi}=(\psi^1,\dots,\psi^J)\in{\mathfrak H}:= V^1_0({\mathbb R}^3_-)^{K\times J}.
\end{equation}
\tag{2.45}
Here the bilinear forms {\mathfrak E} and {\mathfrak Q} are given by
\begin{equation}
{\mathfrak E}(\overrightarrow{w},\overrightarrow{\psi}) =\sum_{j=1}^J {\mathcal E}^j(w^j,\psi^j)
\end{equation}
\tag{2.46}
and
\begin{equation}
\begin{aligned} \, &{\mathfrak Q}(\overrightarrow{w},\overrightarrow{\psi}) =\sum_{j=1}^J \biggl(w^j-\mathbf p(P^j)\mathbf M^{-1}\sum_{n=1}^J |\varpi_n|\mathbf p(P^n)^\top{\mathcal Q}(P^n){\overline{w}}^{n}, {\mathcal Q}(P^j)\psi^j\biggr)_{\varpi_j} \\ &\quad=\sum_{j=1}^J (w^j,\psi^j)_{\varpi_j}- \sum_{j=1}^J |\varpi_j|(\mathbf p(P^j)^\top{\mathcal Q}(P^j){\overline{\psi}}^{j})^\top \mathbf M^{-1}\sum_{n=1}^J \mathbf p(P^n)^\top{\mathcal Q}(P^n){\overline{w}}^{n}|\varpi_n| \\ &\quad=:{\mathfrak Q}_{\mathrm{int}}(\overrightarrow{w},\overrightarrow{\psi})- {\mathfrak Q}_{\mathrm{alg}}(\overrightarrow{w},\overrightarrow{\psi}). \end{aligned}
\end{equation}
\tag{2.47}
In (2.47) we have distinguished the ‘algebraic component’ (which links the original integral identities with indices j=1,\dots,J) by bearing in mind the definition of the mean value of a function in \varpi_j (see (2.42)). 2.4. The spectrum of the limiting problemFixing the value \beta=0 of the weight index makes of (2.45) a variational problem. Furthermore, as noted in Remark 2, the weighted norm in {\mathfrak H}=V^1_0({\mathbb R}^3_-)^{K\times J} is equivalent to the energy norm
\begin{equation*}
\|\overrightarrow{w};{\mathfrak H}\|=\biggl(\sum_{j=1}^J \bigl({\mathcal E}^j(w^j,w^j)+\|w^j;L^2(\varpi_j)\|^2\bigr)\biggr)^{1/2},
\end{equation*}
\notag
and we can obtain the space {\mathfrak H} by completing the linear space C^\infty_c({\overline{{\mathbb R}^3_-}})^{K\times J} with respect to this norm. Lemma 3. The symmetric bilinear forms (2.46) and (2.45) are positive compact and positive definite, respectively, on the space {\mathfrak H}. Proof. Corollary (2.14) to Hardy’s inequality (2.13) yields the relations
\begin{equation*}
\|w_j;V^1_0({\mathbb R}_-^3)\|^2\leqslant 5\|\nabla_{\xi^j}w^j;L^2({\mathbb R}_-^3)\|^2
\end{equation*}
\notag
and
\begin{equation*}
\|w_j;L^2(\varpi_j)\|^2\leqslant C_j\|\nabla_{\xi^j}w^j;L^2({\mathbb R}_-^3)\|^2.
\end{equation*}
\notag
It remains to note that the fact that the positivity of the form \mathfrak H follows from the estimates
\begin{equation}
\begin{gathered} \, \|{\overline{w}}^{j};{\mathbb R}^K\|^2\leqslant|\varpi_j|\,\|w^j;L^2(\varpi_j)\|^2, \\ {\mathfrak Q}_{\mathrm{alg}}(\overrightarrow{w},\overrightarrow{w})\leqslant\sum_{j=1}^J |\varpi_j|\,\|{\overline{w}}^{j};{\mathbb R}^K\|^2, \end{gathered}
\end{equation}
\tag{2.48}
which are, in fact, versions of the Cauchy-Schwarz-Bunyakovsky inequality, an integral and an algebraic one. The second relation in (2.48) should perhaps be explained. Letting M^{1/2} denote the positive square root of the positive definite symmetric matrix M we obtain
\begin{equation*}
\begin{aligned} \, &{\mathfrak Q}_{\mathrm{alg}}(\overrightarrow{w},\overrightarrow{w})= \sum_{j=1}^J\bigl\|M^{-1/2}\mathbf p(P^j)^\top {\mathcal Q}(P^j) {\overline{w}}^{j}|\varpi_j|;{\mathbb R}^{\mathbf d}\bigr\|^2 \\ &\ \ \leqslant \sum_{n=1}^J\bigl\||\varpi_n|^{1/2}{\mathcal Q}(P^n)^\top\mathbf p(P^n)M^{-1/2}; {\mathbb R}^{K\times\mathbf d}\bigr\|^2\,\sum_{j=1}^J\bigl\|{\overline{w}}^{j}|\varpi_j|^{1/2}; {\mathbb R}^K\bigr\|^2 \\ &\ \ =\biggl\|M^{-1/2}\biggl(\sum_{n=1}^J \mathbf p(P^n)^\top {\mathcal Q}(P^n)|\varpi_n| {\mathcal Q}(P^n)^\top\mathbf p(P^n)\biggr)M^{-1/2}; {\mathbb R}^{\mathbf d\times \mathbf d}\biggr\| \sum_{j=1}^J|\varpi_j|\,|{\overline{w}}^{j}|^2. \end{aligned}
\end{equation*}
\notag
By the definitions (2.43) and (2.42) the first factor on the right is the norm of the \mathbf d\times\mathbf d identity matrix, that is, 1.
The proof is complete. We endow the Hilbert space \mathfrak H with an inner product {\mathfrak E}(\,\cdot\,{,}\,\cdot\,) and an operator {\mathfrak K} by means of the identity
\begin{equation*}
{\mathfrak E}({\mathfrak K}\overrightarrow{w},\overrightarrow{\psi})= {\mathfrak Q}(\overrightarrow{w},\overrightarrow{\psi}) \quad \forall\,\overrightarrow{w},\overrightarrow{\psi}\in{\mathfrak H}.
\end{equation*}
\notag
As the embedding {\mathfrak H}\subset L^2(\varpi_1)^K\times\dots\times L^2(\varpi_J)^K is compact, the essential spectrum of this selfadjoint continuous compact operator consists of the unique point {\mathfrak k}=0, and its discrete spectrum is an infinitesimal positive sequence
\begin{equation}
{\mathfrak k}_1\geqslant{\mathfrak k}_2\geqslant\dots\geqslant{\mathfrak k}_m\geqslant\dotsb\to+0
\end{equation}
\tag{2.49}
(see [3], Theorems 10.1.5 and 10.2.2). Theorem 2. The eigenvalues of problem (2.45) form a positive monotone unbounded sequence
\begin{equation}
0<\mu_1\leqslant\mu_2\leqslant\dots\leqslant\mu_m\leqslant\dotsb\to+\infty.
\end{equation}
\tag{2.50}
The corresponding eigenvectors \overrightarrow{w}_{(1)},\overrightarrow{w}_{(2)},\dots,\overrightarrow{w}_{(m)}, \ldots\in {\mathfrak H} can be chosen so as to satisfy the following conditions of orthogonality and normalization: Proof. It is sufficient to mention that (2.45) is equivalent to the abstract equation
\begin{equation*}
{\mathfrak K}\overrightarrow{w}={\mathfrak k}\overrightarrow{w} \quad\text{in}\ {\mathfrak H},
\end{equation*}
\notag
and the sequence (2.50) consists of the reciprocals of elements of (2.49). Note that for \mu=0, from (2.45) and (2.46) for \overrightarrow{\psi}=\overrightarrow{w} we derive the implication
\begin{equation*}
{\mathcal E}^j(w^j,w^j)=0 \quad \Longrightarrow\quad w^j\in{\mathcal P} \quad \Longrightarrow\quad w^j=0.
\end{equation*}
\notag
(First we have used the polynomial property (1.17) and then noted that no polynomial can occur in V^1_0({\mathbb R}^3_-): the integrals in (2.6) diverge for \beta=0.)
The proof is complete. 2.5. Examples1^\circ. Surface waves. The above scheme assigns to problem (1.30)–(1.32) of surface waves in ice holes (see Figure 2, b) the system (j=1,\dots,J) of problems
\begin{equation}
-\Delta_{\xi^j}w^j(\xi^j)=0, \qquad \xi^j\in{\mathbb R}^3_-,
\end{equation}
\tag{2.52}
\begin{equation}
\frac{\partial w^j}{\partial\xi^j_3}(\xi^{j\prime},0)=0, \qquad \xi^{j\prime} \in{\mathbb R}^2\setminus{\overline{\varpi_j}},
\end{equation}
\tag{2.53}
\begin{equation}
\frac{\partial w^j}{\partial\xi^j_3}(\xi^{j\prime},0)=\mu\biggl(w^j(\xi^{j\prime},0) -\biggl(\sum_{n=1}^J|\varpi_n|\biggr)^{-1}\sum_{n=1}^J \int_{\varpi_k}w^n(\xi^{n\prime},0)\,d\xi^{n\prime} \biggr), \qquad \xi^{j\prime}\in\varpi_j.
\end{equation}
\tag{2.54}
The variational version of the combined problem (2.52)–(2.54) looks as follows:
\begin{equation}
\begin{gathered} \, \notag \sum_{j=1}^J(\nabla_{\xi^j}w^j,\nabla_{\xi^j}\psi^j)_{{\mathbb R}^3_-}= \mu\sum_{j=1}^J\biggl((w^j,\psi^j)_{\varpi_j}-\sum_{k=1}^J \frac{|\varpi_j|\,|\varpi_k|}{|\varpi_1|+\dots+|\varpi_J|} {\overline{w}}^{k}{\overline{\psi}}^{j}\biggr) \\ \forall \, \overrightarrow{\psi}=(\psi^1,\dots,\psi^J)\in V^1_0({\mathbb R}^3_-)^J. \end{gathered}
\end{equation}
\tag{2.55}
Here {\overline{w}}^{j} is the mean value of the function w^j on the set \omega_j=\varpi_j\times\{0\} (cf. (2.42)). The last sums with respect to k=1,\dots,J in (2.54) and (2.55) reflect the interaction of waves on a water surface in ice holes made in the ice field covering the basin \Omega. On the other hand, if an eigenfunction w^j_\# of the separate Steklov problem in {\mathbb R}^3_- involving (2.52), (2.53) and the relation
\begin{equation}
\frac{\partial w^j_\#}{\partial\xi^j_3}(\xi^{j\prime},0)=\mu_\#^jw^j_\#(\xi^{j\prime},0), \qquad \xi^{j\prime}\in\varpi_j,
\end{equation}
\tag{2.56}
vanishes at infinity at a rate of O(|\xi^j|^{-2}) (the coefficient of (2\pi|\xi^j|)^{-1} in representation (2.57)), then \mu_\#^j is an eigenvalue of problem (2.55) with eigenvector {\overrightarrow{w}}_\#= (\delta_{j,1} w^1_\#,\dots,\delta_{j,J} w^J_\#). The first eigenfunction of problem (2.52), (2.53), (2.56), which is positive in {\mathbb R}^3_-, does not have the required rate of decay, but when there is geometric symmetry, the leading terms of the asymptotic expression
\begin{equation}
w^j_\#(\xi^j)=(2\pi|\xi^j|)^{-1}b^j_\# +O(|\xi^j|^{-2}), \qquad |\xi^j|\to+\infty,
\end{equation}
\tag{2.57}
for the other eigenfunctions can indeed vanish, so that b^j_\#=0. For example, in the case of a disc \varpi_j=\{\xi^{j\prime}\colon |\xi^{j\prime}|<\mathbf r_j\} this is the property of the second and third eigenfunctions of the ordinary Steklov problem (2.52), (2.53), (2.56), which can be assumed to be odd with respect to the variables \xi^j_1 and \xi^j_2. Even when there is one ice hole (J=1) the limiting problem is different from a usual Steklov-Neumann problem because for j=1 we add the integro-differential spectral condition
\begin{equation}
\frac{\partial w^1}{\partial\xi^1_3}(\xi^{1\prime},0)=\mu\biggl(w^1(\xi^{1\prime},0) -\frac{1}{|\varpi_1|}\int_{\varpi_1}w^1(\xi^{1\prime},0)\,d\xi^{1\prime} \biggr), \qquad \xi^{1\prime}\in\varpi_1,
\end{equation}
\tag{2.58}
to (2.52) and (2.53). Using the maximin principle (for instance, see [3], Theorem 10.2.2) it is easy to see that eigenvalues \mu^\#_m and \mu^\bullet_m of the usual Steklov problem (2.52), (2.53), (2.56) and problem (2.52), (2.53), (2.58), which is perturbed by an integral operator, are related by For J>1 let us look at the special case \varpi_1=\dots=\varpi_J, when all ice holes have the same shape (are made by the same drill). Let \{\mu^\#_1,w^\#_1\} and \{\mu^\bullet_1,w^\bullet_1\} be the first eigenpairs of the separate problems (2.52), (2.56) and (2.52), (2.58) mentioned above. Direct calculations show that in the linked problem (2.52)–(2.54) (or (2.55), in the variational form) the first eigenpairs look as follows:
\begin{equation*}
\bigl\{\mu^\#_1,(\alpha_1^pw^\#_1(\xi^1),\dots,\alpha_J^pw^\#_1(\xi^J))\bigr\}\in {\mathbb R}_+\times V^1_0({\mathbb R}^3_-)^J, \qquad p=1,\dots,J-1.
\end{equation*}
\notag
Here \alpha^p=(\alpha_1^p,\dots,\alpha_J^p)^\top, and \alpha^1,\dots,\alpha^{J-1} is a basis of the subspace \{\alpha\in{\mathbb R}^J: \alpha_1+\dots+\alpha_j=0\}. We also indicate the eigenpair
\begin{equation*}
\bigl\{\mu^\bullet_1,(w^\bullet_1(\xi^1),\dots,w^\bullet_J(\xi^J))\bigr\}\in {\mathbb R}_+\times V^1_0({\mathbb R}^3_-)^J,
\end{equation*}
\notag
which has an eigenvalue larger than \mu^\#_1 by (2.59). Because there are several eigenfunctions corresponding to the eigenvalue \mu^\#_1 with multiplicity J-1, surface waves in the ice holes \omega^\varepsilon_1,\dots,\omega^\varepsilon_j interact chaotically. In other words, we can hope for a full-scale interaction phenomenon only when the ice holes have different shapes. 2^\circ. Problems in elasticity. As \Theta^jn(P^j) is the basis vector e^j_{(3)} parallel to the \xi_3^j-axis (see the comments to (2.1)), for the projection (1.36) the limiting problem (2.45) assumes the following form
\begin{equation}
\begin{aligned} \, \notag \sum_{j=1}^J {\mathcal E}^j(w^j,\psi^j) &=\mu\sum_{j=1}^J \biggl((w^j,\psi^j)_{\varpi_j} \\ \notag &\qquad -\sum_{k=1}^J |\varpi_j|{\overline{\psi}}^{j}_3n(P^j)^\top d(P^j)\mathbf M^{-1} d(P^k)^\top n(P^k){\overline{w}}_3^{k}|\varpi_k|\biggr) \\ &\qquad \forall\, \overrightarrow{\psi}=(\psi^1,\dots,\psi^J)\in V^1_0({\mathbb R}^3_-)^{3\times J}, \end{aligned}
\end{equation}
\tag{2.60}
which involves the matrix
\begin{equation*}
\mathbf M=\sum_{j=1}^J |\varpi_j|d(P^j)^\top n(P^j) n(P^j)^\top d(P^j)
\end{equation*}
\notag
of size 6\times6, which is positive definite because of the restriction imposed in § 1.3 on the linear span of the columns (1.39) (see the example in Figure 3). For the identity projection (1.37) the right-hand side of (2.60) is replaced by the expression
\begin{equation*}
\mu\sum_{j=1}^J \biggl((w^j,\psi^j)_{\varpi_j}-\sum_{k=1}^J |\varpi_j| ({\overline{\psi}}^{j})^\top n(P^j)^\top d(P^j)\Theta^j\mathbf M^{-1} (d(P^k)\Theta^k)^\top n(P^k){\overline{w}}^{k}|\varpi_k|\biggr).
