Spectral properties of an elliptic operator with double-contrast coefficients near a hyperplane

Abstract

In this paper we study the asymptotic behaviour as $ε \to 0$ of the spectrum of the elliptic operator $A^{ε} = - \frac{1}{b^{ε}} div (a^{ε} \nabla)$ posed in a bounded domain $Ω \subset R^{n}$ ( $n ⩾ 2$ ) subject to Dirichlet boundary conditions on $\partial Ω$ . When $ε \to 0$ both coefficients $a^{ε}$ and $b^{ε}$ become high contrast in a small neighborhood of a hyperplane Γ intersecting Ω. We prove that the spectrum of $A^{ε}$ converges to the spectrum of an operator acting in $L^{2} (Ω) \oplus L^{2} (Γ)$ and generated by the operation $- Δ$ in $Ω ∖ Γ$ , Dirichlet boundary conditions on $\partial Ω$ and certain interface conditions on Γ containing the spectral parameter in a nonlinear manner. The eigenvalues of this operator may accumulate at a finite point. Then we study the same problem, when Ω is an infinite straight strip (“waveguide”) and Γ is parallel to its boundary. We show that $A^{ε}$ has at least one gap in the spectrum when ε is small enough and describe the asymptotic behaviour of this gap as $ε \to 0$ . The proofs are based on methods of homogenization theory.

Keywords

high-contrast coefficients spectrum asymptotics homogenization periodic waveguides spectral gaps

1. Introduction

The problem we are going to study lies on the intersection of spectral theory and homogenization theory for partial differential operators. We recall that one of the central problems of homogenization theory is to study the asymptotic behaviour as $ε \to 0$ of the solution $u^{ε}$ to the problem $\begin{array}{l} A^{ε} u^{ε} = f in Ω, \\ u^{ε} = 0 on \partial Ω, \end{array}$ where $ε > 0$ is a small parameter, $A^{ε} : = - div (a^{ε} (x) \nabla)$ , ${a^{ε} (x)}_{ε > 0}$ is a family of real measurable functions satisfying $\begin{matrix} a_{-}^{ε} ⩽ a^{ε} (x) ⩽ a_{+}^{ε} with positive constants a_{\pm}^{ε} \end{matrix}$ and becoming highly oscillating as $ε \to 0$ . The typical example is $a^{ε} (x) = a (x ε^{- 1})$ , where $a (x)$ is a fixed $Z^{n}$ -periodic function. It is well-known (see, e.g., [1,14]) that if $\begin{array}{l} (1.1) & inf_{ε} a_{-}^{ε} > 0, sup_{ε} a_{+}^{ε} < \infty, \end{array}$ then the family ${u^{ε}}_{ε}$ is compact in $L^{2} (Ω)$ , and if $u^{ε} \to u$ as $ε = ε_{k} \to 0$ then $u (x)$ is a solution of the problem $\begin{array}{l} A u = f in Ω, \\ u = 0 on \partial Ω, \end{array}$ where $A : = - div (A (x) \nabla)$ , $A (x)$ is some bounded matrix-function, bounded away from zero, which depends, in general, on the subsequence $ε_{k}$ . If $a^{ε} (x) = a (x ε^{- 1})$ then the whole sequence $u^{ε}$ converges, and in this case $A (x)$ is a constant matrix. As we see the limit differential operator has qualitatively the same form as the initial one provided (1.1) holds.

If, on the contrary, conditions (1.1) are violated, for example there exist subsets $D^{ε} \subset Ω$ with non-zero measure such that ${lim}_{ε \to 0} {sup}_{x \in D^{ε}} a^{ε} (x) = \infty$ or $\begin{array}{l} (1.2) & lim_{ε \to 0} inf_{x \in D^{ε}} a^{ε} (x) = 0, \end{array}$ then the limit operator may have a more complicated form, which depends essentially on the structure of the domains $D^{ε}$ . We refer to the monograph [28], where various problems of this type are studied. In particular, the authors consider the case, when condition (1.2) holds on the union $D^{ε}$ of thin shells, distributed periodically, with period ε, in the domain Ω. They study the behaviour of linear evolution equations involving such operators $A^{ε}$ . Semi-linear evolution equations were investigated in [13,36,37]. Spectral properties of such operators were studied in [22].

In all papers mentioned above the case of bulk distribution of shells was considered. In the present work we are interested in the case of surface distribution of shells, i.e., the shells are located in a neighbourhood of some hyperplane.

We note that our research is inspired, in particular, by spectral problems for periodic differential operators posed in a waveguide-like domain. These applications are discussed later in the introduction.

Below we briefly present our main results. Let Ω be a bounded domain in $R^{n}$ ( $n ⩾ 2$ ), $ε > 0$ be a small parameter. In Ω we consider the operator $\begin{array}{l} (1.3) & A^{ε} = - \frac{1}{b^{ε} (x)} div (a^{ε} (x) \nabla) \end{array}$ subject to Dirichlet boundary conditions on $\partial Ω$ . Here the functions $a^{ε}$ and $b^{ε}$ are bounded above and bounded away from zero uniformly in ε everywhere except a small neighbourhood of some hyperplane Γ having non-empty intersection with the domain Ω. More precisely, we denote by $D^{ε} = {D_{i}^{ε}}$ a family of thin spherical shells distributed periodically, with period ε, along Γ (counted by the parameter i). Each shell has an external radius $R^{ε} = R ε$ (here $R \in (0, 1 / 2)$ is a constant), the thickness of the shells is $d^{ε} = o (ε)$ as $ε \to 0$ . By $B^{ε} = {B_{i}^{ε}}$ we denote the union of balls surrounded by these shells (see Fig. 1). When $ε \to 0$ the number $N (ε)$ of shells goes to infinity as $ε \to 0$ , namely $\begin{array}{l} (1.4) & N (ε) \sim ε^{1 - n} | Γ | \end{array}$ (hereinafter we will use the same notation $| \cdot |$ for the volume of a domain in $R^{n}$ and as well for the area of an $(n - 1)$ -dimensional hypersurface in $R^{n}$ ). We define the functions $a^{ε} (x)$ and $b^{ε} (x)$ as follows: $a^{ε} (x)$ is equal to 1 for $x \in Ω ∖ D^{ε}$ and equal to the constant $α^{ε} > 0$ for $x \in D^{ε}$ , $b^{ε} (x)$ is equal to 1 for $x \in Ω ∖ B^{ε}$ and equal to the constant $β^{ε} > 0$ for $x \in B^{ε}$ .

Fig. 1.

The domain Ω and the family of shells $D_{i}^{ε}$ .

The operator $A^{ε}$ describes vibrations of a body occupying the domain Ω, the functions $a^{ε} (x)$ and $b^{ε} (x)$ are its stiffness and mass density, respectively. We note, that the most interesting effects occur if $\begin{matrix} α^{ε} = O (d^{ε}), β^{ε} = O (ε^{- 1}) as ε \to 0 . \end{matrix}$ In this case $A^{ε}$ describes vibrations of a body with a lot of tiny heavy inclusions $B_{i}^{ε}$ surrounded by thin soft layers $D_{i}^{ε}$ .

Operators of the form (1.3) occur in many other areas of physics. For example, in the case $n = 3$ the operator $A^{ε}$ governs the propagation of acoustic waves in an inhomogeneous medium with mass density ${(a^{ε} (x))}^{- 1}$ and compressibility $b^{ε} (x)$ .1

Indeed, starting from the basic linear acoustic equations (see, e.g., [32]) $κ \frac{\partial P}{\partial t} = div \vec{v}$ , $ρ \frac{\partial \vec{v}}{\partial t} = - \nabla P$ , where $P (x, t)$ is the pressure, $\vec{v} (x, t)$ is the velocity, $κ (x)$ is the compressibility and $ρ (x)$ is the mass density, we arrive at the following equation for the pressure: $\frac{\partial^{2} P}{\partial t^{2}} + A P = 0$ , where $A = - \frac{1}{κ} div (ρ^{- 1} \nabla)$ .

Asymptotics of eigenvibrations of a body with a mass density perturbed near a hypersurface was studied in a lot of papers – see, e.g., [10–12,26,27,29] and references therein; we refer also to the monographs [35,39]. The case of simultaneously perturbed density and stiffness (“double-contrast”) was investigated in [20] (see also [2], where the case of bulk distribution of double-contrast inclusions was studied). In all these articles the geometry of a set supporting the perturbation differs essentially from that one considered in the current paper.

The spectrum $σ (A^{ε})$ of $A^{ε}$ is purely discrete. Our goal is to describe its behaviour as $ε \to 0$ supposing that the following conditions hold: $\begin{array}{l} (1.5) & lim_{ε \to 0} \frac{{(d^{ε})}^{2}}{α^{ε}} = 0, \\ (1.6) & lim_{ε \to 0} q^{ε} = q \in [0, \infty], where q^{ε} = \frac{α^{ε} n}{R d^{ε} ε β^{ε}}, \\ (1.7) & lim_{ε \to 0} r^{ε} = r \in [0, \infty), where r^{ε} = R^{n} ϰ_{n} ε β^{ε} . \end{array}$ Here by $ϰ_{n}$ we denote the volume of the n-dimensional unit ball. We note, that q is allowed to be infinite. In terms of powers of ε conditions (1.5)–(1.7) can be rewritten as follows. We set $d^{ε} = ε^{γ}$ , $β^{ε} = ε^{b}$ , $α^{ε} = ε^{a}$ . Since $d^{ε} = o (ε)$ then $γ > 1$ . We fix $γ \in (1, \infty)$ as a parameter and look how q and r depend on a and b. Condition (1.5) implies $2 γ - a > 0$ ; condition (1.7) requires $1 + b ⩾ 0$ . Thus, on the $(a, b)$ -plane we are restricted to the set ${a < 2 γ, b ⩾ - 1}$ . The interior of this set is marked by the grey color on Fig. 2. We also introduce the following points on this plane: $A = (- \infty, - 1)$ , $B = (γ, - 1)$ , $C = (2 γ, - 1)$ , $D = (2 γ, γ - 1)$ , $E = (2 γ, \infty)$ . Due to (1.6), we obtain:

$q > 0$ , $r > 0$ hold at the point B,

$q = 0$ , $r > 0$ hold on the open line segment $(B, C)$ ,

$q = \infty$ , $r > 0$ hold on the ray $(B, A)$ ,

$q > 0$ , $r = 0$ hold on the open line segment $(B, D)$ ,

$q = 0$ , $r = 0$ hold in the open set, which is bounded by the triangle $\hat{B C D}$ ,

$q = \infty$ , $r = 0$ hold in the open unbounded domain whose boundary consists of the rays $(B, A)$ and $(D, E)$ and the segment $[B, D]$ .

Fig. 2.

$(a, b)$ -plane.

Condition (1.5) means that the eigenfunctions cannot concentrate on the shells (cf. (3.45)). The case $α^{ε} = O ({(d^{ε})}^{2})$ differs essentially and will not be studied in this paper.

The parameter q characterizes the “strength” of the coupling between the domain $Ω^{ε} : = Ω ∖ ⋃_{i} (\overline{D_{i}^{ε} \cup B_{i}^{ε}})$ and the union of the balls $B_{i}^{ε}$ . We postpone the precise statement to Section 3 (see Remark 3.1) because first we need to introduce some more notations.

The finiteness of r implies the uniform (with respect to ε) boundedness of the “mass” $M_{B^{ε}}$ of $B^{ε}$ , namely, using (1.4) and the fact that $d^{ε} = o (ε)$ , we obtain: $\begin{array}{rcl} M_{B^{ε}} & : = & \int_{⋃_{i} B_{i}^{ε}} b^{ε} (x) d x \\ = & β^{ε} \sum_{i} | B_{i}^{ε} | \\ = & β^{ε} {(R^{ε} - d^{ε})}^{n} ϰ_{n} N (ε) \\ = & r^{ε} \frac{{(R ε - d^{ε})}^{n} N (ε)}{R^{n} ε} \sim r | Γ | as ε \to 0 . \end{array}$

In spite of the fact that $A^{ε}$ contains many parameters the behaviour of its spectrum as $ε \to 0$ depends essentially only on q being finite or infinite and r being positive or zero.

At this point we present the results in a formal way; more precise statements are formulated in the next section using the language of operator theory. Below the convergence of spectra is understood in the Hausdorff sense, see Definition 2.1. One has:

Let $q < \infty$ . In this case $σ (A^{ε})$ converges, as $ε \to 0$ , to the union of the point q and the set of eigenvalues of the λ-nonlinear spectral problem $\begin{array}{l} (1.8) & \{\begin{matrix} - Δ u = λ u & in Ω ∖ Γ, \\ [u] = 0, [\frac{\partial u}{\partial n}] = \frac{λ q r}{q - λ} u & on Γ, \\ u = 0 & on \partial Ω, \end{matrix} \end{array}$ where the brackets denote the jump of the enclosed quantities across Γ.

If $r > 0$ then the set of eigenvalues of problem (1.8) consists of two ascending sequences – one of them tends to infinity and the second one tends to q (in the case $q = 0$ the second sequence is not present).

We notice that in the case $Γ \subset \partial Ω$ the interface conditions in (1.8) reduce formally to the boundary conditions $\begin{array}{l} (1.9) & \frac{\partial u}{\partial n} = F (λ) u, \end{array}$ where n in the outward pointing unit normal to $\partial Ω$ , $F (λ) = \frac{λ q r}{q - λ}$ . Boundary conditions of the form (1.9) appear in limit boundary value problems for various homogenization problems – see, e.g., [26,31] (here $F$ has infinitely many poles), [7] (here $F$ has one pole).

Let $q = \infty$ . In this case $σ (A^{ε})$ converges to the set of eigenvalues of the following Steklov-type spectral problem: $\begin{array}{l} (1.10) & \{\begin{matrix} - Δ u = λ u & in Ω ∖ Γ, \\ [u] = 0, [\frac{\partial u}{\partial n}] = λ r u & on Γ, \\ u = 0 & on \partial Ω . \end{matrix} \end{array}$ Note, that (1.10) is a formal limit of (1.8) as $q \to \infty$ .

The following estimate is valid: $\begin{array}{l} (1.11) & sup_{k \in N} (\underset{ε \to 0}{lim^{‾}} λ_{k}^{ε}) ⩽ q, \end{array}$ where $λ_{k}^{ε}$ is the kth eigenvalue of $A^{ε}$ . From (1.11) we immediately obtain $\begin{matrix} \forall k : λ_{k}^{ε} \to 0 as ε \to 0 provided q = 0 . \end{matrix}$

Note, that in the case $r = 0$ the problems (1.8) and (1.10) reduce to the eigenvalue problem for the Dirichlet Laplacian in Ω.

Also we note, that instead of Dirichlet conditions on $\partial Ω$ one can consider Neumann ones. In this case the limit spectral problem will have either the form (1.8) or (1.10), but with Neumann conditions on $\partial Ω$ instead of Dirichlet ones. The proofs are completely analogous. The same is valid for mixed conditions too. The change of boundary conditions on $\partial Ω$ will influence the spectral threshold – in the Dirichlet case it is strictly positive (provided $q > 0$ ), while in the Neumann case it is equal to zero; nevertheless the structure of the spectrum of the limit problem remains the same (cf. Remark 3.2).

In the last part of the paper we consider the same problem for a waveguide-like domain Ω: $\begin{matrix} Ω = R \times (d_{-}, d_{+}), Γ = {x = (x_{1}, 0) : x_{1} \in R} . \end{matrix}$ where $d_{-} < 0$ , $d_{+} > 0$ . Here we are interested in the case $q > 0$ , $r > 0$ only (i.e., we deal with point B on Fig. 2).

Due to the periodicity of $A^{ε}$ its spectrum is a locally finite union of compact intervals (bands). In general the bands may overlap and the natural question arising here is whether gaps open up in the spectrum (i.e., whether there is an open interval $(α, β) \subset (0, \infty)$ such that $(α, β) \cap σ (A^{ε}) = \emptyset$ , while $α, β \in σ (A^{ε})$ ). This problem is interesting for applications since the presence of gaps is important for the description of wave processes which are governed by the differential operators under consideration. Namely, if the wave frequency belongs to a gap, then the corresponding wave cannot propagate in the medium without attenuation. This feature is a dominant requirement for so-called photonic crystals which are materials with periodic dielectric structure attracting much attention in recent years (see, e.g., [16,25]).

It was proved in [22] that in the case $Ω = R^{n}$ and a bulk distribution of shells, the spectrum of $A^{ε}$ has a gap when ε is small enough.

Our goal is to study whether gaps will open up in case of our waveguide-like domain. We will prove that the spectrum of $A^{ε}$ converges in the Hausdorff sense to the spectrum of some operator $A$ which is defined by the same differential expression as in the case of a bounded domain, and its spectrum is as follows: if $q < min {\frac{π^{2}}{d_{-}^{2}}, \frac{π^{2}}{d_{+}^{2}}}$ then $\begin{matrix} σ (A) = [α_{1}, q] \cup [α_{2}, \infty), \end{matrix}$ otherwise $σ (A) = [α_{1}, \infty)$ . Here $α_{1}$ , $α_{2}$ are some positive numbers satisfying $0 < α_{1} < q < α_{2}$ . Thus if the waveguide is thin enough (i.e., if $d_{-} < π q^{- 1 / 2}$ and $d_{+} < π q^{- 1 / 2}$ ) we have a gap in the spectrum of $A$ (and, consequently, $σ (A^{ε})$ has a gap when ε is small enough).

Periodic perturbations of the Laplacian in wavegide-like domains leading to opening of spectral gaps were also studied in [3–5,8,9,19,33,34,41]. In all these papers (except [4,19]) spectral gaps appear because of a perturbation of the boundary of the waveguide (for example by making small holes periodically distributed along the waveguide [9] or by dividing the waveguide into two parts coupled by a periodic system of small windows [5]). In the paper [4] the authors considered small perturbations of the Laplace operator in a cylindrical domain by second-order differential operators with periodic coefficients; they gave sufficient conditions on this perturbation for gap opening. These conditions are not valid for the operators considered in the present work. In the paper [19] the authors perturbed the Laplace operator by a singular potential supported by a family of periodically distributed surfaces.

