Crack growth by vanishing viscosity in planar elasticity

1.   Introduction

In this paper we consider a spectral problem for the Laplace operator in an unbounded strip $\Pi \equiv (-\infty, \infty)\times (0, H)\subset \mathbb{R}^2$ periodically perforated by a family of holes, which are also periodically distributed along a line, the so-called "perforation string". The perforated domain $\Pi^ \varepsilon$ is obtained by removing the double periodic family of holes $\overline{\omega^ \varepsilon}$ from the strip $\Pi$, cf. Figure 1, a), (4)-(6). The diameter of the holes and the distance between them in the string is $O( \varepsilon)$, while the distance between two perforation strings is $1$. $\varepsilon\ll 1$ is a small positive parameter. A Dirichlet condition is prescribed on the whole boundary $\partial \Pi^ \varepsilon$. We study the band-gap structure of the essential spectrum of the problem as $\varepsilon\to 0$.

We provide asymptotic formulas for the endpoints of the spectral bands and show that these bands collapse asymptotically at the points of the spectrum of the Dirichlet problem in a rectangle obtained by gluing the lateral sides of the periodicity cell. These formulas show that the spectrum has spectral bands of length $O( \varepsilon)$ that alternate with gaps of width $O(1)$. In fact, there is a large number of spectral gaps and their number grows indefinitely when $\varepsilon\to +0$.

It should be emphasized that waveguides with periodically perturbed boundaries have been the subject of research in the last decade: let us mention e.g. [34], [21], [22], [2] and [3] and the references therein. However the type of singular perturbation that we study in our paper has never been addressed. We consider a waveguide perforated by a periodic perforation string, which implies using a combination of homogenization methods and spectral perturbation theory.

As usual in waveguide theory, we first apply the Gelfand transform (cf. [6], [30], [33], [26], [11] and (11)) to convert the original problem, cf. (7), into a family of spectral problems depending on the Floquet-parameter $\eta\in[-\pi, \pi]$ posed in the periodicity cell $\varpi^ \varepsilon$ (cf. (13)-(16) and Fig. 1, b). Each one of these problems has a discrete spectrum, cf. (18), which describe the spectrum $\sigma^ \varepsilon$ as the union of the spectral bands, cf. (20) and (9). One of the main distinguishing features of this paper is that each problem constitutes itself a homogenization problem with one perforation string. As a consequence, in the stretched coordinates, cf. (30), there appears a boundary value problem in an unbounded strip $\Xi$ which contains the unit hole $\omega$ (cf. (2), (31)-(33) and Fig. 2).

The above mentioned homogenization spectral problems have different boundary conditions from those considered in the literature (cf. [5], [14] and [16] for an extensive bibliography). Obtaining convergence for their spectra, correcting terms and precise bounds for discrepancies (cf. (10)), as $\varepsilon\to 0$, prove essential for our analysis. We use matched asymptotic expansions methods, homogenization theory and basic techniques from the spectral perturbation theory.

1.1.   Formulation of the problem

Let

be a strip of width $H>0$. Let $\omega$ be a domain in the plane ${\mathbb R}^2$ which is bounded by a simple closed contour $\partial \omega$ which, for simplicity, we assume to be of class $C^\infty$, and that has the compact closure

where $\varpi^0$ is a rectangle, the "limit periodicity" cell in $\Pi$,

We also introduce the strip $\Pi^ \varepsilon$ (see Figure 1, a) perforated by the holes

where $\varepsilon = 1/N$ is a small positive parameter, and $N\in {\mathbb N}$ is a big natural number that we will send to $\infty$. The period of the perforation along the $x_1$-axis in the domain

is made equal to 1 by rescaling, and similarly, the period is made equal to $\varepsilon H$ in the $x_2$-direction. The periodicity cell in $\Pi^ \varepsilon$ takes the form

(see b) in Figure 1). For brevity, we shall denote by $\omega^ \varepsilon$ the union of all the holes in (4), namely,

while $\omega$ is referred to as the "unit hole", cf. (2).

In the domain (5) we consider the Dirichlet spectral problem

The variational formulation of problem (7) refers to the integral identity

where $\left( \cdot, \cdot\right)_{\Pi^ \varepsilon}$ is the scalar product in the space $L^2(\Pi^ \varepsilon)$, and $H^1_0(\Pi^ \varepsilon)$ denotes the completion, in the topology of $H^1(\Pi^ \varepsilon)$, of the space of the infinitely differentiable functions which vanish on $\partial \Pi^ \varepsilon$ and have a compact support in $\overline{\Pi^ \varepsilon}$. Since the bi-linear form on the left of (8) is positive, symmetric and closed in $H_0^1(\Pi^ \varepsilon)$, the problem (8) is associated with a positive self-adjoint unbounded operator ${\mathcal A}^ \varepsilon$ in $L^2(\Pi^ \varepsilon)$ with domain $H^1_0(\Pi^ \varepsilon)\cap H^2(\Pi^ \varepsilon)$ (see Ch. 10 in [1]).

Problem (7) gets a positive cutoff value $\lambda^ \varepsilon_\dagger$ and, therefore, its spectrum $\sigma^ \varepsilon \subset [\lambda^ \varepsilon_\dagger, \infty)$ (cf. (20) and Remark 5). It is known, see e.g. [30], [33], [11] and [26], that $\sigma^ \varepsilon$ has the band-gap structure

where $B^ \varepsilon_n$ are closed connected bounded segments in the real positive axis. The segments $B^ \varepsilon_n$ and $B^ \varepsilon_{n+1}$ may intersect but also they can be disjoint so that a spectral gap becomes open between them. Recall that a spectral gap is a non empty interval which is free of the spectrum but has both endpoints in the spectrum.