\end{equation*}
\notag
Now the symmetric 6\times6 matrix is positive definite, provided that there are three vertices of a nondegenerate triangle among P^1,\dots,P^J.
§ 3. Justifying the asymptotics3.1. A convergence theoremIn Remark 4 we will show that the elements of the sequence (1.12) satisfy
\begin{equation}
\lambda^\varepsilon_m\leqslant C^\unicode{8224}_m\varepsilon^{-1} \quad \text{for } \varepsilon\in(0,\varepsilon_m^\unicode{8224}],
\end{equation}
\tag{3.1}
where m\in{\mathbb N}, and C^\unicode{8224}_m and \varepsilon^\unicode{8224}_m are positive numbers. Thus, for fixed m we have the convergence over a positive infinitesimal sequence \{\varepsilon_m^{(p)}\}_{p\in{\mathbb N}}. Below we drop the indices of \varepsilon for brevity. Clearly, The aim of this section is to verify that for m>\mathbf d the quantity \mu^\bullet_m is the eigenvalue \mu_{m-\mathbf d} in (2.50) and the asymptotic formula (1.14) holds, and to find some information about vector-valued eigenfunctions. We represent the vector-valued eigenfunction u^\varepsilon_{(m)}, normalized in accordance with (1.13), as a sum (1.21), with terms a^\varepsilon_{(m)}\in{\mathbb R}^{\mathbf d} and u^{\varepsilon\bot}_{(m)}\in{\mathcal H}^\varepsilon satisfying (1.22), (1.28) and (1.24)–(1.27), respectively, so that, in particular,
\begin{equation}
\begin{gathered} \, \|a^\varepsilon_{(m)};{\mathbb R}^{\mathbf d}\|^2\leqslant c\varepsilon^{-1}\bigl( {\mathcal E}(u^\varepsilon_{(m)},u^\varepsilon_{(m)};\Omega)+\varepsilon^{-1} \|{\mathcal Q}u^\varepsilon;L^2(\omega^\varepsilon)\|^2\bigr)\leqslant c\varepsilon^{-1}, \\ \|u^{\varepsilon\bot}_{(m)};H^1(\Omega)\|^2+ \varepsilon^{-1}\|u^{\varepsilon\bot}_{(m)};L^2(\omega^\varepsilon_j)\|^2\leqslant c {\mathcal E}(u^\varepsilon_{(m)},u^\varepsilon_{(m)};\Omega)\leqslant c. \end{gathered}
\end{equation}
\tag{3.3}
Here we have taken (1.7) and (1.13) into account. Using Hardy’s one-dimensional inequality (2.13) in neighbourhoods of P^1,\dots,P^J yields the weighted estimates
\begin{equation}
\|(\varepsilon+r)^{-1}u^{\varepsilon\bot}_{(m)};L^2(\Omega)\|\leqslant \|r^{-1}u^{\varepsilon\bot}_{(m)};L^2(\Omega)\|\leqslant c\|u^{\varepsilon\bot}_{(m)};H^1(\Omega)\|.
\end{equation}
\tag{3.4}
Set
\begin{equation}
w^{\varepsilon j}_{(m)}(\eta^j)=\sqrt{\varepsilon}\, \chi_j(x) u^{\varepsilon\bot}_{(m)}(x), \qquad j=1,\dots,J,
\end{equation}
\tag{3.5}
where the \chi_j are the cut-off functions from (2.27), and in the neighbourhood {\mathcal V}^j we have introduced the stretched coordinates
\begin{equation}
\eta^j=(\eta^{j\prime},\eta^j_3)=(\varepsilon^{-1}s^j,\varepsilon^{-1}n^j),
\end{equation}
\tag{3.6}
which are different from the \xi^j=(\xi^{j\prime},\xi^j_3) (cf. § 1.1 and (2.1)). The vector-valued functions w^{\varepsilon 1}_{(m)}, \dots, w^{\varepsilon J}_{(m)} in (3.5) are defined in the half-space {\mathbb R}^3_-. Since and relations (2.2) hold, and since it is a simple calculation that
\begin{equation*}
\begin{aligned} \, \|w^{\varepsilon j}_{(m)};V^1_0({\mathbb R}^3_-)\|^2 &\leqslant \frac{c}{\varepsilon}\,\int_{\Omega\cap{\mathcal V}^j}\bigl(|\nabla_x w^{\varepsilon j}_{(m)}(\eta^j)|^2+\varepsilon^{-2}(1+\varepsilon^{-1}r_j)^{-2} |w^{\varepsilon j}_{(m)}(\eta^j)|^2\bigr)\,dx \\ &\leqslant c\|u^{\varepsilon\bot}_{(m)};H^1(\Omega)\|^2. \end{aligned}
\end{equation*}
\notag
Here we have taken (2.6) with \beta=0, and (3.3) and (3.4) into account. Hence
\begin{equation}
\sqrt{\varepsilon}a^\varepsilon_{(m)}\to a^\bullet_{(m)} \quad\text{in } {\mathbb R}^{\mathbf d}
\end{equation}
\tag{3.9}
and
\begin{equation}
w^{\varepsilon j}_{(m)}\to w^{\bullet j}_{(m)} \quad\text{weakly in } V^1_0({\mathbb R}^3_-)^K \text{ and strongly in } L^2(\varpi_j)^K.
\end{equation}
\tag{3.10}
Thus, taking a vector-valued test function \psi^j\in C^\infty_c({\mathbb R}^3_-)^K we multiply (1.5) by \varepsilon^{-1/2}, make the substitutions (1.14) and (1.21), note that u^{\varepsilon\bot}_{(m)}= \varepsilon^{-1/2}w^{\varepsilon j}_{(m)} on the set \operatorname{supp}\psi^j\cap\Omega, and take the limit as \varepsilon\to+0. Taking the limit relations (3.10) and (3.9) and the inequalities
\begin{equation*}
\bigl|\varepsilon\nabla_{x^j}w^{\varepsilon j}_{(m)}(\eta^j)- \nabla_{\eta^j}w^{\varepsilon j}_{(m)}(\eta^j)\bigr|\leqslant cr_j| \nabla_{\eta^j}w^{\varepsilon j}_{(m)}(\eta^j)|
\end{equation*}
\notag
and
\begin{equation*}
\bigl|{\mathcal Q}(x)w^{\varepsilon j}_{(m)}(\eta^j)- {\mathcal Q}(P^j)w^{\varepsilon j}_{(m)}(\eta^j)\bigr|\leqslant cr_j|w^{\varepsilon j}_{(m)}(\eta^j)|
\end{equation*}
\notag
into account, we see that in the limit we obtain the integral identity
\begin{equation}
\bigl({\mathcal A}{\mathcal D}^j(\nabla_{x^j})w^{\bullet j}_{(m)}, {\mathcal D}^j(\nabla_{x^j})\psi^j\bigr)_{{\mathbb R}^3_-}= \mu^\bullet_j\bigl(w^{\bullet j}_{(m)}+\mathbf p(P^j)a^\bullet_{(m)} , {\mathcal Q}(P^j)\psi^j\bigr)_{\varpi_j}.
\end{equation}
\tag{3.11}
Now we note that a similar limiting procedure under the orthogonality conditions (1.13) for u^\varepsilon_{(n)}=\mathbf p^n, n=1,\dots,\mathbf d, in view of (1.7) and (3.9), (3.10) results in the equalities
\begin{equation}
\sum_{j=1}^J\bigl(w^{\bullet j}_{(m)}+\mathbf p(P^j)a^\bullet_{(m)},{\mathcal Q}(P^j) \mathbf p^n(P^j)\bigr)_{\varpi_j}=0, \qquad n=1,\dots,\mathbf d.
\end{equation}
\tag{3.12}
Furthermore, because of the strong convergence (3.10) the normalization (1.13) of u^\varepsilon_{(m)} ensures in the limit that
\begin{equation}
\sum_{j=1}^J\bigl({\mathcal Q}(P^j)(w^{\bullet j}_{(m)}+\mathbf p(P^j)a^\bullet_{(m)}), w^{\bullet j}_{(m)}+\mathbf p(P^j)a^\bullet_{(m)}\bigr)_{\varpi_j}=1.
\end{equation}
\tag{3.13}
By the definitions (2.42), (2.43) and (2.47), from (3.12) we derive a relation similar to (2.44),
\begin{equation*}
a^\bullet_{(m)} =-\mathbf M^{-1}\sum_{k=1}^J|\varpi_k|\mathbf p(P^k)^\top {\mathcal Q}(P^k){\overline{w}}^{\bullet j}_{(m)},
\end{equation*}
\notag
which transforms the sum of integral identities (3.11) into the limiting problem (2.45) and transforms (3.13) into the normalization condition (2.51). Thus, we have the following result. Theorem 3. For each m>\mathbf d the limiting procedures (3.2) and (3.10) produce an eigenpair \bigl\{\mu^\bullet_m,{\overrightarrow{w}}^{\bullet}_{(m)}\bigr\} \in{\mathbb R}_+\times V^1_0({\mathbb R}_-^3)^{K\times J} of problem (2.45), where the eigenvector {\overrightarrow{w}}^{\bullet}_{(m)}=(w^{\bullet 1}_{(m)}, \dots,w^{\bullet J}_{(m)}) satisfies the normalization (2.51). 3.2. Asymptotic approximations to an eigenpairIn this subsection we use the following result, known as a lemma on an ‘almost eigenvalue’ and an ‘almost eigenvector’ (see the original paper [37]), which is ensured by the spectral decomposition of the resolvent (for instance, see [3], Ch. 6). We use the notation from § 1.1. Lemma 4. Let \mathbf u^\varepsilon\in{\mathcal H}^\varepsilon and \mathbf k^\varepsilon\in{\mathbb R}_+ satisfy
\begin{equation}
\|\mathbf u^\varepsilon;{\mathcal H}^\varepsilon\|=1\quad\textit{and} \quad \|{\mathcal K}^\varepsilon\mathbf u^\varepsilon-\mathbf k^\varepsilon \mathbf u^\varepsilon;{\mathcal H}^\varepsilon\|=:\delta\in[0,\mathbf k^\varepsilon).
\end{equation}
\tag{3.14}
Then the operator {\mathcal K}^\varepsilon has an eigenvalue \kappa^\varepsilon_{n^\varepsilon} satisfying
\begin{equation}
|\mathbf k^\varepsilon-\kappa^\varepsilon_{n^\varepsilon}|\leqslant\delta.
\end{equation}
\tag{3.15}
Moreover, for any \delta_\ast\in(\delta,\mathbf k^\varepsilon) there exist coefficients \mathbf c^\varepsilon_{\mathbf N^\varepsilon},\dots,\mathbf c^\varepsilon_{\mathbf N^\varepsilon +\mathbf X^\varepsilon-1} such that
\begin{equation}
\biggl\|\mathbf u^\varepsilon-\sum_{l=\mathbf N^\varepsilon}^{\mathbf N^\varepsilon +\mathbf X^\varepsilon-1}\mathbf c^\varepsilon_lu^\varepsilon_p;{\mathcal H}^\varepsilon \biggr\|\leqslant2\frac{\delta}{\delta_\ast}, \qquad \sum_{l=\mathbf N^\varepsilon}^{\mathbf N^\varepsilon +\mathbf X^\varepsilon-1}|\mathbf c^\varepsilon_l|^2=1,
\end{equation}
\tag{3.16}
where \kappa^\varepsilon_{\mathbf N^\varepsilon},\dots,\kappa^\varepsilon_{\mathbf N^\varepsilon +\mathbf X^\varepsilon-1} is the set of all eigenvalues of {\mathcal K}^\varepsilon on the closed interval [\mathbf k^\varepsilon-\delta_\ast, \mathbf k^\varepsilon+\delta_\ast], and the corresponding eigenvectors u^\varepsilon_{\mathbf N^\varepsilon},\dots,u^\varepsilon_{\mathbf N^\varepsilon +\mathbf X^\varepsilon-1} satisfy conditions (1.13) of orthogonality and normalization. Let \mu_m be an eigenvalue of problem (2.45) with multiplicity \varkappa_m, that is,
\begin{equation}
\mu_{m-1}<\mu_m=\dots=\mu_{m+\varkappa_m-1}<\mu_{m+\varkappa_m}.
\end{equation}
\tag{3.17}
As almost eigenvalues we take \varkappa_m copies of the quantity and we define almost eigenvectors by the formulae
\begin{equation}
\mathbf u^\varepsilon_{(q)}=\|\mathbf v^\varepsilon_{(q)};{\mathcal H}^\varepsilon\|^{-1} \mathbf v^\varepsilon_{(q)}, \qquad q=m,\dots,m+\varkappa_m-1,
\end{equation}
\tag{3.19}
and
\begin{equation}
\mathbf v^\varepsilon_{(q)}(x)=\mathbf p(x)\mathbf a_{(q)}+\sum_{j=1}^J \chi_j(x)w^j_{(q)}(\eta^j)+\varepsilon{\widehat{v}}_{(q)}(x).
\end{equation}
\tag{3.20}
Here the {\overrightarrow{w}}_{(q)}=(w^1_{(q)},\dots,w^J_{(q)}) are eigenvectors of the limiting problem (2.45) corresponding to the eigenvalue \mu_m and satisfying (2.51), and the {\widehat{v}}_{(q)}\in V^1_\beta(\Omega)^K are the solutions of problems (2.30) for f=0, g=0 and for b^j=b^j_{(q)} equal to the columns of coefficients in the expansions (2.9) of the w^j_{(q)}. We stress that \beta is an arbitrary exponent in the interval (2.8), and the conditions (2.30) for the solvability of the problem for {\widehat{v}}_{(q)} are satisfied by the constructions in § 2.3; moreover, all freedom in the choice of a solution to it is eliminated by the orthogonality conditions (2.26). Lemma 5. The vector-valued functions (3.20) satisfy
\begin{equation}
\bigl|\langle\mathbf v^\varepsilon_{(p)},\mathbf v^\varepsilon_{(q)}\rangle_\varepsilon- \varepsilon(1+\mu_m)\bigr|\leqslant c_m(\beta)\varepsilon^{\beta+3/2},
\end{equation}
\tag{3.21}
where p,q=m,\dots,m+\varkappa_m-1, \beta\in(-1/2,1/2), \varepsilon\in(0,\varepsilon_m(\beta)], and \varepsilon_m(\beta) and c_m(\beta) are some positive numbers. Proof. As explained above, when we make the change of variables x^j\mapsto\eta^j the fact that \partial\Omega is curved results in slight distortions of inner products. So taking (2.2) and (3.7), (3.8) into account we obtain
\begin{equation*}
\begin{gathered} \, {\mathcal E}(\chi_jw^j_{(p)},\chi_jw^j_{(q)};\Omega)=\varepsilon {\mathcal E}^j(w^j_{(p)},w^j_{(q)})+O(\varepsilon^2), \qquad {\mathcal E}(\mathbf p\mathbf a_{(p)},\mathbf v^\varepsilon_{(q)};\Omega)=0, \\ |{\mathcal E}(\varepsilon{\widehat{v}}_{(p)},\mathbf v^\varepsilon_{(q)};\Omega)| \leqslant c\varepsilon \|{\widehat{v}}_{(p)};V^1_{-\beta}(\Omega)\|\, \|\mathbf v^\varepsilon_{(q)};V^1_\beta(\Omega)\|\leqslant C\varepsilon\varepsilon^{\beta+1/2}. \end{gathered}
\end{equation*}
\notag
Here we bear in mind that both \beta and -\beta lie in the interval (-1/2,1/2) and use the inequalities
\begin{equation*}
\begin{gathered} \, \|\chi_jw^j_{(p)};V^1_\beta(\Omega)\|^2\leqslant c\varepsilon^{1+2\beta} \|\eta^j\mapsto\chi_j(x)w^j_{(p)}(\eta^j);V^1_\beta({\mathbb R}^3_-)\|^2\leqslant C\varepsilon^{1+2\beta}, \\ |w^j_{(p)}(\eta^j)|\leqslant c_p|\eta^j|^{-1}\leqslant C_p\varepsilon \quad\text{for } x\in \sigma_j=\operatorname{supp}|\nabla_x\chi_j|, \end{gathered}
\end{equation*}
\notag
as well as the fact that {\mathcal D}(\nabla_x)\mathbf p^\ell=0 (see (1.18)). For the same reasons we have
\begin{equation*}
({\mathcal Q}\mathbf v^\varepsilon_{(p)},\mathbf v^\varepsilon_{(q)})_{\omega^\varepsilon_j} =\varepsilon^2\bigl({\mathcal Q}(P^j)(w^j_{(p)}+\mathbf p(P^j)\mathbf a_{(p)}),\, w^j_{(q)}+\mathbf p(P^j)\mathbf a_{(q)}\bigr)_{\varpi_j}+O(\varepsilon^3).