Let us note that we restrict ourselves to the two-dimensional case since our study is originally motivated by applications to photonic crystals. Namely, it is well-known (see, e.g., [16, Chapter 1]), that in a $3 D$ crystal, which is homogeneous in one direction, the propagation of suitably polarized electro-magnetic waves is governed by scalar $2 D$ operators of the form (1.3) posed on crystal cross-section.

Nevertheless, it will be visible from the proof that there are no mathematical peculiarities in the two-dimensional case and the results remain qualitatively the same for higher dimensions.

The paper is organized as follows. In Section 2 we set the problem and formulate the main result (Theorem 2.1), also we prove the estimate (1.11) (Theorem 2.2). Theorem 2.1 will be proven in Section 3: the case $q < \infty$ in Section 3.2 and the case $q = \infty$ in Section 3.4. Finally, in Section 4 we consider the case of a waveguide.

2. Setting of the problem and main result

2.1. The operator $A^{ε}$

In what follows we denote the Cartesian coordinates in $R^{n}$ by $x = (x_{1}, \dots, x_{n})$ . Let $Ω \subset R^{n}$ be a bounded Lipschitz domain ( $n ⩾ 2$ ). It is supposed that $0 \in Ω$ . We denote by Γ the intersection of Ω with the hyperplane ${x_{n} = 0}$ : $\begin{matrix} Γ = {x \in Ω : x_{n} = 0} . \end{matrix}$

Let $ε > 0$ be a small parameter. We denote by $x^{i, ε}$ , $i = (i_{1}, \dots, i_{n}) \in Z^{n - 1}$ the family of points, distributed periodically, with period ε, on the plane ${x_{n} = 0}$ : $\begin{matrix} x^{i, ε} = (ε i_{1}, ε i_{2}, \dots, ε i_{n - 1}, 0) \end{matrix}$ and introduce the following sets (see Fig. 1): $\begin{array}{l} D_{i}^{ε} = {x \in R^{n} : R^{ε} - d^{ε} < | x - x^{i, ε} | < R^{ε}}, B_{i}^{ε} = {x \in R^{n} : | x - x^{i, ε} | < R^{ε} - d^{ε}} . \end{array}$ Here $\begin{matrix} R^{ε} = R ε, where R \in (0, \frac{1}{2}), q d^{ε} = o (ε) as ε \to 0 . \end{matrix}$

We denote by $I^{ε}$ the set of all multiindices $i \in Z^{n - 1}$ satisfying $\begin{array}{l} (2.1) & x^{i, ε} \in Γ, dist (x^{i, ε}, \partial Ω) ⩾ ε \frac{\sqrt{n}}{2} . \end{array}$ The inequality in (2.1) implies that the cube with center at $x^{i, ε}$ , side length ε, being oriented along the coordinate axes, belongs to $\overline{Ω}$ whenever $i \in J^{ε}$ . This condition is technical and is needed only to simplify the proof presentation.

We introduce the piecewise constant functions $\begin{matrix} (2.2) & \begin{matrix} a^{ε} (x) = \{\begin{matrix} 1, & x \in Ω ∖ ⋃_{i \in I^{ε}} D_{i}^{ε}, \\ α^{ε}, & x \in ⋃_{i \in I^{ε}} D_{i}^{ε}, \end{matrix} \\ b^{ε} (x) = \{\begin{matrix} 1, & x \in Ω ∖ ⋃_{i \in I^{ε}} B_{i}^{ε}, \\ β^{ε}, & x \in ⋃_{i \in I^{ε}} B_{i}^{ε}, \end{matrix} \end{matrix} \end{matrix}$ where $α^{ε}$ , $β^{ε}$ are positive constants.

Now, let us define accurately the operator formally given by $\begin{matrix} A^{ε} = - \frac{1}{b^{ε}} div (a^{ε} \nabla) \end{matrix}$ (subject to Dirichlet boundary conditions on $\partial Ω$ ). By $H^{ε}$ we denote the Hilbert space of functions from $L^{2} (Ω)$ endowed with a scalar product $\begin{array}{l} (2.3) & {(u, v)}_{H^{ε}} = \int_{Ω} u (x) \overline{v (x)} b^{ε} (x) d x . \end{array}$ By $η^{ε}$ we denote the sesquilinear form in $H^{ε}$ defined by $\begin{array}{l} (2.4) & η^{ε} [u, v] = \int_{Ω} a^{ε} (x) \nabla u \cdot \nabla \bar{v} d x \end{array}$ with $dom (η^{ε}) = H_{0}^{1} (Ω)$ . The form $η^{ε}$ is densely defined, closed, positive and symmetric, whence (cf. [21, Chapter 6, Theorem 2.1]) there exists a unique self-adjoint and positive operator $A^{ε}$ associated with the form $η^{ε}$ , i.e., $\begin{array}{l} {(A^{ε} u, v)}_{H^{ε}} = η^{ε} [u, v], \forall u \in dom (A^{ε}), \forall v \in dom (η^{ε}) . \end{array}$

The domain of $A^{ε}$ consists of all functions $u \in H_{0}^{1} (Ω)$ with the corresponding restrictions belonging to the spaces $H^{2} (D_{i}^{ε})$ , $H^{2} (B_{i}^{ε})$ (for all $i \in I^{ε}$ ), $H^{2} (Ω ∖ ⋃_{i \in I^{ε}} (\overline{D_{i}^{ε} \cup B_{i}^{ε}}))$ , and satisfying the following conditions on $\partial D_{i}^{ε}$ : $\begin{array}{l} (2.5) & \{\begin{matrix} {(u)}^{+} = {(u)}^{-} and {(\frac{\partial u}{\partial n})}^{+} = α^{ε} {(\frac{\partial u}{\partial n})}^{-}, & x \in \partial D_{i}^{ε} ∖ \partial B_{i}^{ε}, \\ {(u)}^{+} = {(u)}^{-} and α^{ε} {(\frac{\partial u}{\partial n})}^{+} = {(\frac{\partial u}{\partial n})}^{-}, & x \in \partial B_{i}^{ε}, \end{matrix} \end{array}$ where + (respectively, −) denote the traces of the function u and its normal derivative taken from the exterior (respectively, interior) side of either $\partial D_{i}^{ε} ∖ \partial B_{i}^{ε}$ or $\partial B_{i}^{ε}$ .

The spectrum $σ (A^{ε})$ of the operator $A^{ε}$ is purely discrete. Our goal is to describe the behaviour of $σ (A^{ε})$ as $ε \to 0$ under the assumption that conditions (1.5)–(1.7) hold.

2.2. The limit operator

Next we introduce the limit operator $A_{q, r}$ .

Let $r > 0$ . We denote by $H_{r}$ the Hilbert space of functions from $L^{2} (Ω) \oplus L^{2} (Γ)$ endowed with the scalar product $\begin{array}{l} (2.6) & {(U, V)}_{H_{r}} = \int_{Ω} u_{1} (x) \overline{v_{1} (x)} d x + r \int_{Γ} u_{2} (x) \overline{v_{2} (x)} d s, U = (u_{1}, u_{2}), V = (v_{1}, v_{2}) . \end{array}$ Hereinafter, we use the standard notation $d s$ for the density of the measure generated on Γ (or any other $(n - 1)$ -dimensional hypersurface) by the Euclidean metric in $R^{n}$ .

For $q < \infty$ we introduce the sesquilinear form $η_{q, r}$ in $H_{r}$ by the formula $\begin{array}{l} (2.7) & η_{q, r} [U, V] = \int_{Ω} \nabla u_{1} \cdot \nabla \overline{v_{1}} d x + q r \int_{Γ} (u_{1} - u_{2}) (\overline{v_{1} - v_{2}}) d s, U = (u_{1}, u_{2}), V = (v_{1}, v_{2}) \end{array}$ with $dom (η_{q, r}) = H_{0}^{1} (Ω) \oplus L^{2} (Γ)$ . Here we use the same notation for the functions $u_{1}$ , $v_{1}$ and their traces on Γ.

For $q = \infty$ we introduce the sesquilinear form $η_{\infty, r}$ in $H_{r}$ by the formula $\begin{array}{l} η_{\infty, r} [U, V] = \int_{Ω} \nabla u_{1} \cdot \nabla \overline{v_{1}} d x, U = (u_{1}, u_{2}), V = (v_{1}, v_{2}) \end{array}$ with $dom (η_{\infty, r}) = {U = (u_{1}, u_{2}) \in H_{0}^{1} (Ω) \oplus L^{2} (Γ) : u_{1} |_{Γ} = u_{2} on Γ}$ .

The forms $η_{q, r}$ ( $q < \infty$ ) and $η_{\infty, r}$ are densely defined, closed, positive and symmetric. Then we defined the limit operator $A_{q, r}$ by $\begin{matrix} A_{q, r} = \{\begin{matrix} the operator acting in H_{r} and associated with the form η_{q, r}, & q < \infty, r > 0, \\ the operator acting in H_{r} and associated with the form η_{\infty, r}, & q = \infty, r > 0, \\ (- Δ_{Ω}) \oplus q I, acting in L^{2} (Ω) \oplus L^{2} (Γ), & q < \infty, r = 0, \\ (- Δ_{Ω}), acting in L^{2} (Ω), & q = \infty, r = 0, \end{matrix} \end{matrix}$ where $Δ_{Ω}$ is the Dirichlet Laplacian in Ω with domain $H^{2} (Ω) \cap H_{0}^{1} (Ω)$ , I is the identity operator.

2.3. Spectrum of the operator $A_{q, r}$

Let $q < \infty$ , $r > 0$ . If $q = 0$ then $A_{q, r}$ is a direct sum of the operator $- Δ_{Ω}$ and the null operator in $L^{2} (Γ)$ , implying $σ (A_{0, r}) = σ (- Δ_{Ω}) \cup {0}$ . In the case $q > 0$ one has the following lemma.

Lemma 2.1.
Let $q > 0$ . Then the spectrum of the operator $A_{q, r}$ has the form $\begin{array}{l} σ (A_{q, r}) = {q} \cup (⋃_{k \in N} {λ_{k}^{-}}) \cup (⋃_{k \in N} {λ_{k}^{+}}) . \end{array}$ The points $λ_{k}^{\pm}$ , $k \in N$ belong to the discrete spectrum, q is a point of the essential spectrum and $\begin{array}{l} (2.8) & 0 < λ_{1}^{+} ⩽ λ_{2}^{+} ⩽ \dots ⩽ λ_{k}^{+} ⩽ \dots \underset{k \to \infty}{\to} q < λ_{1}^{-} ⩽ λ_{2}^{-} ⩽ \dots ⩽ λ_{k}^{-} ⩽ \dots \underset{k \to \infty}{\to} \infty . \end{array}$

We will prove this lemma in Section 3.3.

It is easy to see that the operator $A_{\infty, r}$ ( $r > 0$ ) has compact resolvent in view of the trace theorem and the Rellich embedding theorem. Therefore the spectrum of $A_{\infty, r}$ is purely discrete.

Finally, $\begin{matrix} σ (A_{q, 0}) = σ (- Δ_{Ω}) \cup {q} . \end{matrix}$ Remark 2.1.
Let $r > 0$ . It is easy to see that
if $q < \infty$ then $\begin{array}{rcl} λ is an eigenvalue of A_{q, r}, U = (u_{1}, u_{2}) is the corresponding eigenfunction \\ ⟺ u_{2} = \frac{q u_{1} |_{Γ}}{q - λ} and (λ, u_{1}) is an eigenpair of (1.8); \end{array}$

if $q = \infty$ then $\begin{array}{rcl} λ is an eigenvalue of A_{\infty, r}, U = (u_{1}, u_{2}) is the corresponding eigenfunction \\ ⟺ u_{2} = u_{1} |_{Γ} and (λ, u_{1}) is an eigenpair of (1.10) . \end{array}$
Note, that problem (1.10) is a formal limit of problem (1.8) as $q \to \infty$ ; that justifies the notation $A_{\infty, r}$ . Also, setting $r = 0$ in (1.8) or (1.10), we arrive at the eigenvalue problem for the Dirichlet Laplacian in Ω; that justifies the notation $A_{q, 0}$ .

2.4. The main results

In what follows speaking about the convergence of spectra we will use the concept of Hausdorff convergence.

Definition 2.1.
The Hausdorff distance between two compact sets $X, Y \subset R$ is defined as follows: $\begin{matrix} {dist}_{H} (X, Y) : = max {sup_{x \in X} inf_{y \in Y} | x - y |; sup_{y \in Y} inf_{x \in X} | y - x |} . \end{matrix}$ The sequence of compact sets $X^{ε} \subset R$ converges to the compact set $X \subset R$ in the Hausdorff sense if $\begin{matrix} {dist}_{H} (X^{ε}, X) \to 0 as ε \to 0 . \end{matrix}$

Now, we are in position to formulate the main result of this paper.
Theorem 2.1.
Let $l \subset R$ be an arbitrary compact interval. Then the set $σ (A^{ε}) \cap l$ converges in the Hausdorff sense as $ε \to 0$ to the set $σ (A_{q, r}) \cap l$ .
Remark 2.2.
It is straightforward to show that the claim of Theorem 2.1 is equivalent to the following two properties: $\begin{array}{l} (A) & if λ^{ε} \in σ (A^{ε}) and lim_{ε \to 0} λ^{ε} = λ then λ \in σ (A_{q, r}), \\ (B) & for any λ \in σ (A_{q, r}) there exist λ^{ε} \in σ (A^{ε}) such that lim_{ε \to 0} λ^{ε} = λ . \end{array}$

Before starting the proof of Theorem 2.1 we obtain an estimate concerning the behaviour of the kth eigenvalue of $A^{ε}$ . Note, that in the case $q > 0$ , $r > 0$ this estimate follows easily from Theorem 2.1 and Lemma 2.1. Theorem 2.2.
One has $\begin{array}{l} (2.9) & sup_{k \in N} (\underset{ε \to 0}{lim^{‾}} λ_{k}^{ε}) ⩽ q, \end{array}$ where ${λ_{k}^{ε}}_{k \in N}$ is the sequence of eigenvalues of the operator $A^{ε}$ written in ascending order and repeated according to multiplicity.
Proof.
Below we assume that $q < \infty$ , otherwise the theorem is trivial. One has the following min–max principle (cf. [15, §4.5]): $\begin{array}{l} (2.10) & λ_{k}^{ε} = inf_{L \in L_{k}} (sup_{0 \neq v \in L} \frac{η^{ε} [v, v]}{‖ v ‖_{H^{ε}}^{2}}) . \end{array}$ Here $L_{k}$ is the set of all k-dimensional subspaces of $dom (η^{ε}) = H_{0}^{1} (Ω)$ .

We fix k arbitrary pairwise different indices $i_{j}^{ε} \in I^{ε}$ , $j = 1, \dots, k$ and introduce the following continuous and piecewise smooth functions: $\begin{array}{l} v_{j}^{ε} (x) = \{\begin{matrix} 1, & x \in B_{i_{j}^{ε}}^{ε}, \\ G (| x - x^{i, ε} |), & x \in D_{i_{j}^{ε}}^{ε}, \\ 0, & otherwise . \end{matrix} \end{array}$ where the function $G : R \to R$ is defined by $\begin{matrix} G (ρ) = \{\begin{matrix} \frac{ρ^{2 - n} - {(R^{ε})}^{2 - n}}{{(R^{ε} - d^{ε})}^{2 - n} - {(R^{ε})}^{2 - n}}, & n > 2, \\ \frac{ln ρ - ln R^{ε}}{ln (R^{ε} - d^{ε}) - ln R^{ε}}, & n = 2 . \end{matrix} \end{matrix}$

We denote $\begin{matrix} L^{'} : = span {v_{j}^{ε}, j = 1, \dots, k} . \end{matrix}$ Obviously, $L^{'} \subset H_{0}^{1} (Ω)$ . Moreover, since $supp (v_{j_{1}}^{ε}) \cap supp (v_{j_{2}}^{ε}) = \emptyset$ as $j_{1} \neq j_{2}$ , we conclude that $dim (L^{'}) = k$ . Therefore $L^{'} \in L_{k}$ .

Taking into account that $d^{ε} = o (ε)$ we obtain the following asymptotics as $ε \to 0$ : $\begin{array}{l} η^{ε} [v_{j}^{ε}, v_{j}^{ε}] = \{\begin{matrix} α^{ε} (n - 2) {({(R^{ε} - d^{ε})}^{2 - n} - {(R^{ε})}^{2 - n})}^{- 1} ω_{n - 1}, & n > 2 \\ α^{ε} {(ln R^{ε} - ln (R^{ε} - d^{ε}))}^{- 1}, & n = 2 \end{matrix}\} \sim \frac{α^{ε} R^{n - 1} ω_{n - 1} ε^{n - 1}}{d^{ε}}, \\ {‖ v_{j}^{ε} ‖}_{H^{ε}}^{2} = β^{ε} R^{n} ε^{n} ϰ_{n} + O (d^{ε} ε^{n - 1}), \end{array}$ where $ω_{n - 1}$ is the area of the $(n - 1)$ -dimensional unit sphere. Taking into account that $ω_{n - 1} = n ϰ_{n}$ and using (1.5), we obtain easily: $\begin{array}{l} (2.11) & \frac{η^{ε} [v_{j}^{ε}, v_{j}^{ε}]}{‖ v_{j}^{ε} ‖_{H^{ε}}^{2}} = \frac{α^{ε} n}{R d^{ε} ε β^{ε} (1 + O (\frac{d^{ε}}{ε β^{ε}}))} = \frac{α^{ε} n}{R d^{ε} ε β^{ε} (1 + q^{ε} \frac{{(d^{ε})}^{2}}{α^{ε}} O (1))} \sim q as ε \to 0 . \end{array}$ Since the supports of $v_{j}^{ε}$ are pairwise disjoint, (2.11) holds true for any arbitrary $v \in L^{'}$ instead of $v_{j}^{ε}$ .

Finally, using (2.10) and (2.11), we obtain $\begin{array}{l} λ_{k}^{ε} ⩽ sup_{v \in L^{'}} \frac{η {[v, v]}_{L^{2} (Ω)}^{2}}{‖ v ‖_{H^{ε}}^{2}} \sim q . \end{array}$ The theorem is proved. □
Corollary 2.1.
For each $k \in N$ $λ_{k}^{ε} \to 0$ as $ε \to 0$ provided $q = 0$ .
Remark 2.3.
From Corollary 2.1 one may see that the Hausdorff limit and indexwise limit (i.e., the set ${{lim}_{ε \to 0} λ_{k}^{ε}, k \in N}$ ) can be essentially different. Usually, such peculiar behavior arises, when the limit operator has non-empty essential spectrum – see, e.g., [30] and references therein.