1.2.   On the results and structure of the paper

In Section 2 we address the setting of the Floquet parametric family of problems (13)-(16), obtained by applying the Gelfand transform (11) to the original problem (7). They are homogenization spectral problems in a perforated domain, the periodicity cell $\varpi^ \varepsilon$, with quasi-periodicity conditions (15)-(16) on the lateral sides of $\varpi^ \varepsilon$. Obviously, each problem of the parametric family (13)-(16) depends on the Floquet-parameter $\eta$, cf. (11), (19) and (20). For a fixed $\eta\in[-\pi, \pi]$, the problem has the discrete spectrum $\Lambda_i^ \varepsilon(\eta), \, i = 1, 2, \cdots$, cf. (18). Section 2.2 contains a first approach to the eigenpairs (i.e., eigenvalues and eigenfunctions) of this problem via the homogenized problem, cf. (27). To get this homogenized problem, we use the energy method combined with techniques from the spectral perturbation theory. We show that its eigenvalues $\Lambda_i^0, \, i = 1, 2, \cdots$ do not depend on $\eta$, since they constitute the spectrum of the Dirichlet problem in $\upsilon = (0, 1)\times(0, H)$, cf. (24). In particular, Theorem 2.1 shows that

However, this result does not give information on the spectral gaps.

Using the method of matched asymptotic expansions for the eigenfunctions of the homogenization problems (cf. Section 4) we are led to the unit cell boundary value problem (31)-(33), the so-called local problem, that is, a problem to describe the boundary layer phenomenon. Section 3 is devoted to the study of this stationary problem for the Laplace operator, which is independent of $\eta$ and it is posed in an unbounded strip $\Xi$ which contains the unit hole $\omega$. Its two solutions, with a polynomial growth at the infinity, play an important role when determining correctors for the eigenvalues $\Lambda_i^ \varepsilon(\eta)$, $i = 1, 2, \cdots$. Further specifying, the definition and the properties of the so-called polarization matrix $p(\Xi)$, which depend on the "Dirichlet hole" $\omega$, cf. (38) and Section 3.1, are related with the first term of the Fourier expansion of certain solutions of the unit cell problem (cf. (39) and (42)). The correctors $\varepsilon\Lambda_i^1(\eta)$ depend on the polarization matrix and the eigenfunctions of the homogenized problem, and we prove that for sufficiently small $\varepsilon$,

with some $c_i>0$ independent of $\eta$. These bounds are obtained in Section 5, see Theorems 5.1 and 5.2 depending on the multiplicity of the eigenvalues of (24). $\Lambda^{1}_{i}(\eta)$ is a well determined function of $\eta$ (see formulas (61), (62), (68), (69), (71) and Remarks 3 and 4); it is identified by means of matched asymptotic expansions in Section 4.

As a consequence, we deduce that the bands $B_i^ \varepsilon = \{\Lambda_{i}^ \varepsilon(\eta), \, \eta \in [-\pi, \pi]\}$ are contained in intervals

of length $O( \varepsilon)$, where $B^i_- , \, B^i_+$ are also well determined values for each eigenvalue $\Lambda^{0}_{i}$ of (24) (cf. Corollaries 5.1 and 5.2 depending on the multiplicity). All of this together gives that for each $i$ such that $\Lambda_i^0 <\Lambda_{i+1}^0$, cf. (23), the spectrum $\sigma^ \varepsilon$ opens a gap of width $O(1)$ between the corresponding spectral bands $B^ \varepsilon_{i }$ and $B^ \varepsilon_{i +1}$.

Dealing with the precise length of the band, we note that the results rely on the fact that the elements of the antidiagonal of the polarization matrix do not vanish (cf. (70)-(75)), but this is a generic property for many geometries of the unit hole $\omega$ (see, e.g., (47) and (51)). Also note, that for simplicity, we have considered that $\omega$ has a smooth boundary but most of the results hold in the case where $\omega$ has a Lipschitz boundary or even when $\omega$ is a vertical crack, cf. Section 3.1.

Summarizing, Section 2 addresses some asymptotics for the spectrum of the Floquet-parameter family of spectral problems; Section 3 considers the unit cell problem; Section 4 deals with the asymptotic expansions; in Section 5.1, we formulate the main asymptotic results of the paper, while the proofs are performed in Section 5.2.

2.   The Floquet-parameter family of spectral problems

In this section, we deal with the setting of the Floquet-parameter dependent spectral problems and the limit behavior of their spectra, cf. Sections 2.1 and 2.2, respectively.

2.1.   The model problem on the periodicity cell

The Floquet-Bloch-Gelfand transform (FBG-transform, in short)

see [6] and, e.g., [30], [33], [11], [26] and [4], converts problem (7) into a $\eta$-parametric family of spectral problems in the periodicity cell

see Figure 1, b. Note that $x\in \Pi^ \varepsilon$ on the left of (11), while $x\in \varpi^ \varepsilon$ on the right. For each $\eta\in [-\pi, \pi]$, the spectral problem of the family is defined by the equations

where $\Gamma^ \varepsilon = {\partial \varpi^ \varepsilon} \cap \partial \Pi^ \varepsilon$, $\eta$ is the dual variable, i.e., the Floquet-parameter, while $\Lambda^ \varepsilon(\eta)$ and $U^ \varepsilon(\cdot;\eta)$ denote the spectral parameter and an eigenfunction, respectively. If no confusion arises, they can be denoted by $\Lambda^ \varepsilon$ and $U^ \varepsilon$, respectively. Conditions (15)-(16) are the quasi-periodicity conditions on the lateral sides $\{\pm \frac{1}{2}\}\times(0, H)$ of $\varpi^ \varepsilon$.