\end{equation*}
\notag
Because of (4.4) and (2.45), these relations imply the required bounds (3.21).
The verification of the lemma is complete. 3.3. Dealing with discrepanciesWe estimate the quantities \delta_q obtained by the second formula in (3.14) from the vector functions (3.19). We see from the definitions (1.7) and (1.8) that
\begin{equation}
\begin{aligned} \, \notag \delta_q &=\sup\bigl|\langle{\mathcal K}^\varepsilon\mathbf u^\varepsilon_{(q)}- \mathbf k^\varepsilon_m\mathbf u^\varepsilon_{(q)},\psi^\varepsilon \rangle_\varepsilon\bigr| \\ \notag &=\|\mathbf v^\varepsilon_{(q)};{\mathcal H}^\varepsilon\|^{-1} \mathbf k^\varepsilon_m\sup\bigl|\varepsilon^{-1}(1+\mu_m)({\mathcal Q} \mathbf v^\varepsilon_{(q)},\psi^\varepsilon)_{\omega^\varepsilon} \\ \notag &\qquad -{\mathcal E}^\varepsilon(\mathbf v^\varepsilon_{(q)},\psi^\varepsilon;\Omega)-\varepsilon^{-1}({\mathcal Q} \mathbf v^\varepsilon_{(q)},\psi^\varepsilon)_{\omega^\varepsilon}\bigr| \\ &=\|\mathbf v^\varepsilon_{(q)};{\mathcal H}^\varepsilon\|^{-1} \mathbf k^\varepsilon_m\sup\bigl| {\mathcal E}^\varepsilon(\mathbf v^\varepsilon_{(q)},\psi^\varepsilon ;\Omega)-\varepsilon^{-1}\mu_m({\mathcal Q} \mathbf v^\varepsilon_{(q)},\psi^\varepsilon)_{\omega^\varepsilon}\bigr|. \end{aligned}
\end{equation}
\tag{3.22}
Here we take the supremum over the unit ball in {\mathcal H}^\varepsilon, that is, \|\psi;{\mathcal H}^\varepsilon\|\leqslant1, so that the calculations made in the proof of Lemma 1 result in the following bounds (cf. (1.25), (1.27) and (1.29)) for the components \alpha^\varepsilon and \psi^{\varepsilon\bot} of the representation (1.21) for the test function \psi^\varepsilon:
\begin{equation}
\|\psi^{\varepsilon\bot};H^1(\Omega)\|+ \varepsilon^{-1/2}\|\psi^{\varepsilon\bot};L^2(\omega^\varepsilon)\|\leqslant C, \qquad \|\alpha^\varepsilon;{\mathbb R}^{\mathbf d}\|\leqslant C\varepsilon^{-1/2}.
\end{equation}
\tag{3.23}
Our immediate aim is to process the expression {\mathcal J}^\varepsilon(\psi^\varepsilon) under the last modulus sign in (3.22), namely,
\begin{equation}
\begin{aligned} \, \notag {\mathcal J}^\varepsilon(\psi^\varepsilon) &=\sum_{j=1}^J\biggl( {\mathcal E}(\chi_j w^j_{(q)},\psi^\varepsilon;\Omega)-\frac{\mu_m}{\varepsilon} ({\mathcal Q}(w^j_{(q)}+\mathbf p\mathbf a_{(q)}),\psi^\varepsilon)_{\omega^\varepsilon_j} \biggr) \\ &\qquad+\varepsilon{\mathcal E}({\widehat{v}}_{(q)},\psi^\varepsilon;\Omega) +\mu_m({\mathcal Q}{\widehat{v}}_{(q)},\psi^\varepsilon)_{\omega^\varepsilon}. \end{aligned}
\end{equation}
\tag{3.24}
We transform the terms
\begin{equation}
{\mathcal E}(\chi_jw^j_{(q)},\psi^\varepsilon;\Omega)-\varepsilon^{-1}\mu_m\bigl( {\mathcal Q}(w^j_{(q)}+\mathbf p\mathbf a_{(q)}),\psi^\varepsilon\bigr)_{\omega^\varepsilon_j}
\end{equation}
\tag{3.25}
of the first sum as follows. First, we carry over the cut-off function \chi_j from w^j_{(q)} to \psi^\varepsilon. Then we go over to the curvilinear variables s^j and \eta^j, after which we freeze the coefficients at P^j and estimate the resulting errors to obtain an integral identity (2.33) with vector-valued test function \chi_j\psi^\varepsilon and to obtain (2.37) for {\widehat{v}}_{(q)}. Finally, we take account of the fact that \bigl\{\mu_m,{\overrightarrow{w}}_{(q)}\bigr\} is an eigenpair of problem (2.45) and, what is important, \mathbf a_{(q)} has been found using (4.4). First of all, we write the inequalities
\begin{equation}
\begin{aligned} \, \notag &\bigl| {\mathcal E}(\chi_jw^j_{(q)},\psi^{\varepsilon\bot};\Omega)- {\mathcal E}( w^j_{(q)},\chi_j\psi^{\varepsilon\bot};\Omega) \\ \notag &\ \ \qquad +({\mathcal A}({\mathcal D}(\nabla_x) \chi_j)\Phi^jb^j_{(q)}, {\mathcal D}(\nabla_x)\psi^{\varepsilon\bot};\Omega) -({\mathcal A}{\mathcal D}(\nabla_x)\Phi^jb^j_{(q)}, ({\mathcal D}(\nabla_x) \chi_j)\psi^{\varepsilon\bot};\Omega)\bigr| \\ \notag &\ \ =\bigl|\bigl({\mathcal A}({\mathcal D}(\nabla_x)\chi_j){\widetilde{w}}^{j}_{(q)}, {\mathcal D}(\nabla_x)\psi^{\varepsilon\bot}\bigr)_\Omega- ({\mathcal A}({\mathcal D}(\nabla_x){\widetilde{w}}^{j}_{(q)},( {\mathcal D}(\nabla_x)\chi_j)\psi^{\varepsilon\bot})_\Omega\bigr| \\ &\ \ \leqslant c\Bigl(\sup_{x\in\sigma_j} |{\widetilde{w}}^{j}_{(q)}(\eta^j)|+\varepsilon^{-1}\sup_{x\in\sigma_j} |\nabla_{\eta^j}{\widetilde{w}}^{j}_{(q)}(\eta^j)| \Bigr)\|\psi^{\varepsilon\bot};H^1(\Omega)\| \leqslant c\varepsilon^2. \end{aligned}
\end{equation}
\tag{3.26}
Apart from the first relation in (3.23), here we take account of the rates O(\rho_j^{-2}) and O(\rho_j^{-3}) of decay of the remainder term {\widetilde{w}}^{j}_{(q)}(\eta^j) and its gradient (see Remark 3): on the support \sigma_j={\operatorname{supp}}|\nabla_x\chi_j| of the matrix-valued function {\mathcal D}(\nabla_x)\chi_j, which lies away from P^j, these infinitesimal quantities become O(\varepsilon^2) and O(\varepsilon^3), respectively. In addition, by (2.2) and (3.8) the change of variables \eta^j\mapsto \xi^j, which straightens the boundary, results in O(r_j)-errors, which are infinitesimals of order O(\varepsilon) since the vector function w^j_{(q)} decays at infinity. Keeping the notation w^j_{(q)} and \chi_j\psi^{\varepsilon\bot} after switching to the new variables, we see that
\begin{equation}
\bigl|{\mathcal E}( w^j_{(q)},\chi_j\psi^{\varepsilon\bot};\Omega)- \varepsilon{\mathcal E}^j( w^j_{(q)},\chi_j\psi^{\varepsilon\bot})\bigr| \leqslant c\varepsilon^2.
\end{equation}
\tag{3.27}
Second, treating the third and fourth term under the modulus sign on the left-hand side of (3.26) in a similar way and taking (2.36) into account we see that
\begin{equation}
\begin{aligned} \, \notag &\bigl|({\mathcal A}({\mathcal D}(\nabla_x) \chi_j)\Phi^jb^j_{(q)}, {\mathcal D}(\nabla_x)\psi^{\varepsilon\bot};\Omega) \\ &\qquad\qquad -({\mathcal A}{\mathcal D}(\nabla_x)\Phi^jb^j_{(q)},({\mathcal D} (\nabla_x) \chi_j)\psi^{\varepsilon\bot};\Omega) -\varepsilon{\mathcal F}^j (\psi^{\varepsilon\bot})\bigr| \leqslant c\varepsilon^2. \end{aligned}
\end{equation}
\tag{3.28}
Thus, because of (2.37) and (2.38) the last subtrahend in (3.28) coincides with the term {\mathcal E}({\widehat{v}}_{(q)},\psi^\varepsilon; \Omega) in (3.24), so that they annihilate in the expression for {\mathcal J}^\varepsilon(\psi^\varepsilon). It remains to note that {\widehat{v}}_{(q)}\in H^2(\Omega)^K and, using (3.23), deduce the estimate
\begin{equation}
\begin{aligned} \, \notag &\mu_m\bigl|({\mathcal Q}{\widehat{v}}_{(q)},\psi^{\varepsilon\bot}+ \mathbf p\alpha^\varepsilon)_{\omega^\varepsilon}\bigr| \\ &\qquad \leqslant c|\omega^\varepsilon|^{1/2} \max_{x\in\partial\Omega}|{\widehat{v}}_{(q)}(x)|\,\bigl(\|\psi^{\varepsilon\bot} ;L^2(\omega^\varepsilon)\|+|\omega^\varepsilon|^{1/2}|\alpha^\varepsilon|\bigr) \leqslant c\varepsilon^{3/2}. \end{aligned}
\end{equation}
\tag{3.29}
The worst bound (in comparison with (3.26)–(3.28)) occurs in (3.29); in § 4.2 we explain that the last term in (3.24) participates in the correction term \varepsilon\mu' in the asymptotic ansatz (1.14) for an eigenvalue of problem (1.5). To summarize: by (3.21) and (3.18) the coefficients multiplying the last supremum in (3.22) do not exceed c_m\varepsilon^{-1/2}\varepsilon=c_m\varepsilon^{1/2}, so that we have established the inequalities
\begin{equation*}
\delta_q\leqslant C_m\varepsilon^{1/2}\varepsilon^{3/2}=C_m \varepsilon^2, \qquad q=m,\dots,m+\varkappa_m-1,
\end{equation*}
\notag
with some coefficient C_m. Now, by Lemma 4 the C_m\varepsilon^2-neighbourhood of the point (3.18) contains the eigenvalues \kappa^\varepsilon_{Q^\varepsilon_m(m)}, \dots, \kappa^\varepsilon_{Q^\varepsilon_m(m+\varkappa_m-1)} of {\mathcal K}^\varepsilon, so that these eigenvalues satisfy
\begin{equation}
|\mathbf k^\varepsilon_m-\kappa^\varepsilon_{Q^\varepsilon_m(q)}|\leqslant C_m\varepsilon^2, \qquad q=m,\dots,m+\varkappa_m-1.
\end{equation}
\tag{3.30}
3.4. A theorem on the asymptotic behaviour of eigenvaluesFirst we verify that we have found different eigenvalues. To do this we use the second part of Lemma 4, where we take \delta_\ast=\tau C_m\varepsilon^2 with a positive parameter \tau. We start by assuming simply that \tau>1, but then we fix it at a sufficiently large value. We verify that the number \mathbf X^\varepsilon of the eigenvalues of {\mathcal K}^\varepsilon on the interval
\begin{equation*}
[\mathbf k^\varepsilon_m-\tau C_m\varepsilon^2,\mathbf k^\varepsilon_m+\tau C_m\varepsilon^2]
\end{equation*}
\notag
is at least \varkappa_m. Let \Sigma_{(q)}^\varepsilon be the sums over l=\mathbf N^\varepsilon,\dots, \mathbf N^\varepsilon+\mathbf X^\varepsilon-1 in the first formula in (3.16), and let \mathbf C^\varepsilon_{(q)} be the corresponding orthonormalized coefficient columns of linear combinations of the eigenvectors {\mathbf u}^\varepsilon_{\mathbf N^\varepsilon},\dots,{\mathbf u}^\varepsilon_{\mathbf N^\varepsilon+\mathbf X^\varepsilon-1} from Lemma 4. By conditions (1.13) and inequalities (3.16) and (3.21) we have
\begin{equation}
\begin{aligned} \, \notag &|(\mathbf C^\varepsilon_{(p)})^\top \mathbf C^\varepsilon_{(q)}- \delta_{p,q}|= |\langle(\Sigma^\varepsilon_{(q)},\Sigma^\varepsilon_{(p)}\rangle_\varepsilon- \delta_{p,q}| \\ \notag &\qquad\leqslant|\langle \Sigma^\varepsilon_{(q)}-\mathbf u^\varepsilon_{(q)}, \Sigma^\varepsilon_{(q)}\rangle_\varepsilon| +|\langle\mathbf u^\varepsilon_{(q)},\Sigma^\varepsilon_{(p)}-\mathbf u^\varepsilon_{(p)}\rangle_\varepsilon| +|\langle\mathbf u^\varepsilon_{(q)},\mathbf u^\varepsilon_{(p)}\rangle_\varepsilon- \delta_{p,q}| \\ &\qquad \leqslant4\tau^{-1}+c_m\varepsilon. \end{aligned}
\end{equation}
\tag{3.31}
For small \varepsilon and large \tau this means that the columns \mathbf C^\varepsilon_{(m)},\dots, \mathbf C^\varepsilon_{(m+\varkappa_m-1)} \in{\mathbb R}^{\mathbf X^\varepsilon} are ‘almost orthonormalized’, which is possible only for \mathbf X^\varepsilon\geqslant\varkappa_m. Thus, by fixing a suitable \tau>1 we can assume that after the substitution formula (3.30) will involve the eigenvalues \kappa^\varepsilon_{Q^\varepsilon_m}, \dots,\kappa^\varepsilon_{Q^\varepsilon_m+\varkappa_m-1} (new notation) starting from some initial index Q_m. Now we increase C_m in (3.30) in this way. Remark 4. For each element \mu_m of the sequence (2.50) that satisfies (3.17), for \varepsilon\in(0,\varepsilon_m], where \varepsilon_m is a certain positive quantity, we have found at least \varkappa_m eigenvalues of {\mathcal K}^\varepsilon in a neighbourhood of (1+\mu_m)^{-1}. Thus, if \varepsilon\leqslant \min\{\varkappa_1,\dots,\varkappa_m\} then at least \mathbf d+m+\varkappa_m-1 elements of the sequence (1.9) occur on the interval [\varepsilon(1+m_m)^{-1}-C_m\varepsilon^2,1], so that \kappa^\varepsilon_m\geqslant c_m\varepsilon. Furthermore, for q=m,\dots,m+\varkappa_m- 1, using (1.11), (3.18) and (3.30) we deduce that
\begin{equation}
\begin{aligned} \, \notag &\varepsilon\bigl|(1+\varepsilon\lambda^\varepsilon_{Q^\varepsilon_m+q})^{-1}-(1+\mu_m)^{-1}\bigr| \leqslant C_m\varepsilon^2 \\ \notag &\quad \Longrightarrow\quad \begin{cases} |\lambda^\varepsilon_{Q^\varepsilon_m+q})-\varepsilon^{-1}\mu_m| \leqslant C_m(1+\varepsilon\lambda^\varepsilon_{Q^\varepsilon_m+q})(1+\mu_m), \\ 1+\varepsilon\lambda^\varepsilon_{Q_m+q}\leqslant(1+\mu_m)(1+C_m(1+ \varepsilon\lambda^\varepsilon_{Q^\varepsilon_m+q})) \end{cases} \\ &\quad \Longrightarrow\quad |\lambda^\varepsilon_{Q^\varepsilon_m+q})-\varepsilon^{-1}\mu_m|\leqslant 2C_m(1+\mu_m)^2 \quad\text{for } \varepsilon\leqslant(2C_m(1+\mu_m))^{-1}. \end{aligned}
\end{equation}
\tag{3.33}
Hence (3.1) holds indeed. Theorem 4. For each m\in{\mathbb N} there exist positive quantities \varepsilon_m and c_m such that for \varepsilon\in(0,\varepsilon_m] elements of the sequences (1.12) and (2.50) of eigenvalues of problems (1.5) and (2.45), respectively, are related by
\begin{equation*}
\lambda^\varepsilon_{\mathbf d+m}=\varepsilon^{-1}\mu_m+ {\widetilde{\lambda}}^{\varepsilon}_{\mathbf d+m}
\end{equation*}
\notag
and
\begin{equation}
|{\widetilde{\lambda}}^{\varepsilon}_{\mathbf d+m}|\leqslant c_m.