In the current work we do not treat the question of the indexwise convergence of $σ (A^{ε})$ for $q \neq 0$ ; nevertheless, following the known results (for example, the paper of one of the authors [23]) we can make the following conjecture:
$q = \infty$ : $λ_{k}^{ε} \to λ_{k}$ for each $k \in N$ , where $λ_{k}$ is the kth eigenvalue of the operator $A_{\infty, r}$ ,

$0 < q < \infty$ , $r > 0$ : $λ_{k}^{ε} \to λ_{k}^{+}$ (see Lemma 2.1) for each $k \in N$ ,

$0 < q < \infty$ , $r = 0$ : $λ_{k}^{ε} \to λ_{k}$ for $1 ⩽ k ⩽ k^{'}$ and $λ_{k} \to q$ for $k > k^{'}$ , where $λ_{k}$ is the kth eigenvalue of the Dirichlet Laplacian in Ω and $k^{'} = max {k \in N : λ_{k} < q}$ .

Note, that if $λ^{ε} = λ_{k (ε)}^{ε} \to λ$ as $ε \to 0$ and $λ > q$ then, due to (2.9), $k (ε) \to \infty$ as $ε \to 0$ .

3. Proof of Theorem 2.1

We present the proof for the case $n ⩾ 3$ only. For the case $n = 2$ the proof needs some small modifications (for example in (3.22) the function $| x - x^{i, ε} |^{2 - n}$ has to be replaced by $- ln | x - x^{i, ε} |$ ).

3.1. Preliminaries

In what follows by $C, C_{1}, C_{2}, \dots$ we denote generic constants that do not depend on ε.

By ${⟨ u ⟩}_{B}$ we denote the mean value of the function $u (x)$ over the domain B: $\begin{matrix} {⟨ u ⟩}_{B} = \frac{1}{| B |} \int_{B} u (x) d x . \end{matrix}$ If $Σ \subset R^{n}$ is an $(n - 1)$ -dimensional surface then the Euclidean metric in $R^{n}$ induces on Σ the Riemannian metric and measure. As before we denote by $d s$ the density of this measure, and by ${⟨ u ⟩}_{Σ}$ the mean value of the function u over Σ, i.e., ${⟨ u ⟩}_{Σ} = \frac{1}{| Σ |} \int_{Σ} u d s$ , $| Σ | = \int_{Σ} d s$ .

We introduce the following sets (the sets $D_{i}^{ε}$ , $B_{i}^{ε}$ were introduced above in Section 2):

$Ω^{ε} = Ω ∖ ⋃_{i \in I^{ε}} (\overline{D_{i}^{ε} \cup B_{i}^{ε}})$ ,

$S_{i}^{ε, +} = {x \in R^{n} : | x - x^{i, ε} | = R^{ε}}$ ,

$S_{i}^{ε, -} = {x \in R^{n} : | x - x^{i, ε} | = R^{ε} - d^{ε}}$ ,

$Y_{i}^{ε} = {x = (x_{1}, \dots, x_{n}) \in R^{n} : | x_{k} - {(x^{i, ε})}_{k} | < \frac{ε}{2}, k = 1, \dots, n}$ , where ${(x^{i, ε})}_{k}$ is the kth coordinate of $x^{i, ε}$ ,

$Γ_{i}^{ε} = Y_{i}^{ε} \cap Γ$ .

One has $\begin{array}{l} (3.1) & ⋃_{i \in I^{ε}} Y_{i}^{ε} \subset Ω, \\ (3.2) & ⋃_{i \in I^{ε}} Γ_{i}^{ε} \subset Γ, lim_{ε \to 0} | Γ ∖ ⋃_{i \in I^{ε}} Γ_{i}^{ε} | = 0 \end{array}$ (recall that the set $I^{ε}$ consists of $i \in Z^{n - 1}$ satisfying $x^{i, ε} \in Γ$ and $dist (x^{i, ε}, \partial Ω ∖ Γ) ⩾ ε \frac{\sqrt{n}}{2}$ , whence one can easily obtain (3.1)–(3.2)).

By $u_{1}^{ε}, u_{2}^{ε}, \dots, u_{k}^{ε} \dots$ we denote a sequence of eigenfunctions of $A^{ε}$ corresponding to the non-decreasing sequence ${λ_{k}^{ε}}_{k \in N}$ of eigenvalues and normalized by the condition $\begin{matrix} {(u_{k}^{ε}, u_{l}^{ε})}_{H^{ε}} = δ_{k l} . \end{matrix}$

The following lemmata will be frequently used throughout the proof.

Lemma 3.1.
One has the following estimates: $\forall u \in H^{1} (Y_{i}^{ε})$ $\begin{array}{l} (3.3) & {| {⟨ u ⟩}_{S_{i}^{ε, +}} - {⟨ u ⟩}_{Γ_{i}^{ε}} |}^{2} ⩽ C ε^{2 - n} ‖ \nabla u ‖_{Y_{i}^{ε}}^{2}, \\ (3.4) & {| {⟨ u ⟩}_{S_{i}^{ε, -}} - {⟨ u ⟩}_{B_{i}^{ε}} |}^{2} ⩽ C ε^{2 - n} ‖ \nabla u ‖_{B_{i}^{ε}}^{2} . \end{array}$
Proof.
Using a standard trace inequality and rescaling arguments one can easily obtain the following inequality: $\begin{array}{l} (3.5) & ‖ v ‖_{L^{2} (S_{i}^{ε, -})}^{2} ⩽ C (ε^{- 1} ‖ v ‖_{L^{2} (B_{i}^{ε})}^{2} + ε ‖ \nabla v ‖_{L^{2} (B_{i}^{ε})}^{2}), \forall v \in H^{1} (B_{i}^{ε}) . \end{array}$

We set $v : = u - {⟨ u ⟩}_{B_{i}^{ε}}$ . Using (3.5), the Cauchy–Schwarz inequality and the Poincaré inequality $\begin{matrix} {‖ u - {⟨ u ⟩}_{B_{i}^{ε}} ‖}_{L^{2} (B_{i}^{ε})}^{2} ⩽ C ε^{2} ‖ \nabla u ‖_{L^{2} (B_{i}^{ε})}^{2}, \end{matrix}$ we obtain: $\begin{array}{rcl} {| {⟨ u ⟩}_{S_{i}^{ε, -}} - {⟨ u ⟩}_{B_{i}^{ε}} |}^{2} & = & {| {⟨ v ⟩}_{S_{i}^{ε, -}} |}^{2} ⩽ \frac{1}{| S_{i}^{ε, -} |} ‖ v ‖_{L^{2} (S_{i}^{ε, -})}^{2} \\ ⩽ & C (ε^{- n} {‖ u - {⟨ u ⟩}_{B_{i}^{ε}} ‖}_{L^{2} (B_{i}^{ε})}^{2} + ε^{2 - n} ‖ \nabla u ‖_{L^{2} (B_{i}^{ε})}^{2}) ⩽ C_{1} ε^{2 - n} ‖ \nabla u ‖_{L^{2} (B_{i}^{ε})}^{2} \end{array}$ and (3.4) is proved.

In the same way we prove the estimates $\begin{matrix} (3.6) & \begin{matrix} {| {⟨ u ⟩}_{S_{i}^{ε, +}} - {⟨ u ⟩}_{Y_{i}^{ε}} |}^{2} ⩽ C ε^{2 - n} ‖ \nabla u ‖_{Y_{i}^{ε}}^{2}, \\ {| {⟨ u ⟩}_{Γ_{i}^{ε}} - {⟨ u ⟩}_{Y_{i}^{ε}} |}^{2} ⩽ C ε^{2 - n} ‖ \nabla u ‖_{Y_{i}^{ε}}^{2}, \end{matrix} \end{matrix}$ which gives (3.3). The lemma is proved. □
Lemma 3.2.
One has the following inequality: $\forall u \in H^{1} (D_{i}^{ε})$ $\begin{array}{l} (3.7) & {| {⟨ u ⟩}_{S_{i}^{ε, +}} - {⟨ u ⟩}_{S_{i}^{ε, -}} |}^{2} ⩽ C d^{ε} ε^{1 - n} ‖ \nabla u ‖_{L^{2} (D_{i}^{ε})}^{2} . \end{array}$
Proof.
By density arguments it is enough to prove this lemma only for smooth functions.

We introduce in $D_{i}^{ε}$ the spherical coordinates $(ρ, Θ)$ , where $ρ \in (R^{ε} - d^{ε}, R^{ε})$ is the distance to $x^{i, ε}$ , Θ are the angle coordinates. By $S_{n - 1}$ we denote the $(n - 1)$ -dimensional unit sphere, by $d Θ$ we denote the Riemannian measure on $S_{n - 1}$ . One has $\begin{matrix} u (R^{ε}, Θ) - u (R^{ε} - d^{ε}, Θ) = \int_{R^{ε} - d^{ε}}^{R^{ε}} \frac{\partial u}{\partial ρ} (ρ, Θ) d ρ . \end{matrix}$ We integrate this equality over $S_{n - 1}$ (with respect to Θ), divide by $| S_{n - 1} |$ and square. Using the Cauchy–Schwarz inequality and taking into account that $d^{ε} = o (ε)$ , $R^{ε} = R ε$ , we obtain $\begin{array}{rcl} {| {⟨ u ⟩}_{S_{i}^{ε, +}} - {⟨ u ⟩}_{S_{i}^{ε, -}} |}^{2} & = & {| \frac{1}{| S_{n - 1} |} \int_{S_{n - 1}} \int_{R^{ε} - d^{ε}}^{R^{ε}} \frac{\partial u}{\partial ρ} (ρ, Θ) d ρ d Θ |}^{2} \\ ⩽ & C (\int_{S_{n - 1}} \int_{R^{ε} - d^{ε}}^{R^{ε}} {| \frac{\partial u}{\partial ρ} (ρ, Θ) |}^{2} ρ^{n - 1} d ρ d Θ) \cdot (\int_{R^{ε} - d^{ε}}^{R^{ε}} \frac{d ρ}{ρ^{n - 1}}) \\ ⩽ & C_{1} (\frac{1}{{(R^{ε} - d^{ε})}^{n - 2}} - \frac{1}{{(R^{ε})}^{n - 2}}) ‖ \nabla u ‖_{L^{2} (D_{i}^{ε})}^{2} ⩽ C_{2} d^{ε} ε^{1 - n} {‖ \nabla u^{ε} ‖}_{L^{2} (D_{i}^{ε})}^{2} . \end{array}$ The lemma is proved. □

3.2. Proof of Theorem 2.1: The case $q < \infty$

Recall that the claim of Theorem 2.1 is equivalent to the fulfillment of properties (A)–(B) (see Remark 2.2).

3.2.1. Proof of property (A)

Let $λ^{ε} \in σ (A^{ε})$ and $λ^{ε} \to λ$ as $ε \to 0$ . We have to prove that $λ \in σ (A_{q, r})$ .

We denote by $k (ε)$ the index corresponding to $λ^{ε}$ (i.e., $λ^{ε} = λ_{k (ε)}^{ε}$ ). By $u^{ε} = u_{k (ε)}^{ε} \in H_{0}^{1} (Ω)$ we denote the corresponding eigenfunction. One has $\begin{array}{l} (3.8) & 1 = {‖ u^{ε} ‖}_{H^{ε}} = {‖ u^{ε} ‖}_{L^{2} (Ω^{ε})}^{2} + \sum_{i \in I^{ε}} {‖ u^{ε} ‖}_{L^{2} (D_{i}^{ε})}^{2} + β^{ε} \sum_{i \in I^{ε}} {‖ \nabla u^{ε} ‖}_{L^{2} (B_{i}^{ε})}^{2}, \\ (3.9) & λ^{ε} = η^{ε} [u^{ε}, u^{ε}] = {‖ \nabla u^{ε} ‖}_{L^{2} (Ω^{ε})}^{2} + α^{ε} \sum_{i \in I^{ε}} {‖ \nabla u^{ε} ‖}_{L^{2} (D_{i}^{ε})}^{2} + \sum_{i \in I^{ε}} {‖ u^{ε} ‖}_{L^{2} (B_{i}^{ε})}^{2} . \end{array}$

In order to describe the behavior of $u^{ε}$ as $ε \to 0$ we need some additional operators. It is known (see, e.g., [28, Chapter 4, §4.2]) that there exists an extension operator $Π_{1}^{ε} : H^{1} (Ω^{ε}) \to H^{1} (Ω)$ uniformly bounded with respect to ε, i.e., the following conditions hold: $\begin{matrix} (3.10) & \begin{matrix} [Π_{1}^{ε} u] (x) = u (x), \forall x \in Ω^{ε}, \\ {‖ Π^{ε} u ‖}_{H^{1} (Ω)} ⩽ C ‖ u ‖_{H^{1} (Ω^{ε})}, \end{matrix} \end{matrix}$ where the constant C is independent of both u and ε.

Also we introduce the operator $Π_{2}^{ε} : L^{2} (⋃_{i} B_{i}^{ε}) \to L^{2} (Γ)$ : $\begin{array}{l} Π_{2}^{ε} u (x) = \{\begin{matrix} {⟨ u ⟩}_{B_{i}^{ε}} \sqrt{r^{ε}}, & x \in Γ_{i}^{ε}, \\ 0, & x \in Γ ∖ ⋃_{i} Γ_{i}^{ε} . \end{matrix} \end{array}$ Using the Cauchy–Schwarz inequality and the definition (1.7) of $r^{ε}$ we obtain $\begin{array}{rcl} {‖ Π_{2}^{ε} u ‖}_{L^{2} (Γ)}^{2} & ⩽ & r^{ε} \sum_{i \in I^{ε}} \frac{| Γ_{i}^{ε} |}{| B_{i}^{ε} |} \int_{B_{i}^{ε}} {| u (x) |}^{2} d x \\ ⩽ & C \sum_{i \in I^{ε}} \int_{B_{i}^{ε}} β^{ε} {| u (x) |}^{2} d x \\ (3.11) & ⩽ & C_{1} ‖ u ‖_{H^{ε}}^{2} . \end{array}$

Remark 3.1.
As we already mentioned in the introduction, the parameter q characterizes the “strength” of the coupling between $Ω^{ε}$ and the union of the balls $B_{i}^{ε}$ . If $q = \infty$ the coupling is strong: given a family ${w^{ε} \in H^{1} (Ω)}_{ε > 0}$ satisfying $\begin{array}{l} (3.12) & η^{ε} [w^{ε}, w^{ε}] ⩽ C \end{array}$ one can observe that the behaviour as $ε \to 0$ of $w^{ε}$ on $⋃_{i} B_{i}^{ε}$ is determined by the one in $Ω^{ε}$ . Namely (cf. (3.99)), $\begin{matrix} lim_{ε \to 0} {‖ \sqrt{r^{ε}} (Π_{1}^{ε} w^{ε}) |_{Γ} - Π_{2}^{ε} w^{ε} ‖}_{L^{2} (Γ)} = 0 . \end{matrix}$ On the other hand, if q is finite then the coupling is weak: for arbitrary smooth functions $w_{1}$ , $w_{2}$ on Γ there exists a family ${w^{ε} \in H^{1} (Ω)}_{ε > 0}$ satisfying (3.12) and $\begin{matrix} (Π_{1} w^{ε}) |_{Γ} \to w_{1}, Π_{2}^{ε} w^{ε} \to w_{2} in L^{2} (Γ) as ε \to 0 . \end{matrix}$ The function $w^{ε}$ can be constructed, for example, by the formula (3.27) below.

In view of (3.8)–(3.11) the functions $Π_{1}^{ε} u^{ε}$ and $Π_{2}^{ε} u^{ε}$ are bounded in $H^{1} (Ω)$ and $L^{2} (Γ)$ , respectively, uniformly in ε. Then, using the Rellich embedding theorem and the trace theorem, we conclude that there is a subsequence (for convenience, still denoted by $u_{ε}$ ) and $u_{1} \in H^{1} (Ω)$ , $u_{2} \in L^{2} (Γ)$ such that $\begin{array}{l} (3.13) & Π_{1}^{ε} u^{ε} \underset{ε \to 0}{⇀} u_{1} in H^{1} (Ω), \\ (3.14) & Π_{1}^{ε} u^{ε} \underset{ε \to 0}{\to} u_{1} in L^{2} (Ω), \\ (3.15) & Π_{1}^{ε} u^{ε} \underset{ε \to 0}{\to} u_{1} in L^{2} (Γ), \\ (3.16) & Π_{2}^{ε} u^{ε} \underset{ε \to 0}{⇀} u_{2} in L^{2} (Γ) \end{array}$ (here we use the same notation for the functions $u^{ε}$ , $u_{1}$ and their traces on Γ). It is clear that $Π_{1}^{ε} u^{ε} = 0$ on $\partial Ω$ , whence using the trace theorem we arrive at $u_{1} = 0$ on $\partial Ω$ , i.e., $u_{1} \in H_{0}^{1} (Ω)$ .

Now we consider separately two cases: $u_{1} \neq 0$ and $u_{1} = 0$ . Case 1 ( $u_{1} \neq 0$ ).

We will prove that in this case λ is an eigenvalue of the operator $A_{q, r}$ if $r > 0$ (respectively, of the operator $A_{q, 0}$ if $r = 0$ ) and $U = (u_{1}, r^{- 1 / 2} u_{2})$ (respectively, $U = (u_{1}, u_{2})$ ) is a corresponding eigenfunction.

For an arbitrary $w \in H_{0}^{1} (Ω)$ one has $\begin{array}{l} (3.17) & \int_{Ω} a^{ε} (x) \nabla u^{ε} (x) \cdot \nabla w (x) d x = λ^{ε} \int_{Ω} b^{ε} (x) u^{ε} (x) w (x) d x . \end{array}$ Our strategy of proof will be to plug into (3.17) some specially chosen test-function w depending on ε and then pass to the limit as $ε \to 0$ in order to obtain either the equality $A_{q, r} U = λ U$ ( $r > 0$ ) or the equality $A_{q, 0} U = λ U$ ( $r = 0$ ) written in a weak form.