The variational formulation of the spectral problem (13)-(16) reads:

where $H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)$ is a subspace of $H^1(\varpi^ \varepsilon)$ of functions which satisfy the quasi-periodicity condition (15) and vanish on $\Gamma^ \varepsilon$. In view of the compact embedding $H^1(\varpi^ \varepsilon)\subset L^2(\varpi^ \varepsilon)$, the positive, self-adjoint operator ${\mathcal A}^ \varepsilon(\eta)$ associated with the problem (17) has the discrete spectrum constituting the monotone unbounded sequence of eigenvalues

which are repeated according to their multiplicities (see Ch. 10 in [1] and Ch. 13 in [30]). The eigenfunctions are assumed to form an orthonormal basis in $L^2(\varpi^ \varepsilon)$.

The function

is continuous and $2\pi$-periodic (see, e.g., Ch. 7 of [9]). Consequently, the sets

are closed, connected and bounded intervals of the real positive axis $\overline{ {\mathbb R}_+}$. Results (9) and (20) for the spectrum of the operator ${\mathcal A}^ \varepsilon(\eta)$ and the boundary value problem (7) are well-known in the framework of the FBG-theory (see the above references). As a consequence of our results, we show that in our problem, depending on the geometry of the unit hole, and for certain lower frequency range of the spectrum, the spectral band (20) does not reduce to a point (cf. (72), (47), (70) and (74)).

2.2.   A homogenization result

A first approach to the asymptotics for eigenpairs of (13)-(16) is given by the following convergence result, that we show adapting standard techniques in homogenization and spectral perturbation theory: see, e.g., Ch. 3 in [27] for a general framework and [14] for its application to spectral problems in perforated domains with different boundary conditions. Let us recall $\varpi^0$ which coincides with $\varpi^ \varepsilon$ at $\varepsilon = 0$ (cf. (12), and (3)) and contains the perforation string

Theorem 2.1. Let the spectral problem (13)-(16) and the sequence of eigenvalues (18). Then, for any $\eta\in[-\pi, \pi]$, we have the convergence

where

are the eigenvalues, repeated according to their multiplicities, of the Dirichlet problem

Proof. First, for each fixed $m$, we show that there are two constants $C, C_m$ such that

To obtain the lower bound in (25), it suffices to consider (17) for the eigenpair $(\Lambda^\varepsilon, \, U^ \varepsilon)$ with $\Lambda^\varepsilon\equiv \Lambda_1^\varepsilon(\eta)$ and apply the the Poincaré inequality in $H^1(\varpi^0)$ once that $U^ \varepsilon$ is extended by zero in $\overline \omega^ \varepsilon$. To get $C_m$ in (25) we use the minimax principle,

where the minimum is computed over the set of subspaces $E_m^{ \varepsilon}$ of $H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)$ with dimension $m$. Indeed, let us take a particular $E_m^{ \varepsilon}$ that we construct as follows. Consider the eigenfunctions corresponding to the $m$ first eigenvalues of the mixed eigenvalue problem in the rectangle $(1/4, 1/2)\times (0, H)$, with Neumann condition on the part of the boundary $\{1/2\}\times (0, H)$, and Dirichlet condition on the rest of the boundary. Extend these eigenfunctions by zero for $x \in [0, 1/4]\times (0, H)$, and by symmetry for $x \in [-1/2, 0]\times (0, H)$. Finally, multiplying these eigenfunctions by $e^{i\eta x_1}$ gives $E_m^{ \varepsilon}$ and the rigth hand side of (25).

Hence, for each $\eta$ and $m$, we can extract a subsequence, still denoted by $\varepsilon$ such that

for a certain positive $\Lambda^0_m (\eta)$ and a certain function $U^0_m(\cdot;\eta) \in H^{1, \eta}_{per} (\varpi^0)$, both of which, in principle, can depend on $\eta$. Obviously, $U_m^0(\cdot;\eta)$ vanish on the lower and upper bases of $\varpi^0$. Also, we use the Poincaré inequality in $\varpi^0\supset\omega$, cf. (3),

and we deduce

Now, taking limits as $\varepsilon\to 0$, we get $U_m^0(\cdot, \eta) = 0$ on $\{0\}\times(0, H)$ (cf., e.g., [16] and (25)). Hence, we identify ($\Lambda_m^0(\eta)$, $U^0_m(\cdot;\eta)$) with an eigenpair of the following problem:

where the differential equation has been obtained by taking limits in the variational formulation (17) for $V\in \mathcal{ C}_0^\infty ((-1/2, 0)\times (0, H))$ and for $V\in \mathcal{ C}_0^\infty ((0, 1/2)\times (0, H))$.

Now, from the orthonormality of $U_m^ \varepsilon(\cdot;\eta)$ in $L^2(\varpi^ \varepsilon)$, we get the orthonormality of $U_m^0(\cdot, \eta)$ in $L^2(\varpi^0)$. Also, an argument of diagonalization (cf., e.g., Ch. 3 in [27]) shows the convergence of the whole sequence of eigenvalues (18) towards those of (27) with conservation of the multiplicity, and that the set $\{U_m^0(\cdot; \eta)\}_{m = 1}^\infty$ forms a basis of $L^2(\varpi^0)$.