\end{equation}
\tag{3.34}
In addition, for obvious reasons \lambda^\varepsilon_1,\dots,\lambda^\varepsilon_{\mathbf d}=0, where \mathbf d is the dimension of the space (1.18) of vector polynomials. Proof. It remains to verify that Q^\varepsilon_m=m + \mathbf d for small \varepsilon. In Remark 4 we mentioned that Q^\varepsilon_m\geqslant m + \mathbf d. Suppose that Q^{\varepsilon_{mk}}_m>m + \mathbf d for some positive infinitesimal sequence \{\varepsilon_{mk}\}_{k\in{\mathbb N}}. Then on a half-open interval (0,\varepsilon^{-1}(\mu_m+\delta)], where \delta>0 and \mu_m+\delta<\mu_{m+\varkappa_m}, there is an eigenvalue \lambda^{\varepsilon_{mk}}_{N_{mk}^\varepsilon} such that the corresponding vector-valued eigenfunction u^{\varepsilon_{mk}}_{N_{mk}^\varepsilon} satisfies the equalities ({\mathcal Q}u^{\varepsilon_{mk}}_{N_{mk}^\varepsilon}, u^{\varepsilon_{mk}}_\ell)_{\omega^\varepsilon}=0 for \ell taking m+\varkappa_m-1 different values. By Theorem 3 the limits (3.2) and (3.9), (3.10) define an eigenpair \{\mu^\bullet,{\overrightarrow{w}}^\bullet\} of problem (2.45), and strong convergence in L^2(\omega^\varepsilon)^K leads to a paradoxical conclusion: the eigenvector {\overrightarrow{w}}^{\bullet} normalized in accordance with (2.51) and corresponding to the eigenvalue \mu^\bullet<\mu_{m+\varkappa_m} is orthogonal (in the same sense) to m+\varkappa_m-1 eigenvectors of problem (2.45).
The proof is complete. 3.5. The asymptotic behaviour of vector-valued eigenfunctionsFirst we consider a simple eigenvalue \mu_m of problem (2.45), so that \varkappa_m=1 in (3.17). Theorem 5. Let \mu_m be a simple eigenvalue of problem (2.45), and let \overrightarrow{w}_{(m)} be the corresponding eigenvector satisfying the normalization condition (2.51). Then the eigenvalue \lambda^\varepsilon_{\mathbf d+m} of problem (1.5) is simple, and the corresponding vector eigenfunction u^\varepsilon_{(\mathbf d+m)} subject to (1.13) satisfies the inequality
\begin{equation}
\|u^\varepsilon_{(\mathbf d+m)}-\varepsilon^{-1/2}(1+\mu_m)^{-1/2}\mathbf v^\varepsilon_{(m)}; {\mathcal H}^\varepsilon\|\leqslant c'_m(\beta)\varepsilon^{\beta+1/2} \quad\textit{for } \varepsilon\in(0,\varepsilon'_m(\beta)].
\end{equation}
\tag{3.35}
Here c'_m(\beta) and \varepsilon'_m(\beta) are positive quantities, \beta is a fixed exponent in the interval (2.8), and the asymptotic construct \mathbf v^\varepsilon_{(m)} is defined by (3.20) for the eigenpair \{\mu_m,\overrightarrow{w}_{(m)}\} and the column \mathbf a_{(m)}\in{\mathbb R}^{\mathbf d} defined by (2.44). Proof. We use the second part of Lemma 4 for \delta_\ast=\varepsilon\tau selected so that the interval \mathbf k^\varepsilon_m - \varepsilon\tau,\mathbf k^\varepsilon_m + \varepsilon\tau] contains the eigenvalue \kappa^\varepsilon_m of the operator {\mathcal K}^\varepsilon, but does not contain \kappa^\varepsilon_{m\pm1}: this is possible by Theorem 4 if \varepsilon>0 is sufficiently small. As a result, \mathbf N^\varepsilon=m and \mathbf X^\varepsilon=1, and the first inequality in (3.16) takes the form
\begin{equation*}
\|\mathbf u_{(m)}^\varepsilon-\mathbf c^\varepsilon_mu^\varepsilon_{(\mathbf d+m)}; {\mathcal H}^\varepsilon\| \leqslant C_m\tau^{-1}\varepsilon,
\end{equation*}
\notag
where |\mathbf c^\varepsilon_m|=1, that is, we can assume that \mathbf c^\varepsilon_m=1. Recalling (3.19) and the relation
\begin{equation*}
\|\mathbf v^\varepsilon_{(m)};{\mathcal H}^\varepsilon\|^2 =\varepsilon(1+\mu_m)+O(\varepsilon^{\beta+3/2})
\end{equation*}
\notag
ensured by Lemma 5, we arrive at (3.35).
The proof is complete. Remark 5. By the definition (1.7) of the inner product in {\mathcal H}^\varepsilon the corresponding norms of the terms \mathbf p(x)\mathbf a_{(m)} and \chi_j(x)w^j_{(m)}(\eta^j) in the asymptotic construct \mathbf v^\varepsilon_{(m)}(x) have the order \sqrt{\varepsilon}, and the norm of the term \varepsilon{\widehat{v}}_{(m)}(x) has the order \varepsilon. So we can remove the last term \varepsilon{\widehat{v}}_{(m)} from (3.19), replacing the bound in (3.35) by C'_m\sqrt{\varepsilon}. For an eigenvalue \mu_m of multiplicity \varkappa_m the plan of the proof remains the same, but the resulting asymptotic formula is not so explicit because a basis of the corresponding root subspace of problem (2.45) is not uniquely defined (cf. the end of § 4.2). We present the asymptotic formulae for the set of vector eigenfunction of problem (1.5). Theorem 6. Let \mu_m be an eigenvalue of problem (2.45) such that (3.17) holds, and assume that eigenvectors \overrightarrow{w}_{(m)},\dots,\overrightarrow{w}_{(m+\varkappa_m-1)} satisfy conditions (2.51) of orthogonality and normalization. Then there exist coefficients \mathbf a^{\varepsilon\ell}_q, which form an orthogonal \varkappa_m\times\varkappa_m matrix \mathbf a^\varepsilon, such that the vector-valued eigenfunctions u^\varepsilon_{\mathbf d+m},\dots, u^\varepsilon_{\mathbf d+m +\varkappa_m-1} of problem (1.5) subject to condition (1.13) satisfy the inequalities
\begin{equation}
\begin{aligned} \, \notag &\biggl\|u^\varepsilon_{(\mathbf d+q)}-\varepsilon^{-1/2}(1+\mu_m)^{-1/2} \sum_{\ell=m}^{m+\varkappa_m-1}\mathbf a^{\varepsilon\ell}_q \mathbf v^\varepsilon_{(\ell)}; {\mathcal H}^\varepsilon\biggr\| \\ &\qquad \leqslant C'_m(\beta)\varepsilon^{\beta+1/2} \quad\textit{for } \varepsilon\in(0,\varepsilon'_m(\beta)]. \end{aligned}
\end{equation}
\tag{3.36}
Here q=m,\dots,m+\varkappa_m-1, c'_m(\beta) and \varepsilon'_m(\beta) are positive quantities, \beta\in(-1/2,1/2), and the vector functions \mathbf v^\varepsilon_{(m)},\dots, \mathbf v^\varepsilon_{(m+\varkappa_m-1)} are defined in (3.20). Proof. We verify (3.36) as follows. Lemma 4 produces columns \mathbf c^{\varepsilon\ell}=(\mathbf c^{\varepsilon\ell}_m, \dots, \mathbf c^{\varepsilon\ell}_{m+\varkappa_m-1})^\top ensuring relations (3.16) for the vector functions \mathbf u^\varepsilon_{(\ell)}, \ell=m,\dots, {m+\varkappa_m-1}, in (3.19); the calculations (3.31) show that the \varkappa_m\times\varkappa_m matrix \mathbf c^\varepsilon=(\mathbf c^{\varepsilon m}, \dots,\mathbf c^{\varepsilon m+\varkappa_m-1}) is ‘almost orthogonal’; finally, Lemma 7.1.7 in [9] enables us to transform the inverse matrix (\mathbf c^\varepsilon)^{-1} into the orthogonal matrix \mathbf a^\varepsilon of the coefficients of linear combinations in the final formula (3.36).
The proof is complete. § 4. Versions, generalizations and open questions4.1. Deliberately neglected options1^\circ. The boundary. Since we were dealing with the generalized statement (2.25) of problem (2.18), (2.19) in \Omega, the surface \partial \Omega can be piecewise smooth or even Lipschitz (cf. Figures 2, b, and 3), and only in neighbourhoods of the points P^1,\dots,P^J do we need it to be smooth, in order that the limiting system (2.4) be considered in the half-space {\mathbb R}^3_-. On the other hand the P^j can be conical points or can lie on a smooth edge: then the limiting problems (2.4), (2.5) are stated in the cone {\mathbb K}^j or in the dihedral angle {\mathbb D}^j (which is a special case of a cone with nonsmooth directrix) and the set \varpi_j lies on \partial{\mathbb K}^j or \partial {\mathbb D}^j. As before, the results in [6], §§ 2.1 and § 5.3, ensure representation (2.9), but the rate of decay of the remainder {\widetilde{w}}^{j} in the representation depends on the opening of the cone (dihedral angle), which, of course, has implications for the estimates of asymptotic remainder terms in Theorems 4, 5 and 6. We stress that the vast majority of authors assume that the boundary is flat in a neighbourhood of small singular perturbations of boundary conditions: this simplifies the scheme of the justification of asymptotic formulae and the asymptotic analysis itself (see, for instance. § 4.2). 2^\circ. The coefficients of differential operators. We can deal with the matrix-valued function x\mapsto{\mathcal A}(x) if its properties are stable throughout \overline \Omega. Of course, the entries must have suitable smoothness: the class C^2 is sufficient for constructing the leading term of the asymptotic expression. Moreover, in the definition of the operator
\begin{equation}
{\mathcal L}(x,\nabla_x)={\overline{{\mathcal D}(-\nabla_x)}}^\top {\mathcal A}(x){\mathcal D}(\nabla_x)
\end{equation}
\tag{4.1}
and all other differential operators, we can take for {\mathcal A}(x) any Hermitian positive definite matrix and can assume that the differential expression {\mathcal D}(\nabla_x) has complex coefficients, denoting complex conjugation in (4.1) by overlines. On the other hand, such simple generalizations are of no use for the applications mentioned in § 1.3, so they are superfluous. 3^\circ. The dimension of the space. For d>3 all our arguments, calculations and results hold in general for a domain \Omega\subset{\mathbb R}^d with (d-1)-dimensional boundary (for submanifolds of dimension d'\leqslant d-2 there can be significant differences: see the research [25]). We limit ourselves to d=3 for shorter computations and handier formulae, as well as in order to align with the applied problems in § 1.3 as before. The asymptotic constructions in two-dimensional problems are considerably different from ours here because of logarithms in the (matrix-valued) Green’s function for the Neumann problem. For the scalar problem of surface waves (cf. §§ 1.3, 1^\circ and 2.5, 1^\circ) the requisite asymptotic analysis was presented in [26]. Note that for a plane domain the phenomenon of interaction of small regions of singular perturbations disappears in leading order, but it can be seen in the asymptotic correction of order |{\ln \varepsilon}|^{-1}. 4^\circ. The orders of differential operators. Because of the new expression for Green’s matrix, the algorithm for constructing asymptotic formulae changes significantly for higher-order differential operators, and its description for a general formally selfadjoint Douglis-Nirenberg elliptic system of differential equations is excessively bulky. The simplest example of the biharmonic equation is treated in § 4.6, where we also discuss the new elements required; in particular, constants and the Green’s function switch roles unexpectedly in asymptotic procedures. 4.2. Full asymptotic expansionsFor a simple eigenvalue \mu_m (that is, when \varkappa_m=1 in (3.17)) we can easily construct infinite asymptotic series for the eigenpair \{\lambda^\varepsilon_{\mathbf d+m},u^\varepsilon_{(\mathbf d+m)}\} of problem (1.5): the requisite iterative procedures were already prepared in [38], Chs. 4, 9 and 10. We show the calculation of the main correction terms in the ansätze
\begin{equation}
\lambda^\varepsilon_{\mathbf d+m}=\varepsilon^{-1}\mu_m+\mu'_m+\dotsb
\end{equation}
\tag{4.2}
and
\begin{equation}
\begin{split} u^\varepsilon_{(\mathbf d+m)}(x) &=\sum_{j=1}^J\chi_j(x)\bigl(w^j_{(m)}(\eta^j)+\varepsilon w^{j\prime}_{(m)}(\eta^j)\bigr) +\mathbf p(x)(\mathbf a_{(m)}+\varepsilon \mathbf a_{(m)}') \\ &\qquad +\varepsilon{\widehat{v}}_{(m)}(x)+\varepsilon^2{\widehat{v}}_{(m)}'(x)+\cdots, \end{split}
\end{equation}
\tag{4.3}
where \mu'_m\in{\mathbb R}, {\widehat{a}}_{(m)}\in{\mathbb R}^{\mathbf d} and {\widehat{v}}'_{(m)}\in H^2(\Omega)^K are to be determined, and \mathbf a_{(m)} and \mathbf a_{(m)}' are connected with \overrightarrow{w}_{(m)} and \overrightarrow{w}_{(m)}' by formula (2.44), that is, they have the form
\begin{equation}
\mathbf a_{(q)}=-\mathbf M^{-1}\sum_{j=1}^J|\varpi_j|\mathbf p(P^j){\mathcal Q}(P^j) {\overline{w}}^{j}_{(q)},\quad \mathbf a_{(q)}'=-\mathbf M^{-1}\sum_{j=1}^J|\varpi_j|\mathbf p(P^j){\mathcal Q}(P^j) {\overline{w}}^{j\prime}_{(q)}.