For constructing this special test-function we introduce several additional functions. Let $Φ : R \to R$ be a smooth function such that $Φ (ρ) = 1$ for $ρ ⩽ 1$ and $Φ (ρ) = 0$ for $ρ ⩾ 2$ . For $i \in I^{ε}$ we denote $\begin{array}{l} (3.18) & φ_{i}^{ε} (x) = Φ (\frac{| x - x^{i, ε} | + \frac{ε}{2} - 2 R^{ε}}{\frac{ε}{2} - R^{ε}}) . \end{array}$ It is clear that $\begin{array}{l} (3.19) & φ_{i}^{ε} = 0 in R^{n} ∖ ⋃_{\in I^{ε}} Y_{i}^{ε}, φ_{i}^{ε} = 1 in \overline{D_{i}^{ε} \cup B_{i}^{ε}}, \\ (3.20) & supp (D^{m} φ_{i}^{ε}) \subset \overline{Y_{i}^{ε}} ∖ (D_{i}^{ε} \cup B_{i}^{ε}) (m \neq 0), \\ (3.21) & | D^{m} φ_{i}^{ε} | ⩽ \frac{C}{ε^{| m |}}, m = 1, 2, 3, \dots . \end{array}$

By $v_{i}^{ε} (x)$ we denote the following function: $\begin{array}{l} (3.22) & v_{i}^{ε} (x) = \{\begin{matrix} 1, & | x - x^{i, ε} | ⩽ R^{ε} - d^{ε}, \\ \frac{A^{ε}}{| x - x^{i, ε} |^{n - 2}} + B^{ε}, & R^{ε} - d^{ε} < | x - x^{i, ε} | < R^{ε}, \\ 0, & R^{ε} ⩽ | x - x^{i, ε} |, \end{matrix} \end{array}$ where $\begin{array}{l} (3.23) & A^{ε} = {(\frac{1}{{(R^{ε} - d^{ε})}^{n - 2}} - \frac{1}{{(R^{ε})}^{n - 2}})}^{- 1}, B^{ε} = - \frac{A^{ε}}{{(R^{ε})}^{n - 2}} . \end{array}$ Taking into account that $d^{ε} = o (ε)$ one gets the following asymptotics as $ε \to 0$ : $\begin{array}{l} (3.24) & A^{ε} \sim \frac{{(R^{ε})}^{n - 1}}{(n - 2) d^{ε}} . \end{array}$

It is easy to see that $v_{i}^{ε}$ , $i \in I^{ε}$ , are continuous and piecewise smooth functions. Using (1.5)–(1.7), (3.24) one can easily obtain the following asymptotics as $ε \to 0$ : $\begin{array}{l} (3.25) & \int_{Y_{i}^{ε}} a^{ε} {| \nabla v_{i}^{ε} |}^{2} d x = α^{ε} {(A^{ε})}^{2} (n - 2) (\frac{1}{{(R^{ε} - d^{ε})}^{n - 2}} - \frac{1}{{(R^{ε})}^{n - 2}}) ω_{n - 1} \sim q^{ε} r^{ε} ε^{n - 1}, \\ (3.26) & \int_{Y_{i}^{ε}} {| v_{i}^{ε} |}^{2} b^{ε} d x = β^{ε} | B_{i}^{ε} | + O (d^{ε} ε^{n - 1}) \sim r^{ε} ε^{n - 1} . \end{array}$

Finally, taking arbitrary functions $w_{1} \in C_{0}^{\infty} (Ω)$ , $w_{2} \in C^{\infty} (Γ)$ , we construct the following test function: $\begin{array}{l} (3.27) & w^{ε} (x) = w_{1} (x) + \sum_{i \in I^{ε}} (w_{1} (x^{i, ε}) - w_{1} (x)) φ_{i}^{ε} (x) + \sum_{i \in I^{ε}} v_{i}^{ε} (x) (\frac{w_{2} (x^{i, ε})}{\sqrt{r^{ε}}} - w_{1} (x^{i, ε})) . \end{array}$ It is clear that $w^{ε} \in H_{0}^{1} (Ω)$ .

We plug $w = w^{ε} (x)$ into (3.17). This gives, using (3.19)–(3.20), $\begin{array}{rcl} \underset{I_{1}}{\underset{︸}{\int_{Ω^{ε}} \nabla u^{ε} \cdot \nabla w_{1} d x}} + \underset{I_{2}}{\underset{︸}{\sum_{i \in I^{ε}} \int_{Y_{i}^{ε} ∖ \overline{D_{i}^{ε} \cup B_{i}^{ε}}} \nabla u^{ε} \cdot \nabla ((w_{1} (x^{i, ε}) - w_{1}) φ_{i}^{ε}) d x}} \\ + \underset{I_{3}}{\underset{︸}{\sum_{i \in I^{ε}} (\frac{w_{2} (x^{i, ε})}{\sqrt{r^{ε}}} - w_{1} (x^{i, ε})) \int_{D_{i}^{ε}} α^{ε} \nabla u^{ε} \cdot \nabla v_{i}^{ε} d x}} \\ = λ^{ε} (\underset{I_{4}}{\underset{︸}{\int_{Ω^{ε}} u^{ε} w_{1} d x}} + \underset{I_{5}}{\underset{︸}{\sum_{i \in I^{ε}} \int_{Y_{i}^{ε} ∖ \overline{D_{i}^{ε} \cup B_{i}^{ε}}} u^{ε} (w_{1} (x^{i, ε}) - w_{1}) φ_{i}^{ε} d x}} \\ (3.28) & + \underset{I_{6}}{\underset{︸}{\sum_{i \in I^{ε}} \int_{D_{i}^{ε}} u^{ε} w^{ε} d x}} + \underset{I_{7}}{\underset{︸}{\sum_{i \in I^{ε}} \int_{B_{i}^{ε}} β^{ε} u^{ε} \frac{w_{2} (x^{i, ε})}{\sqrt{r^{ε}}} d x}}) . \end{array}$

Let us study step-by-step the terms $I_{j}$ , $j = 1, \dots, 7$ .

1) Since ${lim}_{ε \to 0} | Ω ∖ Ω^{ε} | = 0$ and $‖ \nabla Π^{ε} u^{ε} ‖_{L^{2} (Ω ∖ Ω^{ε})} ⩽ C$ , we obtain, using (3.13), $\begin{array}{l} (3.29) & I_{1} = \int_{Ω} \nabla (Π_{1}^{ε} u^{ε}) \cdot \nabla w_{1} d x - \int_{Ω ∖ Ω^{ε}} \nabla (Π_{1}^{ε} u^{ε}) \cdot \nabla w_{1} d x \underset{ε \to 0}{\to} \int_{Ω} \nabla u_{1} \cdot \nabla w_{1} d x . \end{array}$

2) Using the estimates $\begin{matrix} | \nabla ((w_{1} (x^{i, ε}) - w_{1}) φ_{i}^{ε}) | ⩽ C, \sum_{i \in I^{ε}} ε^{n - 1} = \sum_{i \in I^{ε}} | Γ_{i}^{ε} | ⩽ | Γ | \end{matrix}$ (the first one follows easily from (3.19)–(3.21)) and taking into account that $‖ \nabla u^{ε} ‖_{L^{2} (Ω^{ε})} ⩽ C$ we conclude that $\begin{array}{l} (3.30) & | I_{2} |^{2} ⩽ C {‖ \nabla u^{ε} ‖}_{L^{2} (Ω^{ε})}^{2} | ⋃_{i \in I^{ε}} Y_{i}^{ε} | ⩽ C_{1} \sum_{i \in I^{ε}} ε^{n} ⩽ C_{2} ε . \end{array}$

3) Integrating by parts and taking into account that $Δ v_{i}^{ε} = 0$ in $D_{i}^{ε}$ we get: $\begin{array}{rcl} I_{3} & = & \sum_{i \in I^{ε}} (\frac{w_{2} (x^{i, ε})}{\sqrt{r^{ε}}} - w_{1} (x^{i, ε})) α^{ε} (\int_{S_{i}^{ε, +}} \frac{\partial v_{i}^{ε}}{\partial | x - x^{i, ε} |} u^{ε} d s - \int_{S_{i}^{ε, -}} \frac{\partial v_{i}^{ε}}{\partial | x - x^{i, ε} |} u^{ε} d s) \\ (3.31) & = & α^{ε} A^{ε} ω_{n - 1} (n - 2) \sum_{i \in I^{ε}} (w_{1} (x^{i, ε}) - \frac{w_{2} (x^{i, ε})}{\sqrt{r^{ε}}}) ({⟨ u^{ε} ⟩}_{S_{i}^{ε, +}} - {⟨ u^{ε} ⟩}_{S_{i}^{ε, -}}) . \end{array}$ Recall that by $ω_{n - 1}$ we denote the volume of the $(n - 1)$ -dimensional unit sphere, and $A^{ε}$ is defined by (3.23).

We introduce the operator $Q^{ε} : C^{1} (Γ) \to L^{2} (Γ)$ by $\begin{array}{l} (3.32) & Q^{ε} w = \{\begin{matrix} w (x^{i, ε}), & x \in Γ_{i}^{ε}, \\ 0, & x \in Γ ∖ ⋃_{i \in I^{ε}} Γ_{i}^{ε} . \end{matrix} \end{array}$ It is easy to see that $\begin{array}{l} (3.33) & \forall w \in C^{1} (Γ) : Q^{ε} w \underset{ε \to 0}{\to} w in L^{2} (Γ) . \end{array}$

We obtain from (3.31): $\begin{array}{rcl} I_{3} & = & α^{ε} A^{ε} ω_{n - 1} (n - 2) \sum_{i \in I^{ε}} (w_{1} (x^{i, ε}) - \frac{w_{2} (x^{i, ε})}{\sqrt{r^{ε}}}) ({⟨ Π_{1}^{ε} u^{ε} ⟩}_{Γ_{i}^{ε}} - {⟨ u^{ε} ⟩}_{B_{i}^{ε}}) + δ (ε) \\ (3.34) & = & \frac{α^{ε} A^{ε} ω_{n - 1} (n - 2)}{ε^{n - 1}} \int_{Γ} (Q^{ε} w_{1} - \frac{1}{\sqrt{r^{ε}}} Q^{ε} w_{2}) (Π_{1}^{ε} u^{ε} - \frac{1}{\sqrt{r^{ε}}} Π_{2}^{ε} u^{ε}) d s + δ (ε), \end{array}$ where the remainder $δ (ε)$ vanishes as $ε \to 0$ . Namely, applying (3.3) for $u : = Π_{1}^{ε} u^{ε}$ and (3.4) for $u : = u^{ε}$ and using the asymptotics $\begin{array}{l} (3.35) & \frac{α^{ε} A^{ε} ω_{n - 1} (n - 2)}{ε^{n - 1}} \sim q^{ε} r^{ε} as ε \to 0 \end{array}$ following from (3.24), we obtain (below by $N (ε)$ we denote the cardinality of the set $I^{ε}$ ; clearly, $N (ε)$ satisfies (1.4)): $\begin{array}{rcl} {| δ (ε) |}^{2} & ⩽ & C {(\frac{a^{ε} A^{ε}}{\sqrt{r^{ε}}})}^{2} N (ε) \sum_{i \in I^{ε}} ({| {⟨ u^{ε} ⟩}_{S_{i}^{ε, +}} - {⟨ Π_{1}^{ε} u^{ε} ⟩}_{Γ_{i}^{ε}} |}^{2} + {| {⟨ u^{ε} ⟩}_{S_{i}^{ε, -}} - {⟨ u^{ε} ⟩}_{B_{i}^{ε}} |}^{2}) \\ ⩽ & C_{1} ε {(q^{ε})}^{2} r^{ε} \sum_{i \in I^{ε}} ({‖ \nabla Π_{1}^{ε} u^{ε} ‖}_{L^{2} (Y_{i}^{ε})}^{2} + {‖ \nabla u^{ε} ‖}_{L^{2} (B_{i}^{ε})}^{2}) ⩽ C_{2} ε . \end{array}$ Thus we obtain, using (3.15), (3.16), (3.33)–(3.35) $\begin{array}{rcl} I_{3} & = & \frac{α^{ε} A^{ε} ω_{n - 1} (n - 2)}{ε^{n - 1}} \int_{Γ} (Q^{ε} w_{1} - \frac{1}{\sqrt{r^{ε}}} Q^{ε} w_{2}) (Π_{1}^{ε} u^{ε} - \frac{1}{\sqrt{r^{ε}}} Π_{2}^{ε} u^{ε}) d s + o (1) \\ (3.36) & \to & \int_{Γ} (q r u_{1} w_{1} - q \sqrt{r} u_{1} w_{2} - q \sqrt{r} u_{2} w_{1} + q u_{2} w_{2}) d s as ε \to 0 . \end{array}$

4) Similarly to $I_{1}$ we arrive at $\begin{array}{l} (3.37) & I_{4} \to \int_{Ω} u_{1} w_{1} d x as ε \to 0 \end{array}$

5), 6) In view of (1.5) and since $| (w_{1} (x^{i, ε}) - w_{1}) φ_{i}^{ε} | < C ε$ and $| w^{ε} | < \frac{C}{\sqrt{r^{ε}}}$ in $⋃_{i \in I^{ε}} D_{i}^{ε}$ we obtain $\begin{array}{rcl} | I_{5} + I_{6} |^{2} & ⩽ & C \sum_{i \in I^{ε}} (ε^{2} | Y_{i}^{ε} ∖ (D_{i}^{ε} \cup B_{i}^{ε}) | + \frac{| D_{i}^{ε} |}{r^{ε}}) \sum_{i \in I^{ε}} {‖ u^{ε} ‖}_{L^{2} (Y_{i}^{ε} ∖ B_{i}^{ε})}^{2} ⩽ C_{1} \sum_{i \in I^{ε}} (ε^{n + 2} + \frac{d^{ε} ε^{n - 1}}{r^{ε}}) \\ (3.38) & ⩽ & C_{2} (ε^{3} + \frac{α^{ε}}{{(d^{ε})}^{2}} q^{ε}) \to 0 as ε \to 0 . \end{array}$

7) Using the equality $| B_{i}^{ε} | = {(R^{ε} - d^{ε})}^{n} ϰ_{n} \sim R^{n} ϰ_{n} ε^{n}$ (recall that by $ϰ_{n}$ we denote the volume of the n-dimensional unit ball) we get: $\begin{array}{l} (3.39) & I_{7} = \frac{β^{ε} | B_{i}^{ε} |}{ε^{n - 1}} \sum_{i \in I^{ε}} {⟨ u^{ε} ⟩}_{B_{i}^{ε}} \frac{w_{2} (x^{i, ε})}{\sqrt{r^{ε}}} ε^{n - 1} = \frac{β^{ε} | B_{i}^{ε} |}{ε^{n - 1} r^{ε}} \int_{Γ} Π_{2}^{ε} u^{ε} Q^{ε} w_{2} d s \to \int_{Γ} u_{2} w_{2} d s . \end{array}$

Combining (3.28)–(3.30), (3.36)–(3.39) we get $\begin{array}{rcl} \int_{Ω} \nabla u_{1} \cdot \nabla w_{1} d x + \int_{Γ} (q r u_{1} w_{1} - q \sqrt{r} u_{1} w_{2} - q \sqrt{r} u_{2} w_{1} + q u_{2} w_{2}) d s \\ (3.40) & = λ (\int_{Ω} u_{1} w_{1} d x + \int_{Γ} u_{2} w_{2} d s), \forall w_{1} \in C_{0}^{\infty} (Ω), w_{2} \in C^{\infty} (Γ) . \end{array}$ By density arguments equality (3.40) is valid for any arbitrary $w_{1} \in H_{0}^{1} (Ω)$ and $w_{2} \in L^{2} (Γ)$ .

If $r > 0$ then (3.40) is equivalent to the equality $\begin{array}{l} η_{q, r} [U, W] = λ {(U, W)}_{H_{r}}, where U = (u_{1}, r^{- 1 / 2} u_{2}), W = (w_{1}, r^{- 1 / 2} w_{2}), \end{array}$ whence, obviously, $\begin{array}{l} U \in dom (A_{q, r}), A_{q, r} U = λ U . \end{array}$ Since $u_{1} \neq 0$ , λ is therefore an eigenvalue of the operator $A_{q, r}$ .

If $r = 0$ then (3.40) implies $\begin{array}{l} U = (u_{1}, u_{2}) \in dom (A_{q, 0}), A_{q, 0} U = λ U, \end{array}$ i.e., λ is an eigenvalue of the operator $A_{q, 0}$ .

Case 2 ( $u_{1} = 0$ ).

We will prove that in this case $λ = q$ (recall that for every $r ⩾ 0$ one has $q \in σ (A_{q, r})$ ).