In addition, extending by $\eta$-quasiperiodicity the eigenfunctions $U^0_m(\cdot;\eta)$,

we obtain a smooth function in the rectangle $\upsilon$, and moreover that the pair ($\Lambda_m^0(\eta)$, $U^0_m(\cdot, \eta)$) satisfies (24). In addition, the orthogonality of $\{U^0_m(\cdot; \eta)\}_{m = 1}^\infty$ in $L^2(\varpi^0)$ implies that the extended functions $\{u^0_m(\cdot; \eta)\}_{m = 1}^\infty$ in (28) form an orthogonal basis in $L^2(\upsilon)$, cf. also (55), and we have proved that $\Lambda_m^0(\eta)$ coincides with $\Lambda_m^0$ in the sequence (23) for any $\eta\in [-\pi, \pi]$. Consequently, the result of the theorem holds.

Remark 1. Note that the eigenpairs of (24) can be computed explicitly

The eigenvalues $\Lambda^0_{n p}$ are numerated with two indexes and must be reordered in the sequence (23); the corresponding eigenfunctions $U^0_{n p}$ are normalized in $L^2(\upsilon)$. Also, we note that if $H^2$ is an irrational number all the eigenvalues are simple.

3.   The unit cell problem and the polarization matrix

In this section, we study the properties of certain solutions of the boundary value problem in the unbounded strip $\Xi$, cf. (31)-(33) and Figure 2. This problem, the so-called unit cell problem, is involved with the homogenization problem (13)-(16) and the periodical distribution of the openings in the periodicity cell $\varpi^ \varepsilon$, but it remains independent of the Floquet-parameter.

In order to obtain a corrector for the approach to the eigenpairs of (13)-(16) given by Theorem 2.1, we introduce the stretched coordinates

which transforms each opening of the string $\omega^\varepsilon(0, k)$ into the unit opening $\omega$. Then, we proceed as usual in two-scale homogenization when boundary layers arise (cf., e.g. [28], [18], [32] and [24]): assuming a periodic dependence of the eigenfunctions on the $\xi_2$-variable, cf. (34), we make the change (30) in (13)-(16), and take into account (22), to arrive at the unit cell problem. This problem consists of the Laplace equation

with the periodicity conditions

and the Dirichlet condition on the boundary of the hole $\overline{\omega}$

Regarding (31)-(33), it should be noted that, for any $\Lambda^ \varepsilon \leq C$, we have

and $\varepsilon^2\Lambda^ \varepsilon \leq C \varepsilon^2$ while the main part $\Delta_\xi$ is involved in (31). Also, the boundary condition (33) is directly inherited from (14), while the periodicity conditions (32) have no relation to the original quasi-periodicity conditions (15)-(16), but we need them to support the standard asymptotic ansatz

for the boundary layer. Here, $w$ is a sufficiently smooth function in $x_2\in[0, H]$ and $W$ is $H$-periodic in $\xi_2 = \varepsilon^{-1}x_2$.

It is worth recalling that, according to the general theory of elliptic problems in domains with cylindrical outlets to infinity, cf., e.g., Ch. 5 in [26], problem (31)-(33) has just two solutions with a linear polynomial growth as $\xi_1\to \pm \infty$. Here, we search for these two solutions $W^\pm(\xi)$ by setting $\pm 1$ for the constants accompanying $\xi_1$ (cf. Proposition 1). In order to do it, let us consider a fixed positive $R$ such that

and define the cut-off functions $\chi_\pm\in C^\infty( {\mathbb R})$ as follows

where the subindex $\pm$ represent the support in $\pm \xi_1\in [0, \infty)$.

Proposition 3.1. There are two normalized solutions of (31)-(33) in the form

where the remainder $\widetilde{W}^\pm(\xi)$ gets the exponential decay rate $O(e^{-|\xi_1|2\pi/H})$, and the coefficients $p_{\tau\pm}\equiv p_{\tau\pm}{ (\Xi)}$, with $\tau = \pm$, which are independent of $R$ and compose a $2\times 2$-polarization matrix

Proof. The existence of two linearly independent normalized solutions $W^\pm$ of (31)-(33) with a linear polynomial behavior $\pm\xi_1+p_{\pm\pm}$, as $\pm \xi_1 \to \infty$, is a consequence of the Kondratiev theory [10] (cf. Ch. 5 in [26] and Sect. 3 [20]). Each solution has a linear growth in one direction and stabilizes towards a constant $p_{\mp\pm}$ in the other direction. In addition, it lives in an exponential weighted Sobolev space which guarantees that, substracting the linear part, the remaining functions have a gradient in $(L^2(\Xi))^2$.

Let us consider the functions

which, obviously, satisfy (32), (33) and

with $F^\pm(\xi) = F^\pm(\xi_1) = \pm \Delta(\chi_\pm(\xi_1)\xi_1) = \pm (\partial_{\xi_1}^2\chi_\pm\, \xi_1 + 2 \partial_{\xi_1}\chi_\pm)$. By construction, $F^\pm$ has a compact support in $\pm \xi_1\in [ R, 2R]$.