\end{equation}
\tag{4.4}
Finally, {\widehat{v}}_{(m)} is a solution of problem (2.27) for f=0 and g=0 satisfying the orthogonality conditions (2.26). For simpler calculations assume that the pieces \partial\Omega\cap{\mathcal V}^j of the boundary of \Omega are flat; in particular, the coordinates (3.6) and (2.1) are the same. Moreover, we assume that the restriction of \mathcal Q to the piece \partial\Omega\cap{\mathcal V}^j is a constant orthogonal matrix {\mathcal Q}(P^j). For changes required in case we abandon these simplifications, the reader can address [38], Ch. 4, for instance. Substituting the ansätze (4.2) and (4.3) into (1.2)–(1.4) we collect the terms expressed in terms of the stretched coordinates (2.1), and the traces on \varpi^\varepsilon_j of the terms of smooth type. Since \{\mu_m,\overrightarrow{w}_{(m)}\} is an eigenpair of (2.45) and the column \mathbf a_{(m)} is obtained by (2.44) from the vector ({\overline{w}}_{(m)}^{1},\dots,{\overline{w}}_{(m)}^{J}) \in{\mathbb R}^{K\times J}, the terms of order \varepsilon^{-1} in the boundary condition (1.4) annihilate, while the terms of order \varepsilon^0=1 produce the relation3
\begin{equation*}
\begin{aligned} \, &{\mathcal N}^j(\nabla_{\xi^j})w^{j\prime}_{(m)}(\xi^{j\prime},0)- \mu_m \mathcal Q (P^j)\bigl(w^{j\prime}_{(m)}(\xi^{j\prime},0) +\mathbf p(P^j)\mathbf a'_{(m)}\bigr) \\ &\qquad =\mu_m \mathcal Q(P^j)\bigl(\widehat{v}_{(m)}(P^j) +\mathbf p(P^j)\widehat{\mathbf a}_{(m)}\bigr) \\ &\qquad\qquad +\mu'_m \mathcal Q(P^j)\bigl(w^j_{(m)}(\xi^{j\prime},0) +\mathbf p(P^j)\mathbf a_{(m)}\bigr), \qquad \xi^{j\prime}\in\varpi_j. \end{aligned}
\end{equation*}
\notag
In combination with the homogeneous Neumann condition (2.5) on {\mathbb R}^2\setminus \overline{\varpi_j} and the system of differential equations (2.4) in {\mathbb R}^3_-, this ensures the integral identity
\begin{equation}
\begin{aligned} \, \notag &\sum_{j=1}^J\bigl( \mathcal E^j(w^{j\prime}_{(m)},\psi^j) -\mu_m(w^{j\prime}_{(m)}+\mathbf p(P^j)\mathbf a'_{(m)}, \mathcal Q(P^j)\psi^j)_{\varpi_j}\bigr) \\ \notag &\qquad =\mu_m\sum_{j=1}^J\bigl(\widehat{v} _{(m)}(P^j) +\mathbf p(P^j)\widehat{\mathbf a}_{(m)},{\mathcal Q}(P^j)\psi^j\bigr)_{\varpi_j} \\ &\qquad\qquad +\mu_m' \sum_{j=1}^J(w^j_{(m)}+\mathbf p(P^j)\mathbf a_{(m)},{\mathcal Q}(P^j)\psi^j)_{\varpi_j} \quad \forall\, \overrightarrow{\psi}\in V^1_0({\mathbb R}^3_-)^{J\times K}. \end{aligned}
\end{equation}
\tag{4.5}
Since \mu_m is a simple eigenvalue by assumption, there is a single compatibility condition in problem (4.5); in view of the normalization (2.51) it has the following form:
\begin{equation}
\begin{aligned} \, \notag \mu_m' &=\mu_m'\sum_{j=1}^J(w^j_{(m)}+\mathbf p(P^j)\mathbf a_{(m)}, {\mathcal Q}(P^j)w^j_{(m)})_{\varpi_j} \\ \notag &=-\mu_m\sum_{j=1}^J \bigl(\bigl({\widehat{v}}_{(m)}(P^j)+ \mathbf p(P^j)\widehat{\mathbf a}_{(m)},{\mathcal Q}(P^j)w^j_{(m)}\bigr)_{\varpi_j}\bigr) \\ \notag &\qquad +\sum_{j=1}^J\bigl({\mathcal E}^j(w^{j\prime}_{(m)},w^j_{(m)}) -\mu_m\bigl(w^{j\prime}_{(m)}+\mathbf p( P^j)\mathbf a_{(m)}',{\mathcal Q}(P^j)w^j_{(m)} \bigr)_{\varpi_j}\bigr) \\ &=-\mu_m\sum_{j=1}^J |\varpi_j|({\overline{w}}^{j}_{(m)})^\top {\mathcal Q}(P^j)\bigl({\widehat{v}}_{(m)}(P^j)+\mathbf p(P^j)\widehat{\mathbf a}_{(m)}\bigr). \end{aligned}
\end{equation}
\tag{4.6}
The sum over j=1,\dots,J containing the bilinear forms {\mathcal E}^1,\dots,{\mathcal E}^J has disappeared because it involves the eigenpair \{\mu_m,\overrightarrow{w}_{(m)}\} of problem (2.45) and relations (4.4) hold. It remains to determine the column \widehat{\mathbf a}_{(m)}\in{\mathbb R}^{\mathbf d}: on the basis of problem (4.5) and its compatibility condition (4.6), from it we can find the matrix function \overrightarrow{w}'_{(m)}\in V^1_0({\mathbb R}^3_-)^{J\times K} and the number \mu_m'\in{\mathbb R} involved in (4.2) and (4.3). Solutions w^{j\prime}_{(m)} have expansions similar to (2.9):
\begin{equation}
w^{j\prime}_{(m)}(\xi^j)=\chi^\infty (\xi^j) \Phi^j(\xi^j)b^{j\prime}_{(m)}+{\widetilde{w}}^{j\prime}_{(m)}(\xi^j).
\end{equation}
\tag{4.7}
We also revise the expansion for vector-valued eigenfunctions:
\begin{equation}
w^j_{(m)}(\xi^j)=\chi^\infty (\xi^j) (\Phi^j(\xi^j)b^j+\Phi^{j\prime}_{(m)}(\xi^j))+{\widetilde{w}}^{j}_{(m)}(\xi^j).
\end{equation}
\tag{4.8}
The remainders satisfy {\widetilde{w}}^{j}_{(m)}(\xi^j)=O(\rho_j^{-3}) and {\widetilde{w}}^{j\prime}_{(m)}(\xi^j)=O(\rho_j^{-2}); moreover, for them and their derivatives outside the ball {\mathbb B}_R we can produce estimates of type (2.17), with suitable exponents (cf. Remark 3). Furthermore, the term
\begin{equation}
\Phi^{j\prime}_{(m)}(\xi^j)=\rho_j^{-2}\Phi^{j\prime}_{(m)}(\rho_j^{-1}\xi^j)
\end{equation}
\tag{4.9}
involves a linear combination of the derivatives with respect to \xi^j_1 and \xi^j_2 of columns of \Phi^j (see [6], § 2.6). Finally, the solution \overrightarrow{w}'_{(m)} is defined up to a term c\overrightarrow{w}_{(m)}, so that we can subject the columns b^{1\prime}_{(m)},\dots,b^{J\prime}_{(m)}\in{\mathbb R}^K in (4.7) to the othogonality condition and by Proposition 2 these columns themselves can be calculated by the formula
\begin{equation}
\begin{aligned} \, \notag b^{j\prime}_{(m)} &=\mu_m|\varpi_j|{\mathcal Q}(P^j) \bigl({\widehat{v}}_{(m)}(P^j)+\mathbf p(P^j)\widehat{\mathbf a}_{(m)}\bigr) \\ &\qquad+ \mu_m|\varpi_j|{\mathcal Q}(P^j)({\overline{w}}^{j\prime}_{(m)} +\mathbf p(P^j)\mathbf a'_{(m)}) +\mu'_m|\varpi_j|{\mathcal Q}(P^j)\bigl({\overline{w}}^{j}_{(m)} +\mathbf p(P^j)\mathbf a_{(m)}\bigr). \end{aligned}
\end{equation}
\tag{4.10}
Now, on the basis of the expansions (4.8) and (4.7) and relations (2.10) and (4.9) we put together a problem for the second correction of regular type. Substituting (4.3) into (1.2) we observe that the sum of the terms of order \varepsilon vanishes by the constructions in § 2.3, and terms of order \varepsilon^2 give rise to problem (2.18), (2.19) for the correction {\widehat{v}}_{(m)}', where the right-hand sides are
\begin{equation*}
f^{j\prime}_{(m)}(x)=-\sum_{j=1}^J \bigl(F^j(x)b^{j\prime}_{(m)}+ F^{j\prime}(x)b^j_{(m)}\bigr)
\end{equation*}
\notag
and
\begin{equation*}
g^{j\prime}_{(m)}(x)=-\sum_{j=1}^J \bigl(G^j(x)b^{j\prime}_{(m)}+ G^{j\prime}(x)b^j_{(m)}\bigr),
\end{equation*}
\notag
while F^j and G^j are defined by (2.19) and, in addition,
\begin{equation*}
F^{j\prime}_{(m)}(x)=\bigl[{\mathcal L}(\nabla_x),\chi_j(x)\bigr]\Phi^{j\prime}_{(m)}(x^j)\quad\text{and} \quad G^{j\prime}_{(m)}(x)=\bigl[{\mathcal N}(x,\nabla_x),\chi_j(x)\bigr]\Phi^{j\prime}_{(m)}(x^j).
\end{equation*}
\notag
In accordance with (2.31), the compatibility conditions (2.28) in this problem have the form of a system of linear constraints
\begin{equation}
\sum_{j=1}^J \mathbf p(P^j)^\top b^{j\prime}_{(m)} +\sum_{j=1}^J \mathbf F^j_{(m)}b^j_{(m)}=0\in{\mathbb R}^{\mathbf d},
\end{equation}
\tag{4.11}
involving the \mathbf d\times K matrices
\begin{equation*}
\mathbf F^j_{(m)}=\int_{\Omega}\mathbf p(x)^\top F^{j\prime}_{(m)}(x)\,dx+ \int_{\partial\Omega}\mathbf p(x)^\top G^{j\prime}_{(m)}(x)\,ds_x.
\end{equation*}
\notag
We substitute the expression for b^{j\prime} into (4.11) and note that by (4.4) the last two terms on the right-hand side of (4.10) annihilate after taking the sum and, by the definitions (2.43), (2.42) of \mathbf M, we can write (4.10) as follows:
\begin{equation*}
\mu_m\mathbf M\widehat{\mathbf a}_{(m)}=-\sum_{j=1}^J\mathbf F^j_{(m)}b^j_{(m)}+ \mu_m\,\sum_{j=1}^J|\varpi_j|\,\mathbf p(P^j)^\top {\mathcal Q}(P^j) {\widehat{v}}_{(m)}(P^j).
\end{equation*}
\notag
Thus, we have found \widehat{\mathbf a}_{(m)}\in {\mathbb R}^{\mathbf d} because \mu_m>0 and \mathbf M is invertible by Lemma 2. Note that the solution {\widehat{v}}_{(m)} of the problem in \Omega is well defined up to a term p\in{\mathcal P}; it is unique if we impose some conditions of orthogonality, for instance, (2.26). At the same time the choice of this term does not affect on the quantity (4.6) because the ansatz (4.3) contains the term \varepsilon \mathbf p(x){\widehat{a}}_{(m)}, which absorbs polynomials {p\in\mathcal P}. This completes the construction of the correction terms in (4.2) and (4.3) in the case of a simple eigenvalue \mu_m. In the case (3.17), when the limiting problem (2.45) has a multiple eigenvalue, we cannot uniquely fix the matrix-valued functions {\overrightarrow{w}}_{(m)},\dots,{\overrightarrow{w}}_{(m+\varkappa_m-1)} in the ansatz (4.3) (or (2.34)) at the first step of the iterative procedure. To specify them we must construct lower-order terms of asymptotic expansions. The corresponding procedure is standard (cf. [38], Chs. 9 and 10), namely, at the second step of the iterative procedure we obtain an algebraic problem and using its eigenvalues we can find the corrections in the expansions (4.2) for the eigenvalues \lambda^\varepsilon_{\mathbf d+m},\dots,\lambda^\varepsilon_{\mathbf d+m+\varkappa_m-1} and find the initial terms in the expansions (4.3) for the vector-valued eigenfunctions u^\varepsilon_{(\mathbf d+m)},\dots, u^\varepsilon_{(\mathbf d+m+\varkappa_m-1)}. If it turns out that the new eigenvalues (4.12) are simple, then we meet with no ambiguities at the subsequent steps, and there are no obstacles to the implementation of the iterative procedure. On the other hand, if some eigenvalues in (4.12) are multiple, then we still have some freedom in the choice of the leading terms of the expansions for vector eigenfunctions, so we must construct the next terms of asymptotic expansions and attempt to ‘split’ the eigenvalues asymptotically. On the other hand there are still no algorithms allowing one to find out whether or not multiple eigenvalues split after finitely many steps. One example of a multiple eigenvalue \lambda^\varepsilon_m is provided by the Steklov problem (1.30)–(1.32) in the domain \Omega shown in Figure 2, b, which, as well as the family \omega^\varepsilon_1,\dots,\omega^\varepsilon_J, has rotational symmetry.We do not present the calculations involved or the results available. 4.3. Dirichlet boundary conditionsOn the surface \partial\Omega we fix a domain \gamma of positive area whose closure lies away from the points P^1,\dots,P^J. We replace the Neumann conditions (1.3) by the mixed boundary conditions
\begin{equation}
{\mathcal N}(x,\nabla_x)u^\varepsilon(x)=0, \qquad x\in\partial\Omega\setminus({\overline{\omega^\varepsilon\cup\gamma}}).
\end{equation}
\tag{4.14}
In this case the limiting problem in \Omega is the system of equations (2.18) with boundary conditions
\begin{equation}
\begin{gathered} \, v(x)=0, \qquad x\in\gamma, \\ {\mathcal N}(x,\nabla_x)v(x)=0, \qquad x\in\partial\Omega\setminus {\overline{\gamma}}. \end{gathered}
\end{equation}
\tag{4.15}
Problem (2.18), (4.15) is uniquely solvable, so the phenomenon of far interaction of the spectral boundary conditions (1.4) disappears in the limit: we cannot add a polynomial p\in{\mathbb P} to the ansatz for a vector-valued eigenfunction, and the spectral problems in {\mathbb R}^3_- assume the form of systems (2.4) with boundary conditions
\begin{equation}
\begin{gathered} \, {\mathcal N}^j(\nabla_{\xi^j})w^j(\xi^{j\prime},0)=\mu^j {\mathcal Q}(P^j)w^j(\xi^{j\prime},0), \qquad \xi^{j\prime}\in\varpi_j, \\ {\mathcal N}^j(\nabla_{\xi^j})w^j(\xi^{j\prime},0)=0, \qquad \xi^{j\prime}\in {\mathbb R}^2\setminus\varpi_j, \end{gathered}
\end{equation}
\tag{4.16}
and become independent: the corresponding integral identities
\begin{equation}
{\mathcal E}^j(w^j,\psi^j)=\mu^j( {\mathcal Q}(P^j)w^j,\psi^j)_{\varpi_j}, \qquad \psi^j\in V^1_0({\mathbb R}^3_-)^K,
\end{equation}
\tag{4.17}
do not contain the mean values {\overline{w}}^{1},\dots,{\overline{w}}^{J} of vector eigenfunctions (cf. (2.47) and (2.45)). Each problem (4.16) has a positive discrete spectrum
\begin{equation}
0<\mu^j_1\leqslant\mu^j_2\leqslant\dots\leqslant\mu^j_m\leqslant\dots \to+\infty.
\end{equation}
\tag{4.18}
The proof of the next theorem on asymptotic behaviour follows the scheme presented in § 3, but it is much simpler. Theorem 7. The eigenvalues (1.12) of problem (1.2), (4.13), (4.14), (1.4) have the representations
\begin{equation*}
\lambda^\varepsilon_m=\varepsilon^{-1}\mu_m+{\widetilde{\lambda}}^{\varepsilon}_m , \qquad \varepsilon\in(0,\varepsilon_m],
\end{equation*}
\notag
where the remainder terms {\widetilde{\lambda}}^{\varepsilon}_m have the bounds (3.34), c_m and \varepsilon_m are positive quantities, and the leading asymptotic terms \mu_m are the eigenvalues from the monotone sequence (2.50) obtained as the union of the spectra (4.18) of the limiting problems (2.4), (4.16) for j=1,\dots,J. Other asymptotic constructions arise when Dirichlet conditions are set on one or several small subsets, of diameter O(\varepsilon), of \partial\Omega. We only discuss the case of one set \gamma=\omega^\varepsilon_0, which is defined for a point P^0\not\in\{P^1,\dots,P^J\} and a domain \varpi_0\subset{\mathbb R}^2, bounded by a smooth contour \partial\varpi_0, in accordance with (1.1). The spectrum (1.12) of the problem formed by system (1.2) and boundary conditions (1.4), (4.13), (4.14) is positive even in the case of a small Dirichlet zone Below we need bounded solutions W^k\not\in V^1_0({\mathbb R}^3_-)^K, k=1,\dots,K, of the homogeneous (h^k=0) problem
\begin{equation}
\begin{gathered} \, {\mathcal L}^0(\nabla_x ) W^k(\xi^0)=0, \qquad \xi^0\in{\mathbb R}^3_-, \\ {\mathcal N}^0(\nabla_{\xi^0})W^k(\xi^{0\prime},0)=0, \qquad \xi^{0\prime}\in {\mathbb R}^2 \setminus{\overline{\omega_0}}, \\ W^k(\xi^{0\prime},0)=h^k(\xi^{0\prime}), \qquad \xi^{0\prime}\in\omega_0, \end{gathered}
\end{equation}
\tag{4.20}
which have a representation
\begin{equation}
W^k(\xi^0)=\mathbf e_{(k)}+{\widehat{W}}^{k}(\xi^0) =\mathbf e_{(k)}-\chi^\infty(\xi^0)\Phi^0(\xi^0)\mathbf T^k+{\widetilde{W}}^{k}(\xi^0).