We express the eigenfunction $u^{ε}$ in the form $\begin{array}{l} (3.41) & u^{ε} = v^{ε} - g^{ε} + δ^{ε}, \end{array}$ where $\begin{array}{l} (3.42) & v^{ε} = \sum_{i \in I^{ε}} {⟨ u^{ε} ⟩}_{B_{i}^{ε}} v_{i}^{ε}, g^{ε} = \sum_{k = 1}^{k (ε) - 1} {(v^{ε}, u_{k}^{ε})}_{H^{ε}} u_{k}^{ε}, \end{array}$ and $δ^{ε}$ is a remainder term. Here the functions $v_{i}^{ε}$ are again defined by (3.22)–(3.23). It is clear that $v^{ε} \in H_{0}^{1} (Ω)$ and $g^{ε} \in dom (A^{ε})$ . Also we note that $\begin{array}{l} (3.43) & v^{ε} = {⟨ u^{ε} ⟩}_{B_{i}^{ε}} in B_{i}^{ε}, v^{ε} = 0 in Ω^{ε} . \end{array}$

At first we obtain some estimates for the eigenfunction $u^{ε}$ . For any $u \in H^{1} (Y_{i}^{ε})$ one has the estimate [22, Lemma 4.3]: $\begin{array}{l} (3.44) & ‖ u ‖_{D_{i}^{ε}}^{2} ⩽ C ({(d^{ε})}^{2} ‖ \nabla u ‖_{L^{2} (D_{i}^{ε})}^{2} + ε d^{ε} ‖ \nabla u ‖_{L^{2} (Y_{i}^{ε} ∖ \overline{D_{i}^{ε} \cup B_{i}^{ε}})}^{2} + ε^{- 1} d^{ε} ‖ u ‖_{L^{2} (Y_{i}^{ε} ∖ \overline{D_{i}^{ε} \cup B_{i}^{ε}})}^{2}) . \end{array}$ Recall that $d^{ε} = o (ε)$ and ${(d^{ε})}^{2} = o (α^{ε})$ . Using this and (3.9), we obtain from (3.44): $\begin{array}{l} (3.45) & lim_{ε \to 0} \sum_{i \in I^{ε}} {‖ u^{ε} ‖}_{L^{2} (D_{i}^{ε})}^{2} = 0 . \end{array}$

Finally, using the Poincaré inequality and (3.9), we obtain: $\begin{array}{l} (3.46) & \sum_{i \in I^{ε}} {‖ u^{ε} - {⟨ u^{ε} ⟩}_{B_{i}^{ε}} ‖}_{L^{2} (B_{i}^{ε})}^{2} = O (ε^{2}) as ε \to 0 . \end{array}$ Since $ε β^{ε} = O (1)$ (in view of (1.7)), (3.46) implies $\begin{array}{l} (3.47) & \sum_{i \in I^{ε}} β^{ε} {‖ u^{ε} - {⟨ u^{ε} ⟩}_{B_{i}^{ε}} ‖}_{L^{2} (B_{i}^{ε})}^{2} = O (ε) as ε \to 0 . \end{array}$

Using the fact that $\begin{array}{l} (3.48) & {‖ u^{ε} ‖}_{L^{2} (Ω^{ε})} ⩽ {‖ Π_{1}^{ε} u^{ε} ‖}_{L^{2} (Ω)} \underset{ε \to 0}{\to} ‖ u_{1} ‖_{L^{2} (Ω)} = 0 \end{array}$ and (3.8), (3.45), (3.47) we obtain $\begin{array}{rcl} 1 & = & {‖ u^{ε} ‖}_{H^{ε}}^{2} = \sum_{i \in I^{ε}} {| {⟨ u^{ε} ⟩}_{B_{i}^{ε}} |}^{2} | B_{i}^{ε} | β^{ε} + o (1) \\ = & ε β^{ε} \frac{| B_{i}^{ε} |}{ε^{n}} \sum_{i \in I^{ε}} {| {⟨ u^{ε} ⟩}_{B_{i}^{ε}} |}^{2} ε^{n - 1} + o (1) (ε \to 0), \end{array}$ whence, taking into account that $ε β^{ε} \frac{| B_{i}^{ε} |}{ε^{n}} \sim \frac{ε β^{ε} {(R^{ε})}^{n} χ_{n}}{ε^{n}} \sim r^{ε}$ (here we use $d^{ε} = o (ε)$ ), $\begin{array}{l} (3.49) & r^{ε} \sum_{i \in I^{ε}} {| {⟨ u^{ε} ⟩}_{B_{i}^{ε}} |}^{2} ε^{n - 1} \sim 1 as ε \to 0 . \end{array}$

Using (1.5)–(1.7), (3.25), (3.26) and taking (3.49) into account we obtain the following estimates: $\begin{array}{l} (3.50) & η^{ε} [v^{ε}, v^{ε}] = \sum_{i \in I^{ε}} (\int_{Y_{i}^{ε}} a^{ε} (x) {| \nabla v_{i}^{ε} |}^{2} d x) {| {⟨ u^{ε} ⟩}_{B_{i}^{ε}} |}^{2} \sim q as ε \to 0, \\ (3.51) & {‖ v^{ε} ‖}_{H^{ε}}^{2} = \sum_{i \in I^{ε}} (\int_{Y_{i}^{ε}} b^{ε} (x) {| v_{i}^{ε} |}^{2} d x) {| {⟨ u^{ε} ⟩}_{B_{i}^{ε}} |}^{2} \sim 1 as ε \to 0, \\ (3.52) & \sum_{i \in I^{ε}} {‖ v^{ε} ‖}_{L^{2} (D_{i}^{ε})}^{2} ⩽ | D_{i}^{ε} | \sum_{i \in I^{ε}} {| {⟨ u^{ε} ⟩}_{B_{i}^{ε}} |}^{2} ⩽ C_{1} q^{ε} \frac{{(d^{ε})}^{2}}{α^{ε}} r^{ε} \sum_{i \in I^{ε}} {| {⟨ u^{ε} ⟩}_{B_{i}^{ε}} |}^{2} ε^{n - 1} \to 0 as ε \to 0 . \end{array}$

From the Bessel inequality and the orthogonality of the eigenfunctions (namely, ${(u^{ε}, u_{k}^{ε})}_{H^{ε}} = 0$ provided $k < k (ε)$ ) we obtain the following estimates for the function $g^{ε}$ : $\begin{array}{l} (3.53) & {‖ g^{ε} ‖}_{H^{ε}}^{2} = \sum_{k = 1}^{k (ε) - 1} {| {(v^{ε}, u_{k}^{ε})}_{H^{ε}} |}^{2} = \sum_{k = 1}^{k (ε) - 1} {| {(v^{ε} - u^{ε}, u_{k}^{ε})}_{H^{ε}} |}^{2} ⩽ {‖ v^{ε} - u^{ε} ‖}_{H^{ε}}^{2}, \\ (3.54) & η^{ε} [g^{ε}, g^{ε}] = \sum_{k = 1}^{k (ε) - 1} λ_{k}^{ε} {| {(v^{ε}, u_{k}^{ε})}_{H^{ε}} |}^{2} = \sum_{k = 1}^{k (ε) - 1} λ_{k}^{ε} {| {(v^{ε} - u^{ε}, u_{k}^{ε})}_{H^{ε}} |}^{2} ⩽ λ^{ε} {‖ v^{ε} - u^{ε} ‖}_{H^{ε}}^{2} . \end{array}$ In view of (3.43), (3.45), (3.47), (3.48), (3.52) and the fact that $u_{1} = 0$ one has $\begin{array}{rcl} {‖ u^{ε} - v^{ε} ‖}_{H^{ε}}^{2} & = & {‖ u^{ε} ‖}_{L^{2} (Ω^{ε})}^{2} + \sum_{i \in I^{ε}} {‖ u^{ε} - v^{ε} ‖}_{L^{2} (D_{i}^{ε})}^{2} \\ (3.55) & + \sum_{i \in I^{ε}} β^{ε} {‖ u^{ε} - {⟨ u^{ε} ⟩}_{B_{i}^{ε}} ‖}_{L^{2} (B_{i}^{ε})}^{2} \to 0 as ε \to 0 \end{array}$ and therefore, by virtue of (3.53)–(3.54), $\begin{array}{l} (3.56) & {‖ g^{ε} ‖}_{H^{ε}}^{2} + η^{ε} [g^{ε}, g^{ε}] \to 0 as ε \to 0 . \end{array}$

Now let us estimate the remainder $δ^{ε}$ . We denote ${\tilde{v}}^{ε} = v^{ε} - g^{ε}$ . Since ${\tilde{v}}^{ε} \in {u_{1}^{ε}, \dots, u_{k (ε) - 1}^{ε}}^{⊥}$ , one has by the well-known variational characterization of eigenvalues (see, e.g., [38]) $\begin{array}{l} η^{ε} [u^{ε}, u^{ε}] = λ^{ε} = min {\frac{η^{ε} [u, u]}{‖ u ‖_{H^{ε}}^{2}}; {(u, u_{k}^{ε})}_{H^{ε}} = 0 as k = 1, \dots, k (ε) - 1} ⩽ \frac{η^{ε} [{\tilde{v}}^{ε}, {\tilde{v}}^{ε}]}{‖ {\tilde{v}}^{ε} ‖_{H^{ε}}^{2}} . \end{array}$ or equivalently, using $u^{ε} = {\tilde{v}}^{ε} + δ^{ε}$ , $\begin{array}{l} (3.57) & η^{ε} [δ^{ε}, δ^{ε}] ⩽ - 2 η^{ε} [{\tilde{v}}^{ε}, δ^{ε}] + η^{ε} [{\tilde{v}}^{ε}, {\tilde{v}}^{ε}] ({‖ {\tilde{v}}^{ε} ‖}_{H^{ε}}^{- 2} - 1) . \end{array}$

In view of (3.50), (3.51), (3.56) the second term on the right-hand side of (3.57) tends to zero as $ε \to 0$ : $\begin{array}{l} (3.58) & η^{ε} [{\tilde{v}}^{ε}, {\tilde{v}}^{ε}] ({‖ {\tilde{v}}^{ε} ‖}_{H^{ε}}^{- 2} - 1) \to 0 as ε \to 0 . \end{array}$

Now, let us estimate the first term. One has $\begin{array}{l} (3.59) & η^{ε} [{\tilde{v}}^{ε}, δ^{ε}] = η^{ε} [v^{ε}, u^{ε} - v^{ε}] + η^{ε} [v^{ε}, g^{ε}] - η^{ε} [g^{ε}, δ^{ε}] . \end{array}$ Integrating by parts and using (3.22), (3.23) and (3.43) we get: $\begin{array}{rcl} η^{ε} [v^{ε}, u^{ε} - v^{ε}] & = & \sum_{i \in I^{ε}} \int_{D_{i}^{ε}} α^{ε} \nabla v_{i}^{ε} \cdot \nabla (u^{ε} - v^{ε}) d x \\ = & α^{ε} {⟨ u^{ε} ⟩}_{B_{i}^{ε}} \sum_{i \in I^{ε}} (\int_{S_{i}^{ε, +}} \frac{\partial v_{i}^{ε}}{\partial | x - x^{i, ε} |} u^{ε} d s - \int_{S_{i}^{ε, -}} \frac{\partial v_{i}^{ε}}{\partial | x - x^{i, ε} |} (u^{ε} - {⟨ u^{ε} ⟩}_{B_{i}^{ε}}) d s) \\ (3.60) & = & α^{ε} A^{ε} ω_{n - 1} (n - 2) \sum_{i \in I^{ε}} {⟨ u^{ε} ⟩}_{B_{i}^{ε}} (- {⟨ u^{ε} ⟩}_{S_{i}^{ε, +}} + {⟨ u^{ε} ⟩}_{S_{i}^{ε, -}} - {⟨ u^{ε} ⟩}_{B_{i}^{ε}}) . \end{array}$ Then, using the Cauchy–Schwarz inequality, (3.15), (3.35), (3.49), Lemma 3.1 and the fact that $u_{1} = 0$ , we obtain from (3.60): $\begin{array}{l} {| η^{ε} [v^{ε}, u^{ε} - v^{ε}] |}^{2} \\ ⩽ C {(α^{ε} A^{ε})}^{2} {\sum_{i \in I^{ε}} {| {⟨ u^{ε} ⟩}_{B_{i}^{ε}} |}^{2}} {\sum_{i \in I^{ε}} ({| {⟨ u^{ε} ⟩}_{S_{i}^{ε, +}} |}^{2} + {| {⟨ u^{ε} ⟩}_{S_{i}^{ε, -}} - {⟨ u^{ε} ⟩}_{B_{i}^{ε}} |}^{2})} \\ ⩽ C_{1} {(q^{ε})}^{2} r^{ε} ε^{n - 1} \sum_{i \in I^{ε}} ({| {⟨ Π_{1}^{ε} u^{ε} ⟩}_{Γ_{i}^{ε}} |}^{2} + {| {⟨ u^{ε} ⟩}_{S_{i}^{ε, +}} - {⟨ Π_{1}^{ε} u^{ε} ⟩}_{Γ_{i}^{ε}} |}^{2} + {| {⟨ u^{ε} ⟩}_{S_{i}^{ε, -}} - {⟨ u^{ε} ⟩}_{B_{i}^{ε}} |}^{2}) \\ (3.61) & ⩽ C_{2} ({‖ Π_{1}^{ε} u^{ε} ‖}_{L^{2} (Γ)}^{2} + ε {‖ \nabla Π_{1}^{ε} u^{ε} ‖}_{L^{2} (⋃_{i} Y_{i}^{ε})}^{2} + ε {‖ \nabla u^{ε} ‖}_{L^{2} (⋃_{i} B_{i}^{ε})}^{2}) \to 0 as ε \to 0 . \end{array}$ Further, in view of (3.50), (3.56), $\begin{array}{l} (3.62) & lim_{ε \to 0} η^{ε} [v^{ε}, g^{ε}] = 0 . \end{array}$ And finally, using (3.9), (3.50), (3.56), we obtain: $\begin{array}{l} (3.63) & | η^{ε} [g^{ε}, δ^{ε}] | ⩽ | η^{ε} [g^{ε}, u^{ε}] | + | η^{ε} [g^{ε}, v^{ε}] | + | η^{ε} [g^{ε}, g^{ε}] | \to 0 as ε \to 0 . \end{array}$ It follows from (3.59), (3.61)–(3.63) that $\begin{array}{l} (3.64) & lim_{ε \to 0} η^{ε} [{\tilde{v}}^{ε}, δ^{ε}] = 0 . \end{array}$ Combining (3.57), (3.58), (3.64) we conclude that $\begin{array}{l} (3.65) & lim_{ε \to 0} η^{ε} [δ^{ε}, δ^{ε}] = 0 . \end{array}$

Finally, using (3.9), (3.41), (3.50), (3.56), (3.65), we obtain: $\begin{array}{l} λ = lim_{ε \to 0} λ^{ε} = lim_{ε \to 0} η^{ε} [u^{ε}, u^{ε}] = lim_{ε \to 0} η^{ε} [v^{ε}, v^{ε}] = q . \end{array}$

Property (A) is completely proved.

3.2.2. Proof of property (B)

Let $λ \in σ (A_{q, r})$ if $r > 0$ (respectively, $λ \in σ (A_{q, 0})$ if $r = 0$ ). We have to prove that $\begin{array}{l} (3.66) & there exists a family {λ^{ε} \in σ (A^{ε})}_{ε} : λ^{ε} \to λ as ε \to 0 \end{array}$ or, equivalently, $\begin{matrix} \forall δ > 0 \exists ε (δ) such that \forall ε < ε (δ) : (λ - δ, λ + δ) \cap σ (A^{ε}) \neq \emptyset . \end{matrix}$

For proving this indirectly we assume the opposite. Then a positive number δ and a subsequence (for convenience still indexed by ε) exist such that $\begin{array}{l} (3.67) & (λ - δ, λ + δ) \cap σ (A^{ε}) = \emptyset . \end{array}$

Since $λ \in σ (A_{q, r})$ (respectively, $λ \in σ (A_{q, 0})$ ) there exists $F = (f_{1}, f_{2}) \in L^{2} (Ω) \oplus L^{2} (Γ)$ , such that $\begin{array}{l} (3.68) & F \notin range (A_{q, r} - λ I) (respectively, F \notin range (A_{q, 0} - λ I)) . \end{array}$

We introduce the function $f^{ε} \in H^{ε}$ by $\begin{array}{l} f^{ε} (x) = \{\begin{matrix} f_{1} (x), & x \in Ω^{ε}, \\ 0, & x \in ⋃_{i \in I^{ε}} D_{i}^{ε}, \\ \frac{1}{\sqrt{r^{ε}}} {⟨ f_{2} ⟩}_{Γ_{i}^{ε}}, & x \in B_{i}^{ε} . \end{matrix} \end{array}$ Here $f_{2} (x) = \sqrt{r} f_{2} (x)$ if $r > 0$ (respectively, $f_{2} (x) = f_{2} (x)$ if $r = 0$ ).

Taking (1.7) into account we obtain: $\begin{array}{l} {‖ f^{ε} ‖}_{H^{ε}}^{2} = ‖ f_{1} ‖_{L^{2} (Ω^{ε})}^{2} + \frac{1}{r^{ε}} \sum_{i \in I^{ε}} β^{ε} | B_{i}^{ε} | {| {⟨ f_{2} ⟩}_{Γ_{i}^{ε}} |}^{2} ⩽ ‖ f_{1} ‖_{L^{2} (Ω)}^{2} + ‖ f_{2} ‖_{L^{2} (Γ)}^{2} ⩽ C ‖ F ‖_{L^{2} (Ω) \oplus L^{2} (Γ)}^{2} . \end{array}$

By virtue of (3.67) λ is in the resolvent set of $A^{ε}$ . Hence there exists a unique $u^{ε} \in dom (A^{ε})$ satisfying $\begin{matrix} A^{ε} u^{ε} - λ u^{ε} = f^{ε} \end{matrix}$ and the following estimates are valid: $\begin{array}{l} (3.69) & {‖ u^{ε} ‖}_{H^{ε}} ⩽ δ^{- 1} {‖ f^{ε} ‖}_{H^{ε}} ⩽ C_{1}, \\ (3.70) & η^{ε} [u^{ε}, u^{ε}] = λ {‖ u^{ε} ‖}_{H^{ε}}^{2} + {(f^{ε}, u^{ε})}_{H^{ε}} ⩽ C_{2} . \end{array}$ Estimates (3.69)–(3.70) imply the existence of a subsequence (again indexed by ε) and $u_{1} \in H_{0}^{1} (Ω)$ , $u_{2} \in L^{2} (Γ)$ satisfying (3.13)–(3.16).

For an arbitrary $w \in H^{1} (Ω)$ one has the equality: $\begin{array}{l} (3.71) & \int_{Ω} a^{ε} \nabla u^{ε} \cdot \nabla w d x - λ \int_{Ω} b^{ε} u^{ε} w d x = \int_{Ω} b^{ε} f^{ε} w d x . \end{array}$ We plug into (3.71) the function $w = w^{ε} (x)$ defined by (3.27) and pass to the limit as $ε \to 0$ . Using the same arguments as in the proof of property (A) we conclude that $u_{1}$ , $u_{2}$ satisfy $\begin{array}{l} \int_{Ω} \nabla u_{1} \cdot \nabla w_{1} d x + \int_{Γ} (q r u_{1} w_{1} - q \sqrt{r} u_{2} w_{1} - q \sqrt{r} u_{1} w_{2} + q u_{2} w_{2}) d s \\ - λ (\int_{Ω} u_{1} w_{1} d x + \int_{Γ} u_{2} w_{2} d s) \\ (3.72) & = \int_{Ω} f_{1} w_{1} d x + \int_{Γ} f_{2} w_{2} d s, \end{array}$ for an arbitrary $w_{1} \in C_{0}^{\infty} (Ω)$ , $w_{2} \in C^{\infty} (Γ)$ (and by density arguments for any arbitrary $w_{1} \in H_{0}^{1} (Ω)$ , $w_{2} \in L^{2} (Γ)$ ). It follows from (3.2.2) that $\begin{array}{l} if r > 0 then U = (u_{1}, r^{- 1 / 2} u_{2}) \in dom (A_{q, r}) and A_{q, r} U - λ U = F, \\ if r = 0 then U = (u_{1}, u_{2}) \in dom (A_{q, 0}) and A_{q, 0} U - λ U = F . \end{array}$ This contradicts to (3.68). Thus there is $λ^{ε} \in σ (A^{ε})$ such that ${lim}_{ε \to 0} λ^{ε} = λ$ . Property (B) is proved which finishes the proof of Theorem 2.1 for the case $q < \infty$ .