Let $C^\infty_{c\, per}(\overline{\Xi})$ be the space of the infinitely differentiable $H$-periodic functions, vanishing on $\partial \omega$, with compact support in $\overline{\Xi}$. Let us denote by $\mathcal{H}$ the completion of $C^\infty_{c\, per}(\overline{\Xi})$ in the norm

The variational formulation of (40), (32) and (33) reads: to find $\widehat W^\pm \in \mathcal{H}$ satisfying the integral identity

Since $supp(F^\pm)$ is compact, we can apply the Poincaré inequality to the elements of $\{V\in H^1([-2R, 2R])\times(0, H)): \, V\vert_{\partial \omega} = 0\}$, to derive that the right hand side of (41) defines a linear continuous functional on $\mathcal{H}$. In addition, the left-hand side of the integral identity (41) implies a norm in the Hilbert space ${ \mathcal{H }}$, and consequently, the Riesz representation theorem assures that the problem (41) has a unique solution $\widehat W\in { \mathcal{H }}$ satisfying (41).

In addition, since for each $\tau$, $\tau = \pm$, function $\widehat W^\tau (\xi)$ in (39) is harmonic for $\vert \xi_1 \vert>2R$ with gradient in $L^2 (( -\infty, -2R)\times(0, H))\cap L^2 ((2R, +\infty)\times(0, H))$, the Fourier method (cf., e.g. [13] and [26]) ensures that

where the constants $c^\tau_\pm$ are defined by

Obviously, $c^+_{\pm}$ ($c^-_\pm$ respectively) are independent of $R$ and they provide all the constants appearing in (37); namely, $c^\tau_\pm = p_{\tau{\pm }}{ (\Xi)}$. Hence, the result of the proposition holds.

3.1.   Properties of the polarization matrix

In this section, we detect certain properties of the matrix $p(\Xi)$. This matrix represents an integral characteristics of the "Dirichlet hole" $\overline{\omega}$ in the strip $\Pi$. Its definition is quite analogous to the classical polarization tensor in the exterior Dirichlet problem, see Appendix G in [29]. Let us refer to [23] for further properties of matrix $p(\Xi)$ as well as for examples on its dependence on the shape and dimensions of the hole.

Proposition 3.2. The matrix $p(\Xi)+ R\, {\mathbb I}$ is symmetric and positive, where ${\mathbb I}$ stands for the $2\times 2$ unit matrix and $R$ given in (35).

Proof. We represent (37) in the form

The function $W^\pm_0$ still satisfies the periodicity condition of (32) and the homogeneous Dirichlet condition (33) but remains harmonic in $\Xi\setminus\Upsilon^\pm(R)$, $\Upsilon^\pm(R) = \{\xi\in\Xi: \: \pm \xi_1 = R\}$, and its derivative has a jump on the segment $\Upsilon^\pm(R)$, namely

where $[\phi]_\pm(\xi_2) = \phi(\pm R \pm 0, \xi_2)-\phi(\pm R \mp 0, \xi_2)$.

In what follows, we write the equations for $\tau = \pm$. Since $\Delta W_0^\pm = 0$, we multiply it with $W_0^\tau$ and apply the Green formula in $(\Xi\setminus\Upsilon^\pm(R)) \cap \{|\xi_1| <T\}$. Finally, we send $T$ to $+\infty$ and get

On the other hand, on account of (43) and the definition of $W^\tau$, we have

Consequently, we can write

and using again the Green formula for $W^\tau$ and $W_0^\pm$, in a similar way to (44) we get

Here, we have used the following facts: $\partial/\partial|\xi_1|$ is the outward normal derivative at the end of the truncated domain $\{\xi\in{\Xi}:\, |\xi_1|<R\}$, the function $W^\tau_0$ is smooth near $\Upsilon^\pm(R)$, the derivative $\partial W_0^\pm/\partial|\xi_1|$ decays exponentially and, according to (37) and (43), the function $W^\pm_0$ admits the representation when $\pm \xi_1>2R$ (cf. (37))

Considering (44) and (45) we have shown the equality for the Gram matrix

which gives the symmetry and the positiveness of the matrix $p(\Xi)+ R\, {\mathbb I}\,$.

Let us note that our results above apply for Lipschitz domains or even cracks as it was pointed out in Section 2.1. Now, we get the following results in Propositions 3.3 and 3.4 depending on whether $\omega$ is an open domain in the plane with a positive measure $mes_2(\omega)$, or it is a crack with $mes_2(\omega) = 0$.

Proposition 3.3. Let $\omega$ be such that $mes_2(\omega) >0$. Then, the coefficients of the polarization matrix $p(\Xi)$ satisfy

Proof. We consider the linear combination

It satisfies

with the periodicity conditions in the strip, and $\widetilde{W_0}(\xi) = \widetilde{W}^+(\xi)-\widetilde{W}^-(\xi)$ gets the exponential decay rate $O(e^{-|\xi_1|2\pi/H})$. Considering the equations $\Delta W_0 = 0$ and $\Delta (W_0+\xi_1) = 0$ in $\Xi\cap \{\vert \xi_1\vert <T\}$, and $\Delta \xi_1 = 0$ in $\omega$, we apply the Green formula taking into account the boundary condition for $W_0$. Then, taking limits as $T\to \infty$, we have

Remark 2. observe that for a hole $\omega$, which is symmetric with respect to the $x_1$-axis, the matrix $p(\Xi)$ becomes symmetric with respect to the anti-diagonal, namely,

Indeed, this is due to the fact that each one of the two normalized solutions in (37) are related with each other by symmetry. Also, we note that, on account of Proposition 3.2, the symmetry $p_{+-} = p_{-+}$ holds for any shape of the hole $\omega$.