\end{equation}
\tag{4.21}
Here we use the notation analogous to § 2.1. In addition, the term {\widetilde{W}}^{k} belongs to V^1_1({\mathbb R}^3_-)^K, \mathbf T^k\in{\mathbb R}^K is some column, \mathbf e_{(k)}=(\delta_{1,k},\dots, \delta_{K,k})^\top is a canonical unit vector in the Euclidean space {\mathbb R}^K, and {\widehat{W}}^{k} \in V^1_0({\mathbb R}^3_-)^K is the solution of problem (4.20) with right-hand side Proposition 4. The matrix \mathbf T=(\mathbf T^1,\dots,\mathbf T^K)=(\mathbf T^k_\ell)_{k,\ell=1}^K formed by the columns of coefficients of (4.21) is symmetric and positive definite. Proof. Into Green’s formula for a half-ball of large radius we substitute the vector functions {\widehat{W}}^{k} and {\widehat{W}}^{\ell}. Taking account of the Dirichlet data for {\widehat{W}}^{\ell}, the expansion (4.21) for {\widehat{W}}^{k} and (2.11), similarly to (2.16) we obtain
\begin{equation*}
\begin{aligned} \, {\mathcal E}^0({\widehat{W}}^{k},{\widehat{W}}^{\ell}) &= ({\mathcal N}^0(\nabla_{\xi^0}){\widehat{W}}^{k},{\widehat{W}}^{\ell})_{\varpi_0}= -({\mathcal N}^0(\nabla_{\xi^0}){\widehat{W}}^{k},\mathbf e_{(\ell)})_{\varpi_0} \\ &=\lim_{R\to+\infty}\int_{{\mathbb S}^-_R}\mathbf e_{(\ell)}^\top {\mathcal N}^0_{\cup}(\xi^0,\nabla_{\xi^0}){\widehat{W}}^{k}(\xi^0)\,ds_{\xi^0} \\ &=-\int_{{\mathbb S}^-_1}\mathbf e_{(\ell)}^\top {\mathcal N}^0_{\cup}(\xi^0,\nabla_{\xi^0})\Phi^0(\xi^0)\,ds_{\xi^0}\mathbf T^k =\mathbf T_\ell^k. \end{aligned}
\end{equation*}
\notag
Thus, \mathbf T is the Gram matrix constructed from linearly independent vector functions using the inner product {\mathcal E}^0(\,\cdot\,{,}\,\cdot\,) in V^1_0({\mathbb R}^3_-)^K.
The proof is complete. The algorithm in § 2.3 for constructing an asymptotic formula must be modified only slightly and, in particular, the asymptotic ansätze (1.14) and (2.34) for eigenpairs of problem (1.2), (4.13), (4.14), (1.4), (4.13) remain the same in the case (4.19), but the limiting spectral problem (2.45) is significantly reformatted On the right-hand sides of problem (2.30) for the term {\widehat{v}} of regular type in (2.34) we now have new singular components
\begin{equation*}
f(x)=F^0(x)\mathbf Tp(P^0)\quad\text{and} \quad g(x)=G^0(x)\mathbf Tp(P^0),
\end{equation*}
\notag
which involve the unknown vector-valued polynomial p\in{\mathcal P}. These components are produced by the boundary layer
\begin{equation*}
{\widehat{W}}(\xi^0)p(P^0)=({\widehat{W}}^1(\xi^0),\dots,{\widetilde{W}}^K(\xi^0))p(P^0)
\end{equation*}
\notag
localized in a neighbourhood of P^0 and by the expansion (4.21) for entries of the K\times K matrix {\widehat{W}}, formed by solutions of problem (4.20) with right-hand sides (4.22) which decay at infinity. Furthermore, F^0 and G^0 are given by (2.29) for j=0. Taking representation (2.40) for polynomials p\in{\mathcal P} into account, the compatibility conditions (2.28) in the resulting problem transform into the following system of equations for the column \mathbf a\in{\mathbb R}^{\mathbf d}:
\begin{equation*}
\mathbf p(P^0)^\top\mathbf T\mathbf p(P^0)\mathbf a=\mu\sum_{j=1}^J( \mathbf M^j\mathbf a+\mathbf p(P^j)^\top{\mathcal Q}(P^j){\overline{w}}^{j}|\varpi_j|).
\end{equation*}
\notag
In combination with integral identities (2.34), this system makes up the limiting spectral problem, namely,
\begin{equation}
\begin{aligned} \, \notag &\sum_{j=1}^J{\mathcal E}^j(w^j,\psi^j)+\boldsymbol\alpha^\top\mathbf p(P^0)^\top \mathbf T\mathbf p(P^0)\mathbf a \\ \notag &\qquad =\mu\biggl(\sum_{j=1}^J({\mathcal Q}(P^j)w^j,\psi^j)_{\varpi_j} \sum_{j=1}^J|\varpi_j|({\overline{\psi}}^{j})^\top{\mathcal Q}(P^j) \mathbf p(P^j)\mathbf a \\ \notag &\qquad\qquad +\sum_{j=1}^J \boldsymbol\alpha^\top \mathbf p(P^j)^\top {\mathcal Q}(P^j){\overline{\psi}}^{j}|\varpi_j| + \boldsymbol\alpha^\top\mathbf M\mathbf a\biggr) \\ &\qquad \forall\,(\psi^1,\dots,\psi^J)\in V^1_0({\mathbb R}^3_-)^{K\times J}, \qquad \boldsymbol\alpha\in{\mathbb R}^{\mathbf d}. \end{aligned}
\end{equation}
\tag{4.23}
Since the linear space \mathcal P contains all constant columns, the \mathbf d\times\mathbf d matrix {\mathcal T}^0=\mathbf p(P^0)^\top \mathbf T\mathbf p(P^0) has rank K \leqslant\mathbf d by Proposition 4. Hence problem (4.23) has the eigenvalue \mu=0 with multiplicity \mathbf d-K, which is associated with eigenvectors with nontrivial algebraic components \mathbf a\in\ker {\mathcal T}^0 and zero matrix-valued functions \overrightarrow{w}. Calculations similar to the ones in § 2.4, but bulkier because of the presence of algebraic components, show that the rest of the spectrum of problem (4.23) forms an unbounded positive monotonic sequence. As a result, for the eigenvalues of problem (1.2), (4.13), (4.14), (1.4) in the case (4.19) we can establish a result similar to Theorems 7 and 4, which we do not state here because we still do not know the positive leading terms of the first \mathbf d-K elements of the sequence (1.12). Of course, we could set Dirichlet conditions on several small sets \omega^\varepsilon_{-J^\circ},\dots,\omega^\varepsilon_0, imposing conditions similar to (1.19) on their centres P^{-J^\circ},\dots,P^0, which would ensure that the \mathbf d\times\mathbf d matrix {\mathcal T} is positive definite and the entire spectrum of the limiting problem (4.23) is positive. 4.4. A spectral parameter in the system of differential equationsWe introduce a spectral parameter into the system:
\begin{equation}
{\mathcal L}(\nabla_x)u^\varepsilon(x)=\lambda^\varepsilon {\mathcal B}u^\varepsilon(x), \qquad x\in\Omega.
\end{equation}
\tag{4.24}
Here {\mathcal B} is a symmetric positive definite K\times K matrix. Problem (4.24), (1.3), (1.4) in its variational form
\begin{equation}
{\mathcal E}(u^\varepsilon,\psi^\varepsilon;\Omega)=\lambda^\varepsilon \bigl(({\mathcal B} u^\varepsilon,\psi^\varepsilon)_\Omega+({\mathcal Q}u^\varepsilon, \psi^\varepsilon)_{\omega^\varepsilon}\bigr) \quad \forall\,\psi^\varepsilon\in H^1(\Omega)^K
\end{equation}
\tag{4.25}
also has a discrete spectrum, a sequence of the form (1.12). The eigenvalues of the limiting problem in the bounded domain
\begin{equation}
\begin{gathered} \, {\mathcal L}(\nabla_x)u^0(x)=\lambda^0{\mathcal B}u^0(x), \qquad x\in\Omega, \\ {\mathcal N}(x,\nabla_x)u^0(x)=0, \qquad x\in\partial\Omega, \end{gathered}
\end{equation}
\tag{4.26}
also form an unbounded monotone sequence
\begin{equation*}
0=\lambda^0_1=\dots =\lambda^0_\mathbf d<\lambda^0_{\mathbf d+1} \leqslant \lambda^0_{\mathbf d+2}\leqslant\dots\leqslant\lambda^0_m\leqslant\dots \to+\infty.
\end{equation*}
\notag
In the sequence (1.12) the first \mathbf d=\dim{\mathcal P} elements are equal to zero again. Using the asymptotic methods developed in [38], Chs. 4 and 9, 10, we can verify that the other eigenvalues of problems (4.24), (1.3), (1.4) and (4.26) are related by the inequalities
\begin{equation}
|\lambda^\varepsilon_m-\lambda^0_m| \leqslant c_m\varepsilon^{3/2} \quad\text{for } \varepsilon\in(0,\varepsilon_m],
\end{equation}
\tag{4.27}
where m>\mathbf d and \varepsilon_m and c_m are some positive quantities. It can appear at the first glance that (4.27) describes the asymptotic behaviour of the full spectrum of the singularly perturbed problem. Let us show that this is a false impression and the spectra (4.18) of the limiting problems (4.17) still affect the asymptotic structure of the spectrum (1.12). We assign new meaning to the notation in § 1.1: in the space {\mathcal H}^\varepsilon= H^1(\Omega)^K we introduce the inner product
\begin{equation}
\langle u^\varepsilon,\psi^\varepsilon\rangle_\varepsilon= {\mathcal E}(u^\varepsilon,\psi^\varepsilon;\Omega)+({\mathcal B}u^\varepsilon,\psi^\varepsilon)_\Omega +({\mathcal Q}u^\varepsilon,\psi^\varepsilon)_{\omega^\varepsilon}
\end{equation}
\tag{4.28}
and the positive selfadjoint operator {\mathcal K}^\varepsilon,
\begin{equation*}
\langle {\mathcal K}^\varepsilon u^\varepsilon,\psi^\varepsilon\rangle_\varepsilon= ({\mathcal B}u^\varepsilon,\psi^\varepsilon)_\Omega+ ({\mathcal Q}u^\varepsilon,\psi^\varepsilon)_{\omega^\varepsilon} \quad \forall\, u^\varepsilon,\psi^\varepsilon\in {\mathcal H}^\varepsilon.
\end{equation*}
\notag
Then integral identity (4.25) takes the form of an abstract equation (1.10), and the spectrum (1.9) of {\mathcal K}^\varepsilon is connected with the spectrum (1.12) of problem (4.24), (1.3), (1.4) by the relation Now we use the first part of Lemma 4. Into (3.14) we substitute the expressions
\begin{equation*}
\mathbf k^\varepsilon=(1+\varepsilon^{-1}\mu^j_p)^{-1}\quad\text{and} \qquad \mathbf u^\varepsilon=\|\mathbf v^\varepsilon;{\mathcal H}^\varepsilon\|^{-1}\,\mathbf{v}^\varepsilon,
\end{equation*}
\notag
where \mathbf v^\varepsilon(x)=\chi_j(x)w^j_{(p)}(\eta^j). In addition, \{\mu^j_p,w^j_{(p)}\}\in{\mathbb R}_+\times V^1_0({\mathbb R}^3_-)^K is an eigenpair of problem (4.17) (or (2.4), (4.16), in differential form), and it is important for what follows that in the corresponding representation (2.9) the coefficient column b^j is zero, that is,
\begin{equation}
|w^j_{(p)}(\eta^j)|\leqslant c^j_p(1+\rho_j)^{-2}, \qquad \rho_j\geqslant R.
\end{equation}
\tag{4.30}
As explained in § 2.5, 1^\circ, where we presented a simple example, in this case the number \mu^j_p and the vector \overrightarrow{w} with single nonzero component w^j_{(p)} form an eigenpair of the combined spectral problem (2.45). Now we deal with the expression
\begin{equation}
\begin{aligned} \, \notag \delta^j_p &=\|{\mathcal K}^\varepsilon\mathbf u^\varepsilon- \mathbf k^\varepsilon\mathbf u^\varepsilon;{\mathcal H}^\varepsilon\| \\ &=\|\mathbf v^\varepsilon;{\mathcal K}^\varepsilon\|^{-1}\mathbf k^\varepsilon \sup \bigl|{\mathcal E}(\mathbf v^\varepsilon,\psi;\Omega)-\varepsilon^{-1}\mu^j_p ({\mathcal B}\mathbf v^\varepsilon,\psi)_\Omega-({\mathcal Q} \mathbf v^\varepsilon,\psi)_{\omega^\varepsilon}\bigr|. \end{aligned}
\end{equation}
\tag{4.31}
Here the supremum is taken over the vector functions \psi\in {\mathcal H}^\varepsilon with unit norm, and as (4.28) and (3.15) show, we have
\begin{equation*}
\|r_j^{-1}\psi; L^2(\Omega)\|\leqslant c_j\|\psi; H^1(\Omega)\|\leqslant C_j\|\psi; {\mathcal H}^\varepsilon\|=C_j.
\end{equation*}
\notag
A bound \delta^j_p\leqslant C^j_p\varepsilon^2 for the quantity (4.31) can be obtained using the following arguments. First of all, the calculations (3.26) and the equality b^j_{(p)}=0 (cf. (4.30)) show that
\begin{equation*}
\bigl|{\mathcal E}(\chi_jw^j_{(p)},\psi;\Omega)-{\mathcal E}(w^j_{(p)},\chi_j\psi;\Omega) \bigr|\leqslant c\varepsilon^2.
\end{equation*}
\notag
Furthermore, routine changes of coordinates and the integral identities (4.17), which match problem (2.4), (4.16), ensure that
\begin{equation*}
\bigl|{\mathcal E}(w^j_{(p)},\chi_j\psi;\Omega)-\varepsilon^{-1}\mu^j_p({\mathcal Q} w^j_{(p)},\chi_j\psi)_{\omega_j^\varepsilon} \bigr|\leqslant c\varepsilon^{3/2}.
\end{equation*}
\notag
Finally, relying on assumption (4.30) we obtain the key estimate as follows:
\begin{equation}
\begin{aligned} \, \notag &|({\mathcal B}\chi_jw^j_{(p)},\psi)_\Omega|\leqslant \bigl|({\mathcal B}\chi_jw^j_{(p)},\psi)_{\Omega\cap{\mathbb B}_{\varepsilon R}(P^j)}\bigr|+ \bigl|({\mathcal B}\chi_jw^j_{(p)},\psi)_{\Omega\setminus{\mathbb B}_{\varepsilon R}(P^j)}\bigr| \\ \notag &\qquad\leqslant c\biggl(\varepsilon^5\|\rho_jw^j_{(p)};L^2({\mathbb R}^3_-\cap{\mathbb B}_R)\|^2+ \int_{\varepsilon R}^{R_j} (1+\varepsilon^{-1}r_j)^{-4}r_j^2\,dx\biggr)^{1/2} \|r_j^{-1}\psi;L^2(\Omega)\| \\ &\qquad \leqslant c_j(\varepsilon^5+\varepsilon^4)^{1/2}\leqslant C_j\varepsilon^2. \end{aligned}
\end{equation}
\tag{4.32}
Now, the required estimate for \delta^j_p follows by taking account of the factors \|\mathbf v^\varepsilon; {\mathcal K}^\varepsilon\|^{-1}=O(\varepsilon^{-1/2}) and \mathbf k^\varepsilon=O(\varepsilon) on the right-hand side of (4.31), so that using Lemma 4 (see (3.15)) we find an eigenvalue \kappa^\varepsilon_{N(\varepsilon)} of {\mathcal K}^\varepsilon such that
\begin{equation}
\bigl|\kappa^\varepsilon_{N(\varepsilon)}-(1+\varepsilon^{-1}\mu^j_p)^{-1} \bigr|\leqslant c\varepsilon^2.
\end{equation}
\tag{4.33}
Similarly to (3.33), from (4.33) and (4.29) we deduce that
\begin{equation}
\bigl|\lambda^\varepsilon_{N(\varepsilon)}-\varepsilon^{-1}\mu^j_p\bigr|\leqslant C^j_p \quad\text{for } \varepsilon\in(0,\varepsilon^j_p] \text{ and some } \varepsilon^j_p>0.