3.3. Spectrum of operator $A_{q, r}$ ( $0 < q < \infty$ , $r > 0$ )

This subsection is devoted to the proof of Lemma 2.1. First we study the discrete spectrum of the operator $A_{q, r}$ . Let $λ \neq q$ be an eigenvalue of $A_{q, r}$ corresponding to the eigenfunction $U = (u_{1}, u_{2}) \in H_{r}$ . This means that $\begin{array}{l} \int_{Ω} \nabla u_{1} \cdot \nabla \overline{v_{1}} d x + q r \int_{Γ} (u_{1} - u_{2}) \overline{(v_{1} - v_{2})} d s \\ (3.73) & = λ (\int_{Ω} u_{1} \overline{v_{1}} d x + r \int_{Γ} u_{2} \overline{v_{2}} d s), \forall v = (v_{1}, v_{2}) \in dom (η_{q, r}) . \end{array}$ One can easily get from (3.3) (taking first the test-function $(0, v_{2})$ and then $(v_{1}, 0)$ ): $\begin{array}{l} U = (u_{1}, u_{2}) \in dom (η_{q, r}) satisfies (3.3) \\ ⟺ u_{2} = \frac{q u_{1}}{q - λ} and \\ (3.74) & \int_{Ω} \nabla u_{1} \cdot \nabla \overline{v_{1}} d x - \frac{λ q r}{q - λ} \int_{Γ} u_{1} \overline{v_{1}} d s = λ \int_{Ω} u_{1} \overline{v_{1}} d x, \forall v_{1} \in H_{0}^{1} (Ω) . \end{array}$

Let $μ \in R$ . By $η^{μ}$ we denote the sesquilinear form in $L^{2} (Ω)$ defined as follows: $\begin{matrix} η^{μ} [u, v] = \int_{Ω} \nabla u \cdot \nabla \overline{v} d x - μ \int_{Γ} u \overline{v} d s, dom (η^{μ}) = H_{0}^{1} (Ω) . \end{matrix}$ We denote by $A^{μ}$ the operator associated with this form. Formally the eigenvalue problem $A^{μ} u = λ u$ can be written as $\begin{array}{l} (3.75) & \{\begin{matrix} - Δ u = λ u & in Ω ∖ Γ, \\ [u] = 0 & on Γ, \\ [\frac{\partial u}{\partial n}] - μ u = 0 & on Γ, \\ u = 0 & on \partial Ω . \end{matrix} \end{array}$ The spectrum of $A^{μ}$ is purely discrete. We denote by $\begin{matrix} λ_{1} (μ) ⩽ λ_{2} (μ) ⩽ \dots ⩽ λ_{k} (μ) ⩽ \dots \underset{k \to \infty}{\to} \infty \end{matrix}$ the sequence of eigenvalues of $A^{μ}$ repeated according to their multiplicity. By ${u_{k} (μ)}_{k = 1}^{\infty}$ we denote a corresponding sequence of eigenfunctions satisfying ${(u_{k} (μ), u_{l} (μ))}_{L^{2} (Ω)} = δ_{k l}$ .

We denote by $σ_{p} (A_{q, r})$ the set of eigenvalues of the operator $A_{q, r}$ . It follows easily from (3.3) that $\begin{array}{l} (3.76) & σ_{p} (A_{q, r}) ∖ {q} = {λ \in R : λ = λ_{k} (μ) = \frac{q μ}{q r + μ} for some μ \in R and some k \in N} . \end{array}$

We also introduce the operator $A^{D}$ acting in $L^{2} (Ω)$ being associated with the form $η^{D}$ , which is defined as follows: $\begin{matrix} dom (η^{D}) = {u \in H_{0}^{1} (Ω) : u = 0 on Γ}, η^{D} [u, v] = \int_{Ω} \nabla u \cdot \nabla v d x . \end{matrix}$ We denote by ${λ_{k}^{D}}_{k = 1}^{\infty}$ the sequence of eigenvalues of $A^{D}$ written in increasing order and with account of their multiplicity.

Below we establish some properties of the spectrum of the operator $A^{μ}$ .

Proposition 3.1.
One has for each fixed $k \in N$ : $\begin{array}{l} (3.77) & I . the function λ_{k} (\cdot) : R \to R is continuous and monotonically decreasing, \\ (3.78) & II . λ_{k} (μ) \to λ_{k}^{D} as μ \to - \infty, \\ (3.79) & III . λ_{k} (μ) \to - \infty as μ \to \infty . \end{array}$
Proof.
I. One has due to the min-max principle (see, e.g., [15]): $\begin{array}{l} (3.80) & λ_{k} (μ) = min_{L \in L_{k}} max_{u \in L} \frac{η^{μ} [u, u]}{‖ u ‖_{L^{2} (Ω)}^{2}}, k = 1, 2, 3 \dots, \end{array}$ where $L_{k}$ is the set of all k-dimensional subspaces of $H_{0}^{1} (Ω)$ . Then the monotonicity follows easily from (3.80) and the monotonicity (for fixed u) of the function $μ \mapsto η^{μ} [u, u]$ .

Now, let us prove continuity. Let $[μ_{0}, μ_{1}] \subset R$ be an arbitrary compact interval. We choose some $τ > 0$ such that $\begin{array}{l} (3.81) & τ μ_{1} ⩽ \frac{1}{2} \end{array}$ (if $μ_{1} < 0$ we can choose an arbitrary $τ > 0$ ). By a standard trace inequality there exists $C_{τ} > 0$ such that $\begin{array}{l} (3.82) & \forall u \in H^{1} (Ω) : ‖ u ‖_{L^{2} (Γ)}^{2} ⩽ τ ‖ \nabla u ‖_{L^{2} (Ω)}^{2} + C_{τ} ‖ u ‖_{L^{2} (Ω)}^{2} . \end{array}$

Now, let $μ, \tilde{μ} \in [μ_{0}, μ_{1}]$ , $μ ⩽ \tilde{μ}$ . We denote for abbreviation: $\begin{array}{l} (3.83) & α : = 1 + \frac{τ (\tilde{μ} - μ)}{1 - τ \tilde{μ}}, β : = \frac{C_{τ} (\tilde{μ} - μ)}{1 - τ \tilde{μ}} . \end{array}$ In view of (3.81) α and β are positive.

For each $u \in H_{0}^{1} (Ω) ∖ {0}$ we obtain, using (3.82), $\begin{array}{l} (1 - α) ‖ \nabla u ‖_{L^{2} (Ω)}^{2} + (α \tilde{μ} - μ) ‖ u ‖_{L^{2} (Γ)}^{2} \\ ⩽ (1 - α + (α \tilde{μ} - μ) τ) ‖ \nabla u ‖_{L^{2} (Ω)}^{2} + (α \tilde{μ} - μ) C_{τ} ‖ u ‖_{L^{2} (Ω)}^{2} \overset{(3.83)}{=} β ‖ u ‖_{L^{2} (Ω)}^{2} \end{array}$ or, equivalently, $\begin{array}{l} (3.84) & \frac{η^{μ} [u, u]}{‖ u ‖_{L^{2} (Ω)}^{2}} ⩽ α \frac{η^{\tilde{μ}} [u, u]}{‖ u ‖_{L^{2} (Ω)}^{2}} + β . \end{array}$

It follows from (3.80) and (3.84) that for each fixed $k \in N$ $\begin{array}{l} λ_{k} (μ) ⩽ α λ_{k} (\tilde{μ}) + β, \end{array}$ Using (3.81), (3.83) and the monotonicity of $λ_{k} (\cdot)$ , we obtain: $\begin{array}{rcl} 0 & ⩽ & λ_{k} (μ) - λ_{k} (\tilde{μ}) ⩽ \frac{τ λ_{k} (\tilde{μ}) + C_{τ}}{1 - τ \tilde{μ}} (\tilde{μ} - μ) \\ (3.85) & ⩽ & \frac{τ λ_{k} (μ_{0}) + C_{τ}}{1 - τ μ_{1}} (\tilde{μ} - μ) ⩽ 2 (τ λ_{k} (μ_{0}) + C_{τ}) (\tilde{μ} - μ) \end{array}$ which implies the desired continuity on the interval $[μ_{0}, μ_{1}]$ . Since this interval was chosen arbitrarily we obtain the continuity on the whole axis.

II. It is easy to see that $\begin{matrix} dom (η^{D}) = {u \in H_{0}^{1} (Ω) : sup_{μ ⩽ 0} η^{μ} [u, u] < \infty} \end{matrix}$ and $\begin{matrix} η^{D} [u, v] = η^{μ} [u, v] for u, v \in dom (η^{D}) . \end{matrix}$ Then, using the monotonicity (for fixed u) of the function $μ \mapsto η^{μ} [u, u]$ and Theorem 3.1 from [40], we conclude that for each $f \in L^{2} (Ω)$ $\begin{array}{l} (3.86) & {(A^{μ} + I)}^{- 1} f \to {(A^{D} + I)}^{- 1} f in L^{2} (Ω) as μ \to - \infty . \end{array}$ Moreover, since the operators ${(A^{μ} + I)}^{- 1}$ and ${(A^{D} + I)}^{- 1}$ are compact and ${(A^{μ_{1}} + I)}^{- 1} ⩾ {(A^{μ_{2}} + I)}^{- 1} ⩾ 0$ for $μ_{1} ⩾ μ_{2}$ , by virtue of [21, Theorem VIII-3.5] (3.86) can be improved to the norm convergence $\begin{array}{l} ‖ {(A^{μ} + I)}^{- 1} - {(A^{D} + I)}^{- 1} ‖ \to 0 as μ \to - \infty, \end{array}$ implying the convergence of eigenvalues (3.78).

III. Let $m \in N$ . Let $B_{j}$ , $j = 1, \dots, m$ be open balls with centers at some points $z_{j} \in Γ$ and with radius b. It is supposed that $z_{j}$ and b are such that $\begin{array}{l} (3.87) & B_{j} \subset Ω, j = 1, \dots, m and B_{i} \cap B_{j} = \emptyset, i \neq j . \end{array}$

Let $v (x)$ be an arbitrary smooth function such that $v (x) > 0$ for $| x | < b$ and $v (x) = 0$ for $| x | ⩾ b$ . We denote $v_{j} (x) = v (x - z_{j})$ . Since $supp (v_{j}) \subset B_{j}$ , we have $v_{j} \in H_{0}^{1} (Ω)$ . We denote $\begin{matrix} L = span {v_{1}, \dots, v_{m}} . \end{matrix}$

It is clear that $dim (L) = m$ , and thus using (3.80) we get $\begin{array}{l} (3.88) & λ_{m} (μ) ⩽ max_{u \in L} \frac{η^{μ} [u, u]}{‖ u ‖_{L^{2} (Ω)}^{2}} . \end{array}$ Let $0 \neq \tilde{u} \in L$ maximize the quotient on the right-hand side of (3.88). It can be represented in the form $\tilde{u} = \sum_{j = 1}^{m} α_{j} v_{j}$ , where $α_{j} \in R$ , $α : = \sum_{j = 1}^{m} α_{j}^{2} > 0$ , and hence $\begin{array}{l} ‖ \tilde{u} ‖_{L^{2} (Ω)}^{2} = α ‖ v ‖_{L^{2} ({| x | < b})}^{2}, \\ ‖ \tilde{u} ‖_{L^{2} (Γ)}^{2} = α ‖ v ‖_{L^{2} ({| x | < b, x_{n} = 0})}^{2}, \\ ‖ \nabla \tilde{u} ‖_{L^{2} (Ω)}^{2} = α ‖ \nabla v ‖_{L^{2} ({| x | < b})}^{2} . \end{array}$ Then, taking into account that $‖ v ‖_{L^{2} ({| x | < b})}^{2} > 0$ , $‖ v ‖_{L^{2} ({| x | < b, x_{n} = 0})}^{2} > 0$ , we get $\begin{array}{l} (3.89) & λ_{m} (μ) = \frac{‖ \nabla \tilde{u} ‖_{L^{2} (Ω)}^{2} - μ ‖ \tilde{u} ‖_{L^{2} (Γ)}^{2}}{‖ \tilde{u} ‖_{L^{2} (Ω)}^{2}} ⩽ A - μ B, where B > 0 . \end{array}$ Then (3.79) follows directly from (3.89). □

Now, with Proposition 3.1 we can easily establish the properties of the set on the right-hand side of (3.76). We denote by $C$ the curve $\begin{matrix} C = {(λ, μ) \in R^{2} : λ = \frac{q μ}{q r + μ}} . \end{matrix}$ It consists of two branches $C_{\pm} = {(λ, μ) \in C : \pm (μ + q r) > 0}$ . We also introduce the curves $C_{k} = {(λ, μ) \in R^{2} : λ = λ_{k} (μ)}$ , $k \in N$ .

It follows easily from (3.77)–(3.79) that
For each $k \in N$ the curve $C_{k}$ intersects the curve $C_{+}$ exactly in one point. We denote the corresponding value of λ by $λ_{k}^{+}$ .

We denote by $k_{0}$ the integer satisfying $λ_{k_{0}}^{D} ⩽ q$ and $λ_{k_{0} + 1}^{D} > q$ . Then for each $k \in N$ the curve $C_{k_{0} + k}$ intersects the curve $C_{-}$ exactly in one point. We denote the corresponding value of λ by $λ_{k}^{-}$ . For $k ⩽ k_{0}$ the curve $C_{k}$ has no intersections with the curve $C_{-}$ .

(2.8) holds true.
Thus, with (3.76), we conclude that $\begin{array}{l} (3.90) & σ_{p} (A_{q, r}) ∖ {q} = {λ_{k}^{-}, k = 1, 2, 3, \dots} \cup {λ_{k}^{+}, k = 1, 2, 3, \dots} . \end{array}$

Since $λ_{k}^{+} \to q$ as $k \to \infty$ , we have $q \in σ_{ess} (A_{q, r})$ . To complete the proof of Lemma 2.1 we have to show that if $λ \neq q$ then $λ \notin σ_{ess} (A_{q, r})$

We denote by $B$ the following operator acting in $H_{r}$ : $\begin{matrix} dom (B) = H_{0}^{1} (Ω) \oplus L^{2} (Γ), B U = (0, q u_{1} |_{Γ}), where U = (u_{1}, u_{2}) . \end{matrix}$ In view of the embedding and trace theorems the operator $B {(A_{q, r} + I)}^{- 1}$ is compact, that is $B$ is an $A_{q, r}$ -compact operator. Thus (see, e.g., [18, Theorem 1.9]) the operator $\begin{matrix} \tilde{A} : = A_{q, r} + B \end{matrix}$ with $dom (\tilde{A}) : = dom (A_{q, r})$ is closed and $\begin{matrix} σ_{ess} (A_{q, r}) = σ_{ess} (\tilde{A}), \end{matrix}$ where the essential spectrum of the non-self-adjoint operator $\tilde{A}$ is understood in the following sense:2
²
There are several ways how to define the essential spectrum for non-self-adjoint operators. All the definitions can be found, for example, in [18] (for self-adjoint operators they are equivalent). One of the possible ways is to define it by (3.91). The advantage of this definition is twofold: the so-defined essential spectrum can be characterized via singular sequences (see (3.3)) and it is stable under relatively compact perturbations.

$\begin{array}{l} (3.91) & σ_{ess} (\tilde{A}) : = C ∖ {λ : range (\tilde{A} - λ I) is closed and dim (ker (\tilde{A} - λ I)) < \infty} . \end{array}$ Moreover in view of [18, Theorem 1.6] λ belongs to $σ_{ess} (\tilde{A})$ iff $\begin{array}{l} \exists {U^{k} \in dom (\tilde{A})}_{k} such that {‖ U^{k} ‖}_{H_{r}} = 1, \\ (3.92) & U^{k} ⇀ 0 in H_{r}, \tilde{A} U^{k} - λ U^{k} \to 0 as k \to \infty . \end{array}$

Suppose that $λ \neq q$ and let us prove that $λ \notin σ_{ess} (\tilde{A})$ . We assume the opposite. Then there exists a sequence $U^{k} = (u_{1}^{k}, u_{2}^{k}) \in dom (\tilde{A})$ , $k \in N$ satisfying (3.3). Consequently, for an arbitrary sequence $V^{k} = (v_{1}^{k}, v_{2}^{k}) \in H_{0}^{1} (Ω) \oplus L^{2} (Γ)$ satisfying $‖ V^{k} ‖_{H_{r}} ⩽ C$ one has $\begin{matrix} {(\tilde{A} U^{k} - λ U^{k}, V^{k})}_{H_{r}} \to 0 as k \to \infty \end{matrix}$ or, equivalently, $\begin{array}{l} {(\nabla u_{1}^{k}, \nabla v_{1}^{k})}_{L^{2} (Ω)} + q r {(u_{1}^{k}, v_{1}^{k})}_{L^{2} (Γ)} - q r {(u_{2}^{k}, v_{1}^{k})}_{L^{2} (Γ)} + q r {(u_{2}^{k}, v_{2}^{k})}_{L^{2} (Γ)} \\ (3.93) & - λ {(u_{1}^{k}, v_{1}^{k})}_{L^{2} (Ω)} - λ r {(u_{2}^{k}, v_{2}^{k})}_{L^{2} (Γ)} \to 0 as k \to \infty . \end{array}$ Plugging into (3.3) $V^{k} : = (0, u_{2}^{k})$ and taking into account that $λ \neq q$ and $r > 0$ we obtain that $\begin{array}{l} (3.94) & u_{2}^{k} \to 0 as k \to \infty . \end{array}$ Then, taking in (3.3) $V^{k} : = (v, 0)$ , where v is an arbitrary function belonging to $H_{0}^{1} (Ω)$ , we obtain: $\begin{array}{l} (3.95) & ⟨ u_{1}^{k}, v ⟩ - q r {(u_{2}^{k}, v)}_{L^{2} (Γ)} - λ {(u_{1}^{k}, v)}_{L^{2} (Ω)} \to 0 as k \to \infty, \end{array}$ where $⟨ u, v ⟩ : = {(\nabla u, \nabla v)}_{L^{2} (Ω)} + q r {(u, v)}_{L^{2} (Γ)}$ . In view of the Friedrichs and trace inequalities the norm $‖ u ‖ : = ⟨ u, u ⟩$ is equivalent to the standard Sobolev norm in $H_{0}^{1} (Ω)$ . Then, using (3.94) and (3.95) and taking into account that $‖ u_{1}^{k} ‖_{L^{2} (Ω)} ⩽ C$ , we conclude that $\begin{array}{l} | ⟨ u_{1}^{k}, v ⟩ | ⩽ C, \forall v \in H_{0}^{1} (Ω) . \end{array}$ Then by the Banach–Steinhaus theorem $\begin{matrix} ⟨ u_{1}^{k}, u_{1}^{k} ⟩ ⩽ C, \end{matrix}$ i.e., ${u_{1}^{k}}_{k}$ is bounded in $H^{1} (Ω)$ . Thus by the Rellich embedding theorem the sequence $u_{1}^{k}$ is compact in $L^{2} (Ω)$ , which, together with (3.94), contradicts to $‖ U^{k} ‖_{H_{r}} = 1$ , $U^{k} ⇀ 0$ in $H_{r}$ .