Proposition 3.4. Let $\omega$ be the crack ${\omega} = \{\xi\in {\mathbb R}^2:\: \xi_1 = 0, \:\xi_2\in(h, H-h)\}$, where $h< {H}/{2}$. Then,

In addition, $p_{–} = p_{++} = p_{-+} = p_{+-}.$

Proof. First, let us note that due to the symmetry $W^+(\xi_1, \xi_2) = W^-(-\xi_1, \xi_2)$, and the construction (43) when $R = 0$ reads

where $W^*(\xi_1, \xi_2)$ is the function defined in $\Pi^+ = \{\xi: \: \xi_1>0, \, \xi_2\in(0, H)\}$ satisfying the periodicity condition (32) and equations

Indeed, denoting by $\widetilde W^*$ the extension of $W^*$ to $\Pi^- = \{\xi : \: \xi_1<0, \, \xi_2\in(0, H)\}$, in order to verify the representation (48), it suffices to verify that the jump of $\widetilde W^*$ and its the derivative of through $\Upsilon(0) = \{\xi\in\Xi: \: \xi_1 = 0\}$ is given by

and hence, the function on the right hand side of (48) is a harmonic function in $\Xi$.

Now, considering (49), integrating by parts on $(0, T)\times (0, H)$, and taking limits as $T\to +\infty$ provide

Similarly, from (49), we get

Therefore, we deduce

and from the symmetry of $p(\Xi)$ (cf. Proposition 3.2), we obtain (47).

Also, from the definition (48), we have $p_{–}(\Xi) = p_{-+}(\Xi),$ and (cf. (46)) all the elements of the polarization matrix $p(\Xi)$ coincide. Thus, the proposition is proved.

From Proposition 3.4, note that when $\omega$ is a vertical crack, the inequality in Proposition 3.3 must be replaced by $H\left(2p_{+-}-p_{++}-p_{–}\right) = mes_2(\omega) = 0.$ Also, we observe that in order to get property (47) for a domain $\omega$ with a smooth boundary, we may apply asymptotic results on singular perturbation boundaries (cf. [7], Ch. 3 in [8] and Ch. 5 in [17]) which guarantee that for thin ellipses

(47) holds true, for a small $\delta>0$.

4.   Asymptotic analysis in the periodicity cell $\varpi^ \varepsilon$

In this section we construct asymptotic expansions for the eigenpairs ($\Lambda^{ \varepsilon}_{m} (\eta)$, $U^{ \varepsilon}_{m}(\cdot;\eta)$) of problem (13)-(16) on the periodicity cell $\varpi^{ \varepsilon}$. The parameters $m\in \mathbb{N}$ and $\eta \in [-\pi, \pi]$ are fixed in this analysis. In Sections 4.1-4.2 we consider the case in which the eigenvalue $\Lambda^{0}_{m}$ of (24) is simple. Note that for many values of $H$, all the eigenvalues are simple (cf. Remark 1). Section 4.3 contains the asymptotic ansatz for the eigenpairs case where $\Lambda^{0}_{m}$ is an eigenvalue of (24) of multiplicity $\kappa_m\geq 2$.

4.1.   Asymptotic ans${\rm\bf \ddot{a }}$tze

Let $\Lambda^{0}_{m}$ be a simple eigenvalue in sequence (23) and let $U^{0}_{m}$ be the corresponding eigenfunction of problem (24) normalized in $L^2(\upsilon)$. Then, on account of the Theorem 2.1, for the eigenvalue $\Lambda^{ \varepsilon}_{m}$ of problem (13)-(16) we consider the asymptotic ans${\rm \ddot{a }}$tze

To construct asymptotics of the corresponding eigenfunctions $U^{ \varepsilon}_{m}(x;\eta)$, we employ the method of matched asymptotic expansions, see, e.g., the monographs [35] and [8], and the papers [32], [18] and [24] where this method has been applied to homogenization problems. Namely, we take

as the outer expansion, and

as the inner expansion near the perforation string, cf. (4) and (21).

Above, $U^{0}_{m}(x;\eta)$ is built from the eigenfunction $U^{0}_{m}$ of (24) by formula

$W^\pm$ are the solutions (37) to problem (31)-(33), while the functions $U^{1}_{m}$, $w^m_\pm$ and the number $\Lambda^{1}_{m}(\eta)$ are to be determined applying matching principles, cf. Section 4.2. Note that near the perforation string, cf. (4), (21), the Dirichlet condition satisfied by $U^{0}_{m}(x;\eta)$ implies that the term accompanying $\varepsilon^0$ in the inner expansion vanishes (see, e.g., [24]); this is why the first order function in (54) is $\varepsilon$. Also, above and in what follows, the ellipses stand for higher-order terms, inessential in our formal analysis.

4.2.   Matching procedure

First, let us notice that $U^{0}_{m}\in C^\infty(\overline{\upsilon})$, and the Taylor formula applied in the outer expansion (53) yields

where, for second formula (56), we have used (55).

The inner expansion (54) is processed by means of decompositions (37). We have

Recalling relationship between $x_1$ and $\xi_1$, we compare coefficients of $\varepsilon$ and $x_1 = \varepsilon \xi_1$ on the right-hand sides of (56) and (57). As a result, we identify $w^m_\pm$ by

and also obtain the equalities

Formulas (58) define coefficients of the linear combination (54) while formulas (59) are the boundary conditions for the correction term in (53). Moreover, inserting ans${\rm \ddot{a }}$tze (52) and (53) into (13)-(14), we derive that

and the quasi-periodic conditions with $\eta$ (cf. (15)-(16)).