\end{equation}
\tag{4.34}
Thus, under the assumptions made the C^j_p-neighbourhood of \varepsilon^{-1}\mu^j_p contains at least one eigenvalue of problem (4.24), (1.3), (1.4), but its position {N(\varepsilon)} in the ordered sequence (1.12) depends on the small parameter \varepsilon and grows without limit as \varepsilon\to+0. The size of this neighbourhood is much less than \varepsilon^{-1}\mu^j_p, so (4.34) is indeed an asymptotic formula, which shows that the images of at least some of the eigenvalues (4.18) occur in the high-frequency range of the spectrum of problem (4.24), (1.3), (1.4). We presented an example of rapidly decaying vector-valued eigenfunctions in § 2.5, 1^\circ. Remark 6. 1) Constructing lower-order asymptotic terms for the eigenvalues \lambda^\varepsilon_{N(\varepsilon)} is a hard task. We know of only one paper, [19], where full asymptotic expansions are constructed for eigenvalues in both high- and low-frequency parts of the spectrum of a singularly perturbed problem. 2) If the components w^j_{(m)} of an eigenvector \overrightarrow{w}_{(m)} of problem (2.45) decay as O(\rho_j^{-1}) at infinity, then the new bound C_j\varepsilon in (4.32) will be unacceptably large and we will no be able to make the same conclusion as before. On the other hand the conjecture that all the elements of the sequences (4.18) (j=1,\dots,J) are reflected in the high-frequency range of the spectrum (1.12) of problem (4.24), (1.3), (1.4) is quite legitimate, although its verification is an open problem. 4.5. A single set \omega^\varepsilon_1As mentioned in § 2.5, 2^\circ, for the scalar problem of surface waves (1.30)–(1.32), considering the case J=1 does not imply any significant modifications of the limiting problem (1.30), (1.31), (2.58). The reason is that the relevant linear space of polynomials \mathcal P consists of the constants, so that (1.19) holds for each J\geqslant1. In elasticity problems the same condition (1.19) can only hold for J\geqslant3 (or even J\geqslant6; see § 1.3, 2^\circ). Now we show how the limiting problem changes when there is only one contact domain \omega^\varepsilon_1. Assume for simplicity that, first, the projection \mathcal Q is the identity matrix {\mathbb I}_3 and, second, \Omega\cap{\mathcal V}^1 is a part of the half-spaces {\mathbb R}_-^3, moreover, we let P^1 coincide with the origin \mathcal O. In contrast to § 4.2, an asymptotic analysis would be much more complicated without these restrictions. For a smooth solution v of the elasticity problem (2.18), (2.19) in the bounded domain \Omega we have Taylor’s formula
\begin{equation*}
v(x)= a^t+ d'(x)^r a^r +d^\sigma(x)a^\sigma+O(|x|^2), \qquad x\to {\mathcal O},
\end{equation*}
\notag
where a^t and a^r are the translational and rotational displacements of \mathcal O, a=((a^t)^\top,(a^r)^\top)^\top\in {\mathbb R}^6. In addition, a^\sigma\in {\mathbb R}^3 is the tangent stress vector, d^r(x) is a 3\times3 block in (1.38), and d^\sigma(x) is a 3\times3 linear matrix function whose columns produce the unit stresses \sigma_{11}, \sigma_{22} and \sqrt{2}\,\sigma_{12} in the tangent plane (cf. (1.35)). For an isotropic elastic material with stiffness matrix (1.34) the matrix d^\sigma has the form
\begin{equation*}
d^\sigma(x)= \frac{1}{2\boldsymbol\mu(3\boldsymbol\lambda+2\boldsymbol\mu)}\, \begin{pmatrix} 2(\boldsymbol\lambda+\boldsymbol\mu)x_1& -\boldsymbol\lambda x_1&\sqrt{2}(3\boldsymbol\lambda+2\boldsymbol\mu)x_2 \\ -\boldsymbol\lambda x_2&2(\boldsymbol\lambda+\boldsymbol\mu)x_2& \sqrt{2}(3\boldsymbol\lambda+2\boldsymbol\mu)x_1 \\ -\boldsymbol\lambda x_3&-\boldsymbol\lambda x_3&0 \end{pmatrix}.
\end{equation*}
\notag
The solution of the elasticity problem (2.4), (2.5) in {\mathbb R}^3_- whose right-hand side g^1 has finite support, has the representation
\begin{equation}
w^1(\xi^1)=\chi^\infty(\xi^1)\bigl(\Phi^t(\xi^1)b^t+\Phi^r(\xi^1)b^r+ \Phi^\sigma(\xi^1)b^\sigma\bigr)+{\widetilde{ w}}^{1}(\xi^1),
\end{equation}
\tag{4.35}
where the remainder {\widetilde{w}}^{1} decays at infinity as O(\rho_1^{-3}), b^t,b^r\in{\mathbb R}^3 are the principal vector and principal moment of forces applied at infinity, and b^\sigma\in \mathbb{R}^3 is the dipole vector (for instance, see the textbook [39]). In addition, \Phi^t=\Phi is the old 3\times 3 Green’s matrix (2.10), and \Phi^r and \Phi^\sigma are matrix functions of the same size whose entries are functions of \xi^1=(\xi^1_1,\xi^1_2,\xi^1_3) which are positive homogeneous of degree -2. We have the equalities
\begin{equation*}
-\int_{{\mathbb S}^-_R} d^\tau(\xi^1){\mathcal N}_\cup(\xi^1,\nabla_{\xi^1})\Phi^\gamma(\xi^1)\, ds_{\xi^1}= \delta_{\tau,\gamma}{\mathbb I}_3, \qquad \tau,\gamma=t,r,\sigma.
\end{equation*}
\notag
For an isotropic material all these notions can be found, for example, in [40]. In a singular displacement field v which solves the homogeneous problem (2.18), (2.19), smooth in {\overline{\Omega}}\setminus{\mathcal O} and satisfies the asymptotic condition
\begin{equation*}
v(x)=\Phi^t(x)b^t+\Phi^r(x)b^r+ \Phi^\sigma(x)b^\sigma+O(1),\qquad x\to{\mathcal O},
\end{equation*}
\notag
the column b^\sigma can be arbitrary, but the equalities b^t=b^r=0\in{\mathbb R}^3 are necessary.4 In accordance with the above, we keep representation (1.14) for the eigenvalue, but modify the asymptotic ansatz (2.34) for the vector-valued eigenfunction as follows:
\begin{equation}
u^\varepsilon(x)=\varepsilon^{-1}d^r(x)^\top a^r+ a^t+w^1(\varepsilon^{-1}x)+ {\widehat{v}}(x)+\dotsb.
\end{equation}
\tag{4.36}
The coefficient \varepsilon^{-1} of the first term in (4.36) is due to the relation d^r(x)=\varepsilon d^r(\varepsilon^{-1}x). As a result, for the term w^1 of boundary-layer type we obtain the system of equations (2.4) with boundary conditions
\begin{equation}
{\mathcal N}(\nabla_{\xi^1})w^1(\xi^{1\prime},0)= \mu \bigl(w^1(\xi^{1\prime},0)+d(\xi^{1\prime},0) a\bigr) , \qquad \xi^{1\prime}\in\varpi_1,
\end{equation}
\tag{4.37}
and
\begin{equation}
{\mathcal N}(\nabla_{\xi^1})w^1(\xi^{1\prime},0)=0, \qquad \xi^{1\prime}\in{\mathbb R}^2\setminus{\overline{\varpi_1}}.
\end{equation}
\tag{4.38}
The columns b^t and b^r in the expansion (4.35) for the solution of the problem obtained in the half-space vanish if and only if the column a\in{\mathbb R}^6 formed by a^t and a^r satisfies the system of algebraic equations
\begin{equation*}
\mathbf Ma=- \underline{w}^1:= -\int_{\varpi_1}d(\xi^{1\prime},0)^\top w^1(\xi^{1\prime},0)\,d\xi^{1\prime}
\end{equation*}
\notag
with symmetric positive definite 6\times6 matrix
\begin{equation*}
\mathbf M=\int_{\varpi_1}d(\xi^{1\prime},0)^\top d(\xi^{1\prime},0)\,d\xi^{1\prime}.
\end{equation*}
\notag
Thus, the boundary condition (4.37) on \varpi_1 assumes the form
\begin{equation}
{\mathcal N}(\nabla_{\xi^1})w^1(\xi^{1\prime},0)=\mu \bigl(w^1(\xi^{1\prime},0)-d(\xi^{1\prime},0)\mathbf M^{-1}\underline{w}^1\bigr).
\end{equation}
\tag{4.39}
The variational statement of (2.4), (4.39), (4.38) is as follows:
\begin{equation}
{\mathcal E}(w^1,\psi^1)=\mu \bigl((w^1,\psi^1)_{\varpi_1}-(d(\cdot,0)\mathbf M^{-1}\underline{w}^1,d(\cdot,0)\underline{\psi}^1)_{\varpi_1}\bigr) \quad \forall\,\psi^1\in V^1_0({\mathbb R})^3.
\end{equation}
\tag{4.40}
For the same reasons as in § 2.3 (in the verification of Lemma 2) the bilinear form multiplying \mu on the right-hand side of (4.40) is positive definite. Thus the spectrum of the problem obtained is discrete and forms a sequence (2.50). We present an asymptotic formula for the eigenvalues of the problem under consideration, where we however do not examine the order of smallness of the remainder term. To find it, one must improve the approaches used in § 3. Theorem 8. Under the above assumptions the eigenvalues (1.12) of the elasticity problem (1.5) (or, in the differential form, (1.2)–(1.4)) with one (J=1) contact domain \omega^\varepsilon_1=\varpi^\varepsilon_1 satisfy the relations \lambda^\varepsilon_1=\dots=\lambda^\varepsilon_6=0 and
\begin{equation*}
\lambda^\varepsilon_{m+6}=\varepsilon^{-1}\mu_m+O(\varepsilon^{-1/2}),
\end{equation*}
\notag
where m\in{\mathbb N} and \mu_m is an element of the sequence (2.50) of eigenvalues of problem (4.40). 4.6. The biharmonic equationAssume that the boundary \partial \Omega is flat in neighbourhoods of the points P^1,\dots,P^J and consider a boundary value problem for a fourth order equation:
\begin{equation}
{\mathcal N}_2(x,\nabla_x) u^\varepsilon(x)=0, \qquad x\in\partial\Omega,
\end{equation}
\tag{4.42}
\begin{equation}
{\mathcal N}_3(x,\nabla_x) u^\varepsilon(x)=0, \qquad x\in\partial\Omega\setminus{\overline{\omega^\varepsilon}},
\end{equation}
\tag{4.43}
\begin{equation}
{\mathcal N}_3(x,\nabla_x) u^\varepsilon(x)=\lambda^\varepsilon u^\varepsilon(x), \qquad x\in\omega^\varepsilon.
\end{equation}
\tag{4.44}
Here the {\mathcal N}_q(x,\nabla_x) are differential operators of orders q=2 and 3 from Green’s formula
\begin{equation}
{\mathcal E}(u,\psi;\Omega)=(\Delta^2_xu,\psi)_\Omega -({\mathcal N}_3u,\psi)_{\partial\Omega}+({\mathcal N}_2u, \partial_n\psi)_{\partial\Omega},
\end{equation}
\tag{4.45}
where the bilinear form on the left is
\begin{equation}
{\mathcal E}(u,\psi;\Omega)= \sum_{p,q}\biggl(\frac{\partial^2u}{\partial x_p\,\partial x_q} ,\frac{\partial^2\psi}{\partial x_p\,\partial x_q}\biggr)_\Omega;
\end{equation}
\tag{4.46}
it has the polynomial property (1.17) with the four-dimensional linear space
\begin{equation}
{\mathcal P}=\{p=c_0+c_1x_1+c_2x_2+c_3x_3\mid c_\ell\in{\mathbb R}\}.
\end{equation}
\tag{4.47}
This property ensures that problem (4.41)–(4.44) is elliptic, its spectrum (1.12) is discrete and the first four eigenvalues are equal to zero (see [6], § 1). Remark 7. The form on the left-hand side of the simplest Green’s formula
\begin{equation}
(\Delta_xu,\Delta_x\psi)_\Omega=(\Delta^2_xu,\psi)_\Omega- (\partial_n\Delta_xu,\psi)_{\partial\Omega}+(\Delta_xu, \partial_n\psi)_{\partial\Omega}
\end{equation}
\tag{4.48}
does not have the polynomial property, because it vanishes at each harmonic function. Moreover, the pair of boundary operators ({\mathcal N}^\Delta_3, {\mathcal N}^\Delta_2)=(\partial_n\Delta_x,\Delta_x) does not cover the biharmonic operator on the boundary \partial\Omega, and the corresponding boundary value problem is not elliptic (see [2] and [6], § 1.5). So we need a Green’s formula different from (4.48), for instance, (4.45). There are infinitely many suitable Green’s formulae and boundary operators {\mathcal N}_q(x,\nabla_x) produced by them (cf. [41], § 30, for the problem for the Kirchhoff plate), and all the results that follow are independent of a particular choice because we de note need to know the explicit form of the operators {\mathcal N}_3 and {\mathcal N}_2 for an asymptotic analysis: it is only important that they vanish on the subspace (4.47). The limiting problem in the domain \Omega
\begin{equation}
{\mathcal N}_q(x,\nabla_x) v(x)=g_q(x), \qquad x\in\partial\Omega, \quad q=2,3,
\end{equation}
\tag{4.50}
in its variational form
\begin{equation}
{\mathcal E}(u,\psi;\Omega)= (f,\psi)_\Omega -(g_3,\psi)_{\partial\Omega}+(g_2, \partial_n\psi)_{\partial\Omega} \quad \forall\, \psi\in H^2(\Omega)
\end{equation}
\tag{4.51}
has a solution v\in H^2(\Omega), which is defined up to a linear function of x=(x_1,x_2,x_3), only if the following four conditions are satisfied:
\begin{equation*}
(f,1)_\Omega-(g_3,1)_{\partial\Omega}=0\quad\text{and} \quad (f,x_k)_\Omega-(g_3,x_k)_{\partial\Omega}+(g_2, \partial_nx_k)_{\partial\Omega}=0, \quad k=1,2,3.
\end{equation*}
\notag
To clear up the asymptotic behaviour of the eigenvalues \lambda^\varepsilon_m for m>4 we require another setting of problem (4.49), (4.50), in which we look for a function u\in V^2_0(\Omega) satisfying the integral identity
\begin{equation}
{\mathcal E}(u,\psi;\Omega)= (f,\psi)_\Omega -(g_3,\psi)_{\partial\Omega}+(g_2, \partial_n\psi)_{\partial\Omega} \quad \forall\, \psi\in V^2_0(\Omega).
\end{equation}
\tag{4.52}
Here V^2_0(\Omega) is the Kondrat’ev space with norm (2.20) for \ell=2, and in accordance with the trace inequality (2.24) for \beta=1,0, the right-hand sides must satisfy the membership relations
\begin{equation}
f\in V^0_{-2}(\Omega)\quad\text{and} \quad g_q\in V^0_{q-3/2}(\partial\Omega), \quad q=2,3.
\end{equation}
\tag{4.53}
By Sobolev’s theorem on the embedding H^2\subset C in {\mathbb R}^3 and by versions of Hardy’s inequality the space V^2_0(\Omega) coincides algebraically and topologically with the subspace It is easy to verify that the sets (4.47) and (4.54) intersect precisely in zero, provided that the set of points P^1,\dots,P^J contains the vertices of a nondegenerate pyramid, that is, J\geqslant4 and
\begin{equation}
p\in{\mathcal P}, \quad p(P^1)=\dots=p(P^J)=0 \quad \Longleftrightarrow\quad p=0.
\end{equation}
\tag{4.55}
The next result is a consequence of Theorem 1.9 in [6]. Proposition 5. Assume that (4.53) and (4.55) hold. Then problem (4.52) has a unique solution v\in V^2_0(\Omega), and its norm is at most cN, where N is the sum of the norms of the functions in (4.53) in the weighted spaces indicated. We endow the Kondrat’ev space V^2_0({\mathbb R}^3_-) with the norm (2.20), where \ell=2 and \beta=0. It is clear that this space contains the constant functions, so that they solve the homogeneous problem (for f^j=0 and g^j_q=0)
\begin{equation}
\Delta_{\xi^j}^2w^j(\xi^j)=f^j(\xi^j), \qquad \xi^j\in{\mathbb R}^3_-,
\end{equation}
\tag{4.56}
\begin{equation}
{\mathcal N}^j_q(\nabla_{\xi^j})w^j(\xi^{j\prime},0)=g^j_q(\xi^{j\prime}), \qquad \xi^{j\prime}\in{\mathbb R}^2, \quad q=2,3.