Lemma 2.1 is proved. Remark 3.2.
It is clear from the proof that the spectrum of the operator $A_{q r}$ subject to Neumann conditions on $\partial Ω$ instead of Dirichlet ones has the same structure – the discrete spectrum consists of two sequences of eigenvalues, one of them converges to infinity, the other one converges to q, which is the only point of the essential spectrum. The only difference is a location of the spectral threshold, which is, obviously, the λ-coordinate of the point $C_{+} \cap C_{1}$ . In the Dirichlet case it is strictly positive, since for $μ = 0$ the operator $A^{μ}$ coincides with the Dirichlet Laplacian in Ω and hence $λ_{1} (μ) > 0$ as $μ = 0$ . In contrast, in the Neumann case the threshold is equal to zero.

3.4. Proof of Theorem 2.1: The case $q = \infty$

Let $λ^{ε} \in σ (A^{ε})$ and $λ^{ε} \to λ$ as $ε \to 0$ . We have to prove that $\begin{array}{l} λ \in σ (A_{\infty, r}) if r > 0, and λ \in σ (A_{\infty, 0}) if r = 0 . \end{array}$

Again by $u^{ε}$ we denote the eigenfunction corresponding to $λ^{ε}$ and satisfying (3.8)–(3.9). In the same way as in the case $q < \infty$ we conclude that there exists $(u_{1}, u_{2}) \in H_{0}^{1} (Ω) \oplus L^{2} (Γ)$ such that (3.13)–(3.15) hold.

For an arbitrary $w \in H_{0}^{1} (Ω)$ one has the equality (3.17). Let $w_{0}$ be an arbitrary function from $C_{0}^{\infty} (Ω)$ . We plug into (3.17) the function $w = w^{ε}$ defined by $\begin{array}{l} (3.96) & w^{ε} (x) = w_{0} (x) + \sum_{i \in I^{ε}} (w_{0} (x^{i, ε}) - w_{0} (x)) φ_{i}^{ε} (x), \end{array}$ where the function $φ_{i}^{ε}$ is defined by (3.18).

Taking into account that the integral $\int_{⋃_{i} (D_{i}^{ε} \cup B_{i}^{ε})} α^{ε} \nabla u^{ε} \cdot \nabla w^{ε} d x$ is equal to zero (since $w^{ε} = w_{0} (x^{i, ε}) = const$ . in $D_{i}^{ε} \cup B_{i}^{ε}$ ) we pass to the limit in (3.17) and via the same arguments as in the case $q < \infty$ we get: $\begin{array}{l} (3.97) & \int_{Ω} \nabla u_{1} \nabla w_{0} d x = λ \int_{Ω} u_{1} w_{0} d x + λ \int_{Γ} \sqrt{r} u_{2} w_{0} d s . \end{array}$

Let us prove that $\sqrt{r} u_{1} |_{Γ} = u_{2}$ . Let w be an arbitrary function from $C^{1} (Γ)$ , the operator $Q^{ε}$ be defined by (3.32). One has, using (1.7), (3.15), (3.16) and (3.33): $\begin{array}{rcl} \int_{Γ} (\sqrt{r} u_{1} - u_{2}) w d s & = & lim_{ε \to 0} \int_{Γ} (\sqrt{r^{ε}} Π_{1}^{ε} u^{ε} - Π_{2}^{ε} u^{ε}) (Q^{ε} w) d s \\ = & lim_{ε \to 0} \sum_{i \in I^{ε}} \int_{Γ_{i}^{ε}} (\sqrt{r^{ε}} Π_{1}^{ε} u^{ε} - Π_{2}^{ε} u^{ε}) (Q^{ε} w) d s \\ (3.98) & = & lim_{ε \to 0} \sum_{i \in I^{ε}} \sqrt{r^{ε}} ({⟨ Π_{1}^{ε} u^{ε} ⟩}_{Γ_{i}^{ε}} - {⟨ u^{ε} ⟩}_{B_{i}^{ε}}) w (x^{i, ε}) | Γ_{i}^{ε} | . \end{array}$

Using (3.3), (3.4), (3.7), the inequality $\sum_{i \in I^{ε}} ε^{n - 1} ⩽ C$ and taking into account (1.6), (1.7), (2.2), (2.4) we obtain from (3.98): $\begin{array}{l} {| \int_{Γ} (\sqrt{r} u_{1} - u_{2}) w d s |}^{2} \\ ⩽ C lim_{ε \to 0} {(\sqrt{r^{ε}} \sum_{i \in I^{ε}} ε^{n - 1} | {⟨ Π_{1}^{ε} u^{ε} ⟩}_{Γ_{i}^{ε}} - {⟨ u^{ε} ⟩}_{B_{i}^{ε}} |)}^{2} \\ ⩽ C lim_{ε \to 0} r^{ε} (\sum_{i \in I^{ε}} ε^{n - 1} {| {⟨ Π_{1}^{ε} u^{ε} ⟩}_{Γ_{i}^{ε}} - {⟨ u^{ε} ⟩}_{B_{i}^{ε}} |}^{2}) \sum_{i \in I^{ε}} ε^{n - 1} \\ ⩽ C_{1} lim_{ε \to 0} \sum_{i \in I^{ε}} r^{ε} ε^{n - 1} ({| {⟨ Π_{1}^{ε} u^{ε} ⟩}_{Γ_{i}^{ε}} - {⟨ u^{ε} ⟩}_{S_{i}^{ε, +}} |}^{2} + {| {⟨ u^{ε} ⟩}_{S_{i}^{ε, +}} - {⟨ u^{ε} ⟩}_{S_{i}^{ε, -}} |}^{2} + {| {⟨ u^{ε} ⟩}_{S_{i}^{ε, -}} - {⟨ u^{ε} ⟩}_{B_{i}^{ε}} |}^{2}) \\ ⩽ C_{2} lim_{ε \to 0} r^{ε} (ε {‖ \nabla Π_{1}^{ε} u^{ε} ‖}_{L^{2} (⋃_{i} Y_{i}^{ε})}^{2} + d^{ε} {‖ \nabla u^{ε} ‖}_{L^{2} (⋃_{i} D_{i}^{ε})}^{2} + ε {‖ \nabla u^{ε} ‖}_{L^{2} (⋃_{i} B_{i}^{ε})}^{2}) \\ ⩽ C_{3} lim_{ε \to 0} (ε r^{ε} + \frac{1}{q^{ε}}) η^{ε} [u^{ε}, u^{ε}] = 0 . \end{array}$ Thus $\int_{Γ} (\sqrt{r} u_{1} - u_{2}) w d s = 0$ for all $w \in C^{1} (Γ)$ , whence $\begin{array}{l} (3.99) & \sqrt{r} u_{1} |_{Γ} = u_{2} . \end{array}$

It follows from (3.97), (3.99) that $\begin{matrix} (3.100) & \begin{matrix} if r > 0 then U = (u_{1}, u_{1} |_{Γ}) \in dom (A_{\infty, r}), A_{\infty, r} U = λ U, \\ if r = 0 then u_{1} \in dom (A_{\infty, 0}), A_{\infty, 0} u_{1} = λ u_{1} . \end{matrix} \end{matrix}$

Finally we prove that $u_{1} \neq 0$ . One has $\begin{array}{l} (3.101) & 1 = {‖ u^{ε} ‖}_{H^{ε}}^{2} = {‖ u^{ε} ‖}_{L^{2} (Ω^{ε})}^{2} + \sum_{i \in I^{ε}} {‖ u^{ε} ‖}_{L^{2} (D_{i}^{ε})}^{2} + \sum_{i \in I^{ε}} β^{ε} {‖ u^{ε} ‖}_{L^{2} (B_{i}^{ε})}^{2} . \end{array}$ Therefore, using (3.14) and taking in mind that $u^{ε} = Π_{1}^{ε} u^{ε}$ in $Ω^{ε}$ , ${lim}_{ε \to 0} | Ω ∖ Ω^{ε} | = 0$ : $\begin{array}{l} (3.102) & lim_{ε \to 0} {‖ u^{ε} ‖}_{L^{2} (Ω^{ε})} ⩽ lim_{ε \to 0} (‖ u_{1} ‖_{L^{2} (Ω^{ε})} + {‖ u^{ε} - u_{1} ‖}_{L^{2} (Ω^{ε})}) = ‖ u_{1} ‖_{L^{2} (Ω)} . \end{array}$ In the same way as in the case $q < \infty$ (see (3.45)) we get: $\begin{array}{l} (3.103) & lim_{ε \to 0} \sum_{i \in I^{ε}} {‖ u^{ε} ‖}_{L^{2} (D_{i}^{ε})}^{2} = 0 \end{array}$ (recall that the proof of (3.45) relies on inequality (3.44) and the condition ${(d^{ε})}^{2} = o (α^{ε})$ ). Finally, using the Poincaré inequality, (3.3), (3.4), (3.7) and the Cauchy–Schwarz inequality, we get $\begin{array}{l} \sum_{i \in I^{ε}} β^{ε} {‖ u^{ε} ‖}_{L^{2} (B_{i}^{ε})}^{2} \\ ⩽ C \sum_{i \in I^{ε}} β^{ε} ({‖ u^{ε} - {⟨ u^{ε} ⟩}_{B_{i}^{ε}} ‖}_{L^{2} (B_{i}^{ε})}^{2} + ε^{n} | {⟨ u^{ε} ⟩}_{B_{i}^{ε}} - {⟨ u^{ε} ⟩}_{S_{i}^{ε, -}} |^{2} \\ + ε^{n} {| {⟨ u^{ε} ⟩}_{S_{i}^{ε, -}} - {⟨ u^{ε} ⟩}_{S_{i}^{ε, +}} |}^{2} + ε^{n} {| {⟨ u^{ε} ⟩}_{S_{i}^{ε, +}} - {⟨ Π_{1}^{ε} u^{ε} ⟩}_{Γ_{i}^{ε}} |}^{2} + ε^{n} {| {⟨ Π_{1}^{ε} u^{ε} ⟩}_{Γ_{i}^{ε}} |}^{2}) \\ ⩽ C_{1} (ε r^{ε} {‖ \nabla u^{ε} ‖}_{L^{2} (⋃_{i} B_{i}^{ε})}^{2} + \frac{1}{q^{ε}} α^{ε} {‖ \nabla u^{ε} ‖}_{L^{2} (⋃_{i} D_{i}^{ε})}^{2} + ε r^{ε} {‖ \nabla Π_{1}^{ε} u^{ε} ‖}_{L^{2} (⋃_{i} Y_{i}^{ε})}^{2}) \\ (3.104) & + C_{1} r^{ε} {‖ Π_{1}^{ε} u^{ε} ‖}_{L^{2} (Γ)}^{2} . \end{array}$ Passing to the limit in (3.4) and taking (3.9) into account we obtain $\begin{array}{l} (3.105) & lim_{ε \to 0} \sum_{i \in I^{ε}} β^{ε} {‖ u^{ε} ‖}_{L^{2} (B_{i}^{ε})}^{2} ⩽ C ‖ u_{1} ‖_{L^{2} (Γ)}^{2} . \end{array}$

It follows from (3.101)–(3.103), (3.105) that $u_{1} \neq 0$ . Therefore in view of (3.100) λ is an eigenvalue of $A_{\infty, r}$ as $r > 0$ , and eigenvalue of $A_{\infty, 0}$ as $r = 0$ .

Property (B) of the Hausdorff convergence is proved in the same way as in the case $q < \infty$ (using the test-function $w^{ε} (x)$ defined before by (3.96) instead of the one defined by (3.27)). This completes the proof of Theorem 2.1 for the case $q = \infty$ .

4. Spectrum of a waveguide

In this section we consider the unbounded waveguide type domain $Ω \subset R^{2}$ : $\begin{matrix} Ω = R \times (d_{-}, d_{+}), d_{-} < 0 < d_{+} . \end{matrix}$ In this case $\begin{matrix} Γ = {x = (x_{1}, x_{2}) : x_{2} = 0} . \end{matrix}$ We again suppose that conditions (1.5)–(1.7) hold. In what follows we consider the case $q > 0$ , $r > 0$ only.

In the same way as before we introduce the Hilbert spaces $H^{ε}$ and $H_{r}$ , and the operators $A^{ε}$ and $A_{q, r}$ . Furthermore, for brevity we will use the notations $H$ and $A$ instead of $H_{r}$ , $A_{q, r}$ , correspondingly.

In order to state the result we need to introduce some additional notations. For fixed $μ \in R$ we denote by $α (μ)$ the smallest eigenvalue of the problem $\begin{array}{l} (4.1) & \{\begin{matrix} - u^{″} = λ u in (d_{-}, d_{+}) ∖ {0}, \\ u (d_{-}) = u (d_{+}) = 0, \\ u (- 0) = u (+ 0), u^{'} (- 0) - u^{'} (+ 0) = μ u (0) . \end{matrix} \end{array}$ Problem (4.1) is a 1-dimensional version of (3.75), in this case $Ω = (d_{-}, d_{+})$ , $Γ = {0}$ . Repeating the proof of Proposition 3.1 for the problem (4.1), we easily infer that (3.77) and (3.78) still hold, while (3.79) is valid for $λ_{1} (μ)$ only. Thus, the function $μ \mapsto α (μ)$ is continuous, monotonically decreasing and moreover $α (μ) \underset{μ \to - \infty}{\to} min {{(\frac{π}{d_{-}})}^{2}, {(\frac{π}{d_{+}})}^{2}}$ 3

This is the smallest eigenvalue of the problem $- u^{″} = λ u$ on $(d_{-}, d_{+}) ∖ {0}$ , $u (d_{-}) = u (0) = u (d_{+})$ .

and

α (μ) \underset{μ \to \infty}{\to} - \infty

Of course, the aforementioned properties can be also obtained via the direct computation of $α (μ)$ : it is straightforward to show that $\begin{array}{l} α (μ) = \{\begin{matrix} the smallest positive root λ \\ of the equation μ = \sqrt{λ} (cot (d_{+} \sqrt{λ}) - cot (d_{-} \sqrt{λ})), & μ < μ_{0}, \\ the unique negative root λ \\ of the equation μ = \sqrt{| λ |} (coth (d_{+} \sqrt{| λ |}) - coth (d_{-} \sqrt{| λ |})), & μ > μ_{0}, \\ 0, & μ = μ_{0}, \end{matrix} \end{array}$ where $μ_{0} = \frac{d_{-} - d_{+}}{d_{-} d_{+}} > 0$ .

Using these properties one can easily conclude that there exists one and only one point $μ_{1} \in (- q r, \infty)$ satisfying $\begin{array}{l} α (μ_{1}) = \frac{q μ_{1}}{q r + μ_{1}}, \end{array}$ and if $q < min {\frac{π^{2}}{d_{-}^{2}}, \frac{π^{2}}{d_{+}^{2}}}$ then there exists one and only one point $μ_{2} \in (- \infty, - q r)$ satisfying $\begin{array}{l} α (μ_{2}) = \frac{q μ_{2}}{q r + μ_{2}}, \end{array}$ moreover $\begin{matrix} 0 < α (μ_{1}) < q < α (μ_{2}) . \end{matrix}$

Now we are able to present the main result of this section.

Theorem 4.1.

Let $l \subset R$ be an arbitrary compact interval. Then the set $σ (A^{ε}) \cap l$ converges in the Hausdorff sense as $ε \to 0$ to the set $σ (A) \cap l$ .

The spectrum of the operator $A$ has the following form: $\begin{array}{l} (4.2) & σ (A) = D : = \{\begin{matrix} [α (μ_{1}), q] \cup [α (μ_{2}), \infty) & if q < min {\frac{π^{2}}{d_{-}^{2}}, \frac{π^{2}}{d_{+}^{2}}}, \\ [α (μ_{1}), \infty) & otherwise . \end{matrix} \end{array}$

Proof.

First let us prove (4.2).

For $L > 0$ we denote $\begin{matrix} Ω^{L} = {x = (x_{1}, x_{2}) \in R^{2} : x_{1} \in (- L, L), x_{2} \in (d_{-}, d_{+})}, Γ^{L} = Γ \cap Ω^{L} . \end{matrix}$

By $H^{L}$ we denote the Hilbert space of functions from $L^{2} (Ω^{L}) \oplus L^{2} (Γ^{L})$ and the scalar product defined by (2.6) with $Ω^{L}$ and $Γ^{L}$ instead of Ω and Γ.