Since $U^0_m(x;\eta)$ is defined by (55), $(\Lambda^{0}_{m}, \, U^{0}_{m}(x))$ is an eigenpair of (24), and $\|U^{0}_{m}; L^2(\upsilon)\| = \|U^{0}_{m}(\cdot;\eta); L^2(\varpi^0)\| = 1$, we multiply by $U^0_m(x;\eta)$ in the differential equation of (60), integrate by parts and obtain

Thus, by (55) and (59), the only compatibility condition in (60) (recall that $\Lambda^{0}_{m}$ is a simple eigenvalue) converts into

where

and it determines uniquely the second term of the ansatz (52). Here and in what follows, the top index $T$ indicates the transpose vector.

Also, from (53), (54) and (57) the composite expansion approaching $U^ \varepsilon_{m}(x;\eta)$ in the whole domain $\varpi_0$ reads

4.3.   The case of a multiple eigenvalue $\Lambda^{0}_{m}$

We address the case where $\Lambda^{0}_{m}$ is an eigenvalue of (24) with multiplicity $\kappa_m\geq 2$. Let us consider $\Lambda^{0}_{m} = \cdots = \Lambda^{0}_{m+\kappa_m-1}$ in the sequence (23) and the corresponding eigenfunctions $U^{0}_{m}, \cdots ,$ $U^{0}_{m+\kappa_m-1}$ which are orthonormal in $L^2(\upsilon)$. On account of Theorem 2.1 there are $\kappa_m$ eigenvalues of problem (13)-(16), which we denote by $\Lambda^ \varepsilon_{m+l}( \eta)$, $l = 0, \cdots, \kappa_m-1$, satisfying

Let $U^ \varepsilon_{m+l}(\cdot;\eta)$, $l = 0, \cdots, \kappa_m-1$, be the corresponding eigenfunctions among the set of the eigenfunctions which form an orthonormal basis in $L^2(\varpi^ \varepsilon)$, cf. (17).

Following Section 4.1, for each $l = 0, \cdots, \kappa_m-1$, we take the ansatz for $\Lambda^ \varepsilon_{m+l}( \eta)$

the outer expansion for $U^ \varepsilon_{m+l}(\cdot;\eta)$

and the inner expansion

where the terms $\Lambda^{1}_{m+l}(\eta)$, $U^{1}_{m+l}(x;\eta)$ and $w^{m+l}_\pm(x_2;\eta)$ have to be determined by the matching procedure, cf. Section 4.2, while $U^{0}_{m+l}(x;\eta)$ is constructed from $U^{0}_{m+l}(x )$ replacing $U^{0}_{m}$ by $U^{0}_{m+l}$ in formula (55), and $W^\pm$ are the solutions (37) to problem (31)-(33).

By repeating the reasoning in Section 4.2, we obtain formulas for the above mentioned terms in (65), (66) and (67) by replacing index $m$ by $m+l$ in (56)-(62), while we realize that the compatibility condition for each $\Lambda_{m+l}^1(\eta)$ is satisfied. Indeed, multiplying by $U^0_{m+l'}(x;\eta)$, $l' = 0, \cdots, \kappa_m-1$ in the partial differential equation satisfied by $U^1_{m+l'}(x;\eta)$ (cf. (60))

and integrating by parts, we obtain

Since the eigenfunctions $U^0_{m+l}$ and $U^0_{m+l'}$ have been computed (cf. the explicit formulas (29) in Remark 1), we conclude now that

and, hence, for each $l = 0, \cdots, \kappa_m-1$, the $\kappa_{m}$ compatibility conditions to be satisfied by the pairs ($\Lambda_{m+l}^1(\eta)$, $U^1_{m+l}(x;\eta)$), cf. (60), provide $\Lambda_{m+l}^1(\eta)$ given by

where $B_{m+l}(x_2;\eta)$ is defined by

Therefore we have determined completely all the terms in the asymptotic ans${\rm \ddot{a }}$tze (65), (66) and (67) for $l = 0, \cdots, \kappa_m-1$.

5.   Justification of asymptotics

In this section, we justify the results obtained by means of matched asymptotic expasions in Section 4. Since the case in which all the eigenvalues of the Dirichlet problem (24) are simple can be a generic property, we first consider this case, cf. Theorem 5.1 and Corollary 5.1, and then the case in which these eigenvalues have a multiplicity greater than $1$, cf. Theorem 5.2 and Corollary 5.2. We state the results in Section 5.1 while we perform the proofs in Section 5.2.

5.1.   Asymptotics of eigenvalues: The results

Theorem 5.1. Let $m\in {\mathbb N}$, let $\Lambda^{0}_{m}$ be a simple eigenvalue of the Dirichlet problem (24), and let $\Lambda^{1}_{m}(\eta)$ be defined in (61) and (62). There exist positive $\varepsilon_m$ and $c_m$ independent of $\eta$ such that, for any $\varepsilon\in(0, \varepsilon_m]$, the eigenvalue $\Lambda^{ \varepsilon}_{m}(\eta)$ of problem (13)-(16) meets the estimate

and there are no other different eigenvalues in the sequence (18) satisfying (70).