\end{equation}
\tag{4.57}
Here the operators {\mathcal N}^j_q(\nabla_{\xi^j}) are obtained from the {\mathcal N}_q(x,\nabla_x) by making the substitution (2.1) and freezing the coefficients at P^j. Thus the eigenvalue zero of the limiting spectral problem in the variational form
\begin{equation}
{\mathcal E}(w^j,\psi^j;{\mathbb R}^3_-)=\mu^j(w^j,\psi^j)_{\varpi_j} \quad \forall\, \psi^j\in V^2_0({\mathbb R}^3_-)
\end{equation}
\tag{4.58}
is associated with the constant eigenfunctions, and only with them, because those elements of the subspace (4.47) (on which the form (4.51) vanishes) that increase linearly at infinity do not occur in V^2_0({\mathbb R}^3_-). Proposition 6. Problem (4.58) has a discrete spectrum
\begin{equation}
0=\mu^j_0<\mu^j_1\leqslant\mu^j_2\leqslant\dots\leqslant\mu^j_m\leqslant\dots\to+\infty,
\end{equation}
\tag{4.59}
and the corresponding eigenfunctions |\varpi_j|^{1/2}=w^j_0,w^j_1, w^j_2,\dots, w^j_m,\ldots\in V^2_0({\mathbb R}^3_-) can be subject to the conditions of orthogonality and normalization We describe the behaviour of solutions of the limiting problems in \Omega and {\mathbb R}^3_- near P^1,\dots,P^J and at infinity. Note that for b_j\ne 0 the product b_j\chi_j does not belong to the space V^2_0(\Omega) with norm (2.20), and the Green’s function of the Neumann problem in the half-space satisfies In addition, \chi_j\Phi^j_0\in V^2_0(\Omega) and a formula similar to (2.11) holds:
\begin{equation}
-\int_{{\mathbb S}^-_R} {\mathcal N}^j_{3\cup}(x^j,\nabla_{x^j})\Phi^j_0(x^j)\,ds_{x^j}=1.
\end{equation}
\tag{4.61}
Moreover, the linear functions x^j_1, x^j_2 and x^j_3 have same order of homogeneity as the Green’s function. Thus, by Kondrat’ev’s theorem on the asymptotic behaviour (see [33] and also [34], Theorem 4.2.1) the solution of problem (4.49), (4.50) with smooth right-hand sides f and g_q that vanish in a neighbourhood of P^1,\dots,P^J takes the form
\begin{equation*}
v(x)=\sum_{j=1}^J \biggl(c^j_0\Phi^j_0(x^j)+\sum_{\ell=1}^3 c^j_\ell x^j_\ell\biggr)+{\widetilde{v}}(x),
\end{equation*}
\notag
where the c^j_q are some constants and the remainder {\widetilde{v}} is a smooth function in \overline \Omega that vanishes at P^1,\dots,P^J together with its gradient \nabla_x{\widetilde{v}}. It is smooth because of our assumption on the flat parts of \partial\Omega. As mentioned already, the constant functions belong to V^2_0({\mathbb R}^3_-). By the calculations in [6], § 2.6, the three other solutions also have the zeroth order of homogeneity:
\begin{equation*}
\Phi^j_\ell(\xi^j)=\Phi^j_\ell(\rho_j^{-1}\xi^j), \qquad \ell=1,2,3.
\end{equation*}
\notag
For these we have
\begin{equation*}
\begin{aligned} \, Q_j(x_k,\Phi^j_\ell) &=-\int_{{\mathbb S}^-_R(P^j)} \biggl(\xi_k{\mathcal N}^j_{3\cup}(x^j,\nabla_{x^j})\Phi^j_\ell(x^j) \\ &\qquad\qquad - \frac{\partial\xi_k}{\partial \rho_j}{\mathcal N}^j_{2\cup}(x^j,\nabla_{x^j})\Phi^j_\ell(x^j) \biggr)\,ds_{x^j}=\delta_{\ell,k} \end{aligned}
\end{equation*}
\notag
for k,\ell=1,2,3. We stress that, because of a disparity in the orders of homogeneity of the pairs \{1,\Phi^j_\ell\} and \{x^j_k,\Phi^j_0\}, we have
\begin{equation*}
Q(1,\Phi^j_\ell)=0, \quad Q(x^j_\ell,\Phi^j_0)=0, \qquad \ell=1,2,3.
\end{equation*}
\notag
We see that in problem (4.52) a constant is an ‘irregular’ component in a neighbourhood of P^j, so that by Proposition 5, for each column a=(a_1,\dots,a_J)^\top\in{\mathbb R}^J the homogeneous problem (4.49), (4.50) (for f=0 and g_q=0) has a solution \zeta(a;\cdot)\in H^2(\Omega) with a representation
\begin{equation}
\zeta(a;x)=\sum_{j=1}^J\chi_j(x)\biggl(a_j+M^0_j(a)\Phi^j(x^j)+\sum_{\ell=1}^3 M^\ell_j(a)x^j_\ell\biggr)+{\widetilde{\zeta}}(a;x).
\end{equation}
\tag{4.62}
This involves a smooth function {\widetilde{\zeta}}(a;\cdot) in \overline \Omega and linear maps
\begin{equation}
{\mathbb R}^6\ni a\,\,\mapsto\,\, M^\ell_j(a)\in {\mathbb R}^6,\qquad \ell=0,1,2,3.
\end{equation}
\tag{4.63}
The next result shows, in particular, that only in exceptional cases are the functions (4.62) in the space H^2(\Omega) solutions of the homogeneous problem (4.51) (for f=0 and g_q=0):
\begin{equation}
p\in{\mathcal P}, \quad a^p=(p(P^1),\dots,p(P^J))^\top \quad\Longrightarrow\quad \zeta(a^p;x)=p(x).
\end{equation}
\tag{4.64}
Proposition 7. For \ell=0 the operator (4.63) is positive and its four-dimensional root subspace has the form
\begin{equation*}
{\mathbb R}^J({\mathcal P})=\bigl\{a=(p(P^1),\dots,p(P^J))^\top\mid p\in{\mathcal P}\bigr\}
\end{equation*}
\notag
(see (4.64) and (4.47)). Proof. For a\in {\mathbb R}^6
\begin{equation*}
\begin{aligned} \, {\mathcal E}(\zeta(a;\cdot),\zeta(a;\cdot);\Omega) &=\lim_{R\to+0} {\mathcal E}\bigl(\zeta(a;\cdot),\zeta(a;\cdot);\Omega\setminus( {\mathbb B}_R(P^1)\cup\dots\cup{\mathbb B}_R(P^J)\bigr) \\ &=-\lim_{R\to+0}\sum_{j=1}^J\int_{{\mathbb S}^-_R(P^j)}\biggl( \zeta(a;x){\mathcal N}^j_{3\cup}(x^j,\nabla_{x^j})\zeta(a;x) \\ &\qquad+\frac{\partial \zeta}{\partial r_j}(a;x){\mathcal N}^j_{2\cup}(x^j,\nabla_{x^j})\zeta(a;x)\biggr)\,ds_x \\ &=\sum_{j=1}^J a_j M^0_j(a). \end{aligned}
\end{equation*}
\notag
Here we have taken (4.62) and (4.61) into account. Thus, the J\,{\times}\,J matrix M^0=(M^0_1,\dots,M^0_J) of the map (4.63) for \ell=0 is a Gram matrix. To complete the proof it remains to mention the polynomial property (1.17), (4.47) of the quadratic form (4.46), which shows that M^0a=0 only for a\in {\mathbb R}^J({\mathcal P}).
The proposition is proved. We proceed to constructing the leading terms of the formal asymptotic expansions
\begin{equation}
\lambda^\varepsilon_p=\varepsilon^{-3}(\mu_p+\varepsilon\mu_p'+\cdots)
\end{equation}
\tag{4.65}
and
\begin{equation}
u^\varepsilon_p(x)=\sum_{j=1}^J\chi_j(x)(w^j_p(\xi^j)+\varepsilon w^{j\prime}_p(\xi^j))+ {\mathcal X}^\varepsilon(x) {\widehat{v}}(x)+\dotsb.
\end{equation}
\tag{4.66}
Note that the power -3 of the small parameter \varepsilon in (4.65) is due to the fact that the spectral boundary condition (4.44) involves the third-order differential operator, and we need the cut-off function {\mathcal X}^\varepsilon\in C^\infty_c({\overline{\Omega}}\setminus\{P^1, \dots,P^J\}) produced from \chi^\infty in (2.9),
\begin{equation}
{\mathcal X}^\varepsilon(x)=1-\sum_{j=1}^J(1-\chi^\infty(\xi^j)),
\end{equation}
\tag{4.67}
because the O(r_j)-singularities of the Green’s functions (4.60) involved in the expansion (4.68) are foreign to eigenfunctions of (4.41)–(4.44) and therefore must be smoothened. First we consider perturbations of the eigenvalue zero of problems (4.58), j=1,\dots,J. We set w^j_p(\xi^j)=b^j_{(p)}\in{\mathbb R}, substitute the ansätze (4.65) and (4.66) into (4.41)–(4.44), interchange differential operators with the сut-off functions \chi_j and collect the terms of order 1=\varepsilon^0 expressed in the ‘slow’ variables x. As a result, for the smooth term {\widehat{v}} in (4.66) we obtain problem (4.49), (4.50) with right-hand sides
\begin{equation*}
f(x)=-\sum_{j=1}^J[\Delta_x^2,\chi_j(x)]b^j_{(p)}\quad\text{and} \quad g_q(x)=-\sum_{j=1}^J[{\mathcal N}_q(x,\nabla_x),\chi_j(x)]b^j_{(p)}.
\end{equation*}
\notag
In accordance with the definition (4.62) and formula (4.64), its unique solution in V^2_0(\Omega) has the following form:
\begin{equation}
\begin{aligned} \, \notag {\widehat{v}}(x) &=\sum_{j=1}^J(\zeta(b_{(p)};x)-\chi_j(x)b^j_{(p)}) \\ &=\sum_{j=1}^J\chi_j(x)\biggl(M^0_j(b_{(p)})\Phi^j(x^j)+\sum_{\ell=1}^3 M^\ell_j(b_{(p)})x^j_\ell\biggr)+{\widetilde{v}}(x). \end{aligned}
\end{equation}
\tag{4.68}
Now we put together problems in the half-space for the correction terms w^{j\prime}. We take the terms of order r_j detached in (4.68) into account, go over to the stretched coordinates (2.1), interchange with the cut-off function (4.67) and collect the terms of order \varepsilon^{-3} in the biharmonic equation (4.41) and of order \varepsilon^{1-q} in the boundary conditions (4.44). As a result, we obtain problem (4.56), (4.57) where the right-hand sides
\begin{equation*}
f^j(\xi^j) =[\Delta^2_{\xi^j},\chi^\infty(\xi^j)]\biggl(M^0_j(b_{(p)})\Phi^j(\xi^j)+\sum_{\ell=1}^3 M^\ell_j(b_{(p)})\xi^j_\ell\biggr)
\end{equation*}
\notag
and
\begin{equation*}
\begin{aligned} \, g_q^j(\xi^{j\prime}) &=\delta_{3,q}\mu_p' \Psi_j(\xi^{j\prime})b^j_{(p)} \\ &\qquad +[{\mathcal N}^j_q(\nabla_{\xi^j}),\chi^\infty(\xi^j)]\biggl(M^0_j(b_{(p)})\Phi^j(\xi^j)+\sum_{\ell=1}^3 M^\ell_j(b_{(p)})\xi^j_\ell\biggr)\bigg|_{\xi^j_3=0} \end{aligned}
\end{equation*}
\notag
have compact supports; here \Psi_j is the characteristic function of the set \varpi_j\subset{\mathbb R}^2=\partial{\mathbb R}^3_-. The compatibility condition in this problem is the orthogonality of the right-hand sides to 1 (=to the constant functions) in the sense of Green’s formula. As usual, taking (4.61) into account we let R\to+\infty:
\begin{equation*}
\begin{aligned} \, 0 &=\lim_{R\to+\infty}\biggl( \int_{{\mathbb R}^3_-\cap{\mathbb B}_R}f^j(\xi^j)\,d\xi^j - \int_{\partial{\mathbb R}^3_-\cap{\mathbb B}_R}g^j_3(\xi^{j\prime})\,d\xi^{j\prime}\biggr) \\ &=-\mu_p' |\varpi_j|b^j_{(p)}-\lim_{R\to+\infty}\int_{{\mathbb S}^-_R}{\mathcal N}^j_{3\cup}(\xi^j,\nabla_{\xi^j})\biggl(\!M^0_j(b_{(p)})\Phi^j(\xi^j)+\sum_{\ell=1}^3 M^\ell_j(b_{(p)})\xi^j_\ell\biggr)\,ds_{\xi^j} \\ &\qquad-\mu_p' |\varpi_j|b^j_{(p)}-\lim_{R\to+\infty}\int_{{\mathbb S}^-_R}{\mathcal N}^j_{3\cup}(\xi^j,\nabla_{\xi^j})\Phi^j(\xi^j)\,ds_{\xi^j} M^0_j(b_{(p)}) \\ &=- \mu_p' |\varpi_j|b^j_{(p)}+M^0_j(b_{(p)}). \end{aligned}
\end{equation*}
\notag
Thus we see that for p=1,\dots,J the leading term \mu_0 in (4.65) is zero and the principal correction terms
\begin{equation}
0=\mu'_1=\mu'_2=\mu'_3=\mu'_4<\mu'_5\leqslant\dots\leqslant\mu'_J
\end{equation}
\tag{4.69}
are eigenvalues of the linear pencil
\begin{equation}
{\mathbb R}\ni\mu'\mapsto M^0\mu'-\operatorname{diag}\{|\varpi_1|,\dots,|\varpi_J|\} \mu'\in {\mathbb R}^J.
\end{equation}
\tag{4.70}
The string (4.69) of four zero and J-4 positive eigenvalues of the pencil (4.70) is a consequence of Proposition 7. In the low-frequency range \{\lambda^\varepsilon_m \leqslant C\varepsilon^{-2}\} of the spectrum (1.12) singular perturbations of the boundary conditions in problem (4.41)–(4.44) for the biharmonic equation interact because the matrix M^0 of the linear operator (4.63) for \ell=0 is constructed from the solutions (4.62) of the homogeneous problem (4.49), (4.50) in the whole of \Omega. However, in the high-frequency range \{\lambda^\varepsilon_m\geqslant c\varepsilon^{-3}\} there is no such interaction in leading order because for p>J the leading term \mu_p of the asymptotic expression (4.65) comes from the positive parts of the sequences (4.59), and when (4.55) holds, the limiting problem (4.52) is uniquely solvable (cf. the reasoning in § 4.3 concerning problem (1.2), (4.13), (4.14), (1.4) with Dirichlet conditions on \gamma). We state a result on the asymptotic behaviour of the eigenvalues; the asymptotic remainders are estimated by following the routine pattern described in § 3, and details of its application to higher-order equations were presented, for instance, in [38], Ch. 5, § 2, and Ch. 6, § 3. Theorem 9. The first four elements of the sequence (1.12) of eigenvalues of problem (4.41)–(4.44) are zeros. Under the assumption (4.55) the leading terms of the asymptotic formulae
\begin{equation*}
\lambda^\varepsilon_p=\varepsilon^{-2} \mu^p+O(\varepsilon^{-3/2}), \qquad p=5,\dots,J,
\end{equation*}
\notag
are the positive eigenvalues (4.69) of the pencil (4.70) constructed from the solutions (4.62) of the homogeneous problem (4.49), (4.50). For the other eigenvalues the formulae
\begin{equation*}
\lambda^\varepsilon_\ell=\varepsilon^{-3} \mu^\ell+O(\varepsilon^{-5/2}), \qquad \ell=J+1,J+2,\dots,
\end{equation*}
\notag
hold, where \{\mu_{\ell-J}\}_{\ell\in{\mathbb N}} is the ordered sequence of positive eigenvalues (4.59) of the independent limiting problems (4.58), j=1,\dots,J. As in the other problems we have considered we do not know the asymptotic structure of the spectrum of the problem if the assumption (4.55) is not imposed. Of the greatest applied interest is a similar two-dimensional problem of a spring fixation of the Kirchhoff plate, but no asymptotic analysis of it has ever been performed. 
Citation in format AMSBIB
\Bibitem{Naz23}
This publication is cited in the following 2 articles:
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