We denote by $η^{L, #}$ the sesquilinear form in $H^{L}$ which is defined by (2.7) (with $Ω^{L}$ and $Γ^{L}$ instead of Ω and Γ) and the domain $\begin{matrix} dom (η^{L, #}) = {(u_{1}, u_{2}) \in H^{1} (Ω^{L}) \oplus L^{2} (Γ^{L}) : u_{1} (- L, \cdot) = u_{1} (L, \cdot), u_{1} (\cdot, d_{-}) = u_{1} (\cdot, d_{+}) = 0} . \end{matrix}$ We define the operator $A^{L, #}$ acting in $H^{L}$ being associated with this form.

In the same way as in Lemma 2.1 (or via direct calculations) we conclude that the spectrum of $A^{L, #}$ consists of the point q (the only point of the essential spectrum) and two sequences of eigenvalues ${λ_{k}^{L, #, \pm}}_{k \in N}$ satisfying (2.8) (with $λ_{k}^{L, #, \pm}$ instead of $λ_{k}^{\pm}$ ). Furthermore, it is easy to see that ${lim}_{L \to \infty} λ_{1}^{L, #, -} = α (μ_{1})$ , while ${lim}_{L \to \infty} λ_{1}^{L, #, -} = α (μ_{2})$ if $q < min {\frac{π^{2}}{d_{-}^{2}}, \frac{π^{2}}{d_{+}^{2}}}$ , otherwise ${lim}_{L \to \infty} λ_{1}^{L, #, -} = q$ . Moreover $λ_{k}^{L, #, +}$ (respectively, $λ_{k}^{L, #, -}$ ) are distributed more and more dense on the interval $[λ_{1}^{L, #, -}, q]$ (respectively, on the ray $[λ_{1}^{L, #, -}, \infty)$ ) as L increases, namely one has the equality $\begin{array}{l} (4.3) & ⋃_{L = 1}^{\infty} σ (A^{L, #}) = D . \end{array}$

Let λ be an eigenvalue of $A^{L, #}$ , U be a corresponding eigenfunction normalized by $‖ U ‖_{H^{L}} = 1$ . We extend U to the whole Ω by periodicity and set $\begin{matrix} U_{N} (x) = \frac{1}{\sqrt{N}} U (x) Φ (N^{- 1} | x_{1} |), \end{matrix}$ where $Φ : R \to R$ is a smooth function such that $Φ (r) = 1$ as $r ⩽ 1$ and $Φ (r) = 0$ as $r ⩾ 2$ . It is easy to show that $\begin{array}{l} U_{N} \in dom (A), ‖ A U_{N} - λ U_{N} ‖_{H} \to 0 as N \to \infty, \\ 0 < C_{1} ⩽ ‖ U_{N} ‖_{H} ⩽ C_{2} \end{array}$ (the constants $C_{1}$ , $C_{2}$ are independent of N) and therefore (see, e.g., [15]) $λ \in σ (A)$ . Thus we have proved that $\begin{array}{l} \forall L > 0 : σ (A^{L, #}) \subset σ (A), \end{array}$ whence, in view of (4.3), $\begin{matrix} D \subset σ (A) . \end{matrix}$

Now, we prove the reverse inclusion. Let $λ \in R ∖ D$ . We have to prove that λ belongs to the resolvent set of $A$ , i.e., for every $F \in H$ there is $U \in dom (A)$ such that $A U - λ U = F$ .

We denote by $η^{L}$ the sesquilinear form which is defined by (2.7) (with $Ω^{L}$ and $Γ^{L}$ instead of Ω and Γ) and the domain $dom (η^{L}) = H_{0}^{1} (Ω^{L}) \oplus L^{2} (Γ^{L})$ . Let $A^{L}$ be the operator acting in $H^{L}$ which is associated with this form.

One can easily calculate that $σ (A^{L}) \subset D$ for every $L > 0$ and therefore, since $λ \notin D$ , for every $F^{L} \in H^{L}$ there is $U^{L} = (u_{1}^{L}, u_{2}^{L}) \in dom (A^{L})$ such that $A^{L} U^{L} - λ U^{L} = F^{L}$ .

Let us fix $F = (f_{1}, f_{2}) \in H$ and set $F^{L} : = (f_{1} |_{Ω^{L}}, f_{2} |_{Γ^{L}})$ . One has $\begin{array}{l} (4.4) & {‖ U^{L} ‖}_{H_{Ω^{L}}} ⩽ dist (λ, D) {‖ F^{L} ‖}_{H^{L}} = dist (λ, D) ‖ F ‖_{H} ⩽ C, \end{array}$ and as a consequence $\begin{array}{l} (4.5) & {‖ \nabla u_{1}^{L} ‖}_{H^{L}} ⩽ C . \end{array}$

We extend $u_{1}^{L}$ (respectively, $u_{2}^{L}$ ) by 0 to Ω (respectively, Γ) using the same notations for the extended functions. Obviously $u_{1}^{L} \in H_{0}^{1} (Ω)$ , $u_{2}^{L} \in L^{2} (Γ)$ . It follows from (4.4)–(4.5) that there exists a subsequence (still indexed by L) and $u_{1} \in H^{1} (Ω)$ and $u_{2} \in L^{2} (Γ)$ such that $\begin{array}{l} (4.6) & u_{1}^{L} ⇀ u_{1} in H^{1} (Ω), u_{2}^{L} ⇀ u_{2} in L^{2} (Γ) as L \to \infty . \end{array}$

Let $(w_{1}, w_{2}) \in C_{0}^{\infty} (Ω) \oplus L^{2} (Γ)$ . When L is large enough then $supp (w_{1}) \subset Ω^{L}$ and therefore one can write: $\begin{array}{l} \int_{Ω} \nabla u_{1}^{L} \cdot \nabla w_{1} d x + \int_{Γ} q r (u_{1}^{L} - u_{2}^{L}) (w_{1} - w_{2}) d s - λ (\int_{Ω} u_{1}^{L} w_{1} d x + r \int_{Γ} u_{2}^{L} w_{2} d s) \\ (4.7) & = \int_{Ω} f_{1} w_{1} d x + r \int_{Γ} f_{2} w_{2} d s . \end{array}$ Using (4.6) we pass to the limit in (4.7) and obtain that $U = (u_{1}, u_{2})$ also satisfies (4.7), i.e., $\begin{matrix} A U - λ U = F . \end{matrix}$ Thus λ belongs to the resolvent set of $A$ . This finishes the proof of (4.2).

Let us now turn to the proof of the Hausdorff convergence. Recall that we have to show the fulfillment of properties (A) and (B).

The proof of property (B) of the Hausdorff convergence repeats word-by-word the proof in case of a bounded domain Ω. Therefore we focus of the proof of property (A): let $λ^{ε} \in σ (A^{ε})$ , ${lim}_{ε \to 0} λ^{ε} = λ$ , and we have to prove that $λ \in σ (A)$ .

For the sake of clarity we suppose that the shells are centered at the points $\begin{array}{l} (4.8) & {\tilde{x}}^{i, ε} = (i ε + \frac{1}{2} ε, 0), \end{array}$ i.e., we shift the shells along Γ by $ε / 2$ . Obviously, this shift does not change the spectrum of $A^{ε}$ .

We denote $\begin{matrix} \tilde{Ω} = {x = (x_{1}, x_{2}) \in R^{2} : x_{1} \in (0, 1), x_{2} \in (d_{-}, d_{+})}, \tilde{Γ} = Γ \cap \tilde{Ω} . \end{matrix}$ It is clear that $\begin{matrix} a^{ε} (x_{1} + 1, x_{2}) = a^{ε} (x_{1}, x_{2}), b^{ε} (x_{1} + 1, x_{2}) = b^{ε} (x_{1}, x_{2}) provided ε^{- 1} \in N, \end{matrix}$ i.e., $A^{ε}$ is periodic with respect to the cell $\tilde{Ω}$ provided $ε^{- 1} \in N$ . Moreover, in view of the shift (4.8), one has for $ε^{- 1} \in N$ : $\begin{matrix} ⋃_{Y_{i}^{ε} \subset \tilde{Ω}} Γ_{i}^{ε} = ⋃_{i = 1}^{N (ε)} Γ_{i}^{ε} = \tilde{Γ}, where N (ε) = ε^{- 1} . \end{matrix}$ Further we will study the subsequence $λ^{ε_{k}}$ , where $ε_{k} = k^{- 1}$ , $k = 1, 2, 3 \dots$ . For convenience we will use the notation ε keeping in mind $ε_{k}$ .

It is well-known from Floquet–Bloch theory (see, e.g., [6,17,24]) that the spectrum of $A^{ε}$ can be expressed as a union of spectra of certain operators on the period cell. By ${\tilde{H}}^{ε}$ we denote the space of functions from $L^{2} (\tilde{Ω})$ and the scalar product defined by (2.3) with $\tilde{Ω}$ instead of Ω. Let $φ \in [0, 2 π)$ . In the space ${\tilde{H}}^{ε}$ we consider the sesquilinear form ${\tilde{η}}^{φ, ε}$ defined by (2.4) (with $\tilde{Ω}$ instead of Ω) and the domain $\begin{matrix} dom ({\tilde{η}}^{φ, ε}) = {u \in H^{1} (\tilde{Ω}) : u (0, \cdot) = e^{i φ} u (1, \cdot), u (\cdot, d_{-}) = u (\cdot, d_{+}) = 0} . \end{matrix}$ By ${\tilde{A}}^{φ, ε}$ we denote the operator associated with this form. The spectrum of ${\tilde{A}}^{φ, ε}$ is purely discrete. We denote by ${{\tilde{λ}}_{k}^{φ, ε}}_{k \in N}$ the sequence of its eigenvalues written in the ascending order and with account to their multiplicity. One has the following representation: $\begin{array}{l} (4.9) & σ (A^{ε}) = ⋃_{k = 1}^{\infty} I_{k}^{ε}, where I_{k}^{ε} = ⋃_{φ \in [0, 2 π)} {{\tilde{λ}}_{k}^{φ, ε}} . \end{array}$ The sets $I_{k}^{ε}$ are compact intervals.

We also introduce the operator ${\tilde{A}}^{φ}$ as the operator acting in $\tilde{H} : = L^{2} (\tilde{Ω}) \oplus L^{2} (\tilde{Γ})$ equipped with a scalar product defined by (2.6) (with $\tilde{Ω}$ , $\tilde{Γ}$ instead of Ω, Γ), being associated with the sesquilinear form ${\tilde{η}}^{φ}$ which is defined by (2.7) (with $\tilde{Ω}$ , $\tilde{Γ}$ instead of Ω, Γ) and domain $dom ({\tilde{η}}^{φ}) = dom (η^{φ, ε}) \oplus L^{2} (\tilde{Γ})$ .

In the same way as in Lemma 2.1 we conclude that $\begin{array}{l} σ ({\tilde{A}}^{φ}) = {q} \cup {{\tilde{λ}}_{k}^{φ, -}, k = 1, 2, 3, \dots} \cup {{\tilde{λ}}_{k}^{φ, +}, k = 1, 2, 3, \dots}, \end{array}$ the points ${\tilde{λ}}_{k}^{φ, \pm}$ , $k = 1, 2, 3 \dots$ belong to the discrete spectrum, q is the only point of the essential spectrum and $\begin{array}{rcl} α (μ_{1}) & ⩽ & {\tilde{λ}}_{1}^{φ, +} ⩽ {\tilde{λ}}_{2}^{φ, +} ⩽ \dots ⩽ {\tilde{λ}}_{k}^{φ, +} ⩽ \dots \underset{k \to \infty}{\to} q < {\tilde{λ}}_{1}^{φ, -} \\ (4.10) & ⩽ & {\tilde{λ}}_{2}^{φ, -} ⩽ \dots ⩽ {\tilde{λ}}_{k}^{φ, -} ⩽ \dots \underset{k \to \infty}{\to} \infty . \end{array}$ Moreover if $q < min {\frac{π^{2}}{d_{-}^{2}}, \frac{π^{2}}{d_{+}^{2}}}$ then $\begin{array}{l} (4.11) & α (μ_{2}) ⩽ {\tilde{λ}}_{1}^{φ, -} . \end{array}$

In view of (4.9) there exists $φ^{ε} \in [0, 2 π)$ such that $λ^{ε} \in σ ({\tilde{A}}^{φ^{ε}, ε})$ . We extract a subsequence (still indexed by ε) such that $\begin{array}{l} (4.12) & φ^{ε} \to φ \in [0, 2 π] as ε \to 0 . \end{array}$

Let $u^{ε}$ be the eigenfunction of ${\tilde{A}}^{φ^{ε}, ε}$ corresponding to $λ^{ε}$ and normalized by the condition $‖ u^{ε} ‖_{{\tilde{H}}^{ε}} = 1$ .

In the same way as in the proof of Theorem 2.1 we conclude that there exists a subsequence (still indexed by ε), $u_{1} \in H^{1} (\tilde{Ω})$ and $u_{2} \in L^{2} (\tilde{Γ})$ such that $\begin{matrix} (4.13) & \begin{matrix} Π_{1}^{ε} u^{ε} ⇀ u_{1} in H^{1} (\tilde{Ω}), Π_{1}^{ε} u^{ε} \to u_{1} in L^{2} (\tilde{Ω}), \\ Π_{1}^{ε} u^{ε} \to u_{1} in L^{2} (\tilde{Γ}), Π_{2}^{ε} u^{ε} ⇀ u_{2} in L^{2} (\tilde{Γ}) \end{matrix} \end{matrix}$ (the operators $Π_{1}^{ε}$ and $Π_{2}^{ε}$ were defined in Section 3.2). In particular, it follows from (4.12)–(4.13) that $U = (u_{1}, u_{2}) \in dom ({\tilde{η}}^{φ})$ .

If $u_{1} = 0$ then $λ = q$ , the proof repeats word-by-word the proof of this fact in Theorem 2.1.

Now, let $u_{1} \neq 0$ . For an arbitrary $w \in dom (η^{φ^{ε}, ε})$ we have $\begin{array}{l} (4.14) & \int_{\tilde{Ω}} a^{ε} \nabla u^{ε} \cdot \nabla \overline{w} d x = λ^{ε} \int_{\tilde{Ω}} b^{ε} u^{ε} \overline{w} d x . \end{array}$

Let $w_{1}$ , $w_{2}$ be arbitrary functions from $C^{\infty} (\tilde{Ω})$ and $C^{\infty} (\tilde{Γ})$ , respectively, moreover satisfying $\begin{matrix} w_{1} (0, \cdot) = e^{i φ} w_{1} (1, \cdot), w_{1} (\cdot, d_{-}) = w_{1} (\cdot, d_{+}) = 0 . \end{matrix}$ Using these functions we construct the function $w^{ε}$ by the formula (3.27). It is clear that $w^{ε} \in dom ({\tilde{η}}^{φ, ε})$ . Finally we set $\begin{matrix} {\hat{w}}^{ε} (x) = w^{ε} (x) ((e^{i (φ^{ε} - φ)} - 1) (1 - x_{1}) + 1), x = (x_{1}, x_{2}) . \end{matrix}$ It is easy to see that ${\hat{w}}^{ε} \in dom ({\tilde{η}}^{φ^{ε}, ε})$ and ${\tilde{η}}^{φ^{ε}, ε} [{\hat{w}}^{ε} - w^{ε}, {\hat{w}}^{ε} - w^{ε}] + ‖ {\hat{w}}^{ε} - w^{ε} ‖_{{\tilde{H}}^{ε}}^{2} \underset{ε \to 0}{\to} 0$ .

Plugging $w = {\hat{w}}^{ε} (x)$ into (4.14) we obtain $\begin{array}{l} (4.15) & \int_{\tilde{Ω}} a^{ε} \nabla u^{ε} \cdot \nabla \overline{w^{ε}} d x + δ (ε) = \int_{\tilde{Ω}} b^{ε} u^{ε} \overline{w^{ε}} d x, \end{array}$ where the remainder $δ (ε)$ vanishes as $ε \to 0$ : $\begin{array}{l} {| δ (ε) |}^{2} ⩽ 2 λ^{ε} η^{φ^{ε}, ε} [{\hat{w}}^{ε} - w^{ε}, {\hat{w}}^{ε} - w^{ε}] + 2 {‖ {\hat{w}}^{ε} - w^{ε} ‖}_{{\tilde{H}}^{ε}}^{2} \underset{ε \to 0}{\to} 0 . \end{array}$

Then passing to the limit $ε \to 0$ in (4.15) in the same way as in the proof of Theorem 2.1 we obtain: $\begin{array}{l} \int_{\tilde{Ω}} \nabla u_{1} \cdot \nabla \overline{w_{1}} d x + \int_{\tilde{Γ}} (q r u_{1} \overline{w_{1}} + q \sqrt{r} u_{1} \overline{w_{2}} + q \sqrt{r} u_{2} \overline{w_{1}} + q u_{2} \overline{w_{2}}) d s \\ (4.16) & = λ (\int_{\tilde{Ω}} u_{1} \overline{w_{1}} d x + \int_{\tilde{Γ}} u_{2} \overline{w_{2}} d s) . \end{array}$ Using density arguments we conclude that (4.16) holds for any arbitrary $(w_{1}, w_{2}) \in dom ({\tilde{η}}^{φ})$ , whence, evidently, $\begin{matrix} {\tilde{A}}^{φ} U = λ U . \end{matrix}$ Since $u_{1} \neq 0$ we obtain $λ \in σ ({\tilde{A}}^{φ})$ . Then in view of (4.10)–(4.11) $λ \in D$ .

Theorem 4.1 is proved. □

Footnotes

Acknowledgements

The authors are grateful to the referee for his/her careful reading of our manuscript and for the useful and constructive criticism. We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through RTG 1294 and CRC 1173.

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Spectral properties of an elliptic operator with double-contrast coefficients near a hyperplane

Abstract

Keywords

1. Introduction

2.1. The operator A ε

2.2. The limit operator

2.3. Spectrum of the operator A q , r

3.1. Preliminaries

3.2.1. Proof of property (A)

Case 2 ( u 1 = 0 ).

3.2.2. Proof of property (B)

3.3. Spectrum of operator A q , r ( 0 < q < ∞ , r > 0 )

4. Spectrum of a waveguide

Footnotes

Acknowledgements

References

2.1. The operator $A^{ε}$

2.3. Spectrum of the operator $A_{q, r}$

Case 2 ( $u_{1} = 0$ ).

3.3. Spectrum of operator $A_{q, r}$ ( $0 < q < \infty$ , $r > 0$ )