Theorem 5.1 shows that $\varepsilon \Lambda^{1}_{m}(\eta)$ provides a correction term for $\Lambda^{ \varepsilon}_{m}(\eta)$ improving the approach to $\lambda^0_m$ shown in Theorem 2.1. In particular, it justifies the asymptotic ansatz (52) and formula (61). This corrector depends on the polarization matrix $p(\Xi)$, which is given by the coefficients $p_{\tau\pm}\equiv p_{\tau\pm}{ (\Xi)}$, with $\tau = \pm$, in the decomposition (37), and on the eigenfunction $U^{0}_{m}$ of problem (24), which corresponds to $\Lambda^{0}_{m}$ and is normalized in $L^2(\upsilon)$ (cf. (61) and (62)).

In order to detect the gaps between consecutive spectral bands (20) it is worthy writing formulas

which are obtained from (61) and (62). Formula (29) demonstrates that

and that the integral in $B_1(m)$ does not vanish. We note that $B_1(m) = 0$ only in the case when $p_{+-} = 0$; if so, $p(\Xi)$ is diagonal and the solutions of (37), $W^\pm$, decay exponentially when $\xi_1\to\mp\infty$, respectively. However, we have given examples of cases where $p_{+-}\not = 0$ (cf. (47) and (51)).

Remark 3. Let us consider that the eigenvalue $\Lambda^{0}_{m}$ coincides with $\Lambda^{0}_{n{ q }}$ in formula (29) for certain natural $n$ and ${q}$. Then, we obtain

and, consequently,

Corollary 5.1. Under the hypothesis of Theorem 5.1, the endpoints $B^{ \varepsilon\pm}(m)$ of the spectral band (20) satisfy the relation

Hence, the length of the band $B^ \varepsilon_m$ is $2 \varepsilon|B_1(m)|+ O( \varepsilon^{3/2})$.

Note that for the holes such that the polarization matrix (38) satisfies $p_{+-}\not = 0$, asymptotically, the bands $B^ \varepsilon_m$ have the precise length $2 \varepsilon|B_1(m)|+ O( \varepsilon^{3/2})$ and they cannot reduce to a point, namely to the point $\Lambda_m^0 + \varepsilon B_0(m)$ (cf. (71) and Remark 3). Also note that if $p_{+-} = 0$, Theorem 5.1 still provides a correction term for $\Lambda^ \varepsilon_m(\eta)$ which however does not depend on $\eta$ (cf. (70), (71) and Remark 3), the width of the band being $O( \varepsilon^{3/2})$. Although the length of the band is shorter than in the cases where $p_{+-}\not = 0$, bounds in Corollary 5.1 may not be optimal (cf. Remark 3) and further information on the corrector depending on $\eta$ can be obtained by constructing higher-order terms in the asymptotic ansatz (53).

Theorem 5.2. Let $m\in {\mathbb N}$, let $\Lambda^{0}_{m}$ be an eigenvalue of the Dirichlet problem (24) with multiplity $\kappa_m>1$. Let $\Lambda^{1}_{m+l}(\eta)$ defined in (68) and (69) for $l = 0, \cdots, \kappa_m-1$. There exist positive $\varepsilon_m$ and $c_m$ independent of $\eta$ such that, for any $\varepsilon\in(0, \varepsilon_m]$, and for each $l = 0, \cdots, \kappa_m-1$, at least one eigenvalue $\Lambda^{ \varepsilon}_{m+l_0}(\eta)$ of problem (13)-(16) satisfying (64) meets the estimate

In addition, when $l\in \{0, 1, \cdots, \kappa_m-1\}$, the total multiplicity of the eigenvalues in (18) satisfying (74) is $\kappa_m$.

Corollary 5.2. Under the hypothesis in Theorem 5.2, the spectral bands $B^ \varepsilon_{m+l}$ associated with $\Lambda^{ \varepsilon}_{m+l }(\eta)$, for $l = 0, \cdots, \kappa_m-1$, cf. (20), are contained in the interval

Hence, the length of the the bands $B^ \varepsilon_{m+l}$ are $O( \varepsilon)$ but they may not be disjoint.

Remark 4. Under the hypothesis of Theorem 5.2, it may happen that, for $l = 0, \cdots, \kappa_m-1$, only the eigenvalue $\Lambda^{ \varepsilon}_{m+l_0}(\eta)$ in the sequence (18) satisfies (74). This depends on the polarization matrix $p(\Xi)$. As a matter of fact, it can be shown by contradiction under the assumption that for two different $l$ the functions $\Lambda^1_{m+l}(\eta)$ do not intersect at any point $\eta$, cf. (71) and (72). For instance, this follows for $\omega$ with $p_{+-}(\Xi) = 0$.

Remark 5. Notice that the positive cutoff value $\lambda^ \varepsilon_\dagger$ such that the spectrum $\sigma^ \varepsilon \subset [\lambda^ \varepsilon_\dagger, \infty)$ is bounded from above by a positive constant, cf. (9), (20) and (25). In addition, from Theorem 5.2 (cf. Remark 1), we have proved that $\lambda^ \varepsilon_\dagger \to \pi^2( 1+H^{-2})$ as $\varepsilon\to 0$.

5.2.   The proofs

In this section we prove the results of Theorems 5.1 and 5.2 and of their respective corollaries.

Proof of Theorem 5.1. Let us fix $\eta$ in $[-\pi, \pi]$. Let us endow the space $H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)$, with the scalar product $\langle U^ \varepsilon, V^ \varepsilon\rangle = (\nabla U^ \varepsilon, \nabla V^ \varepsilon)_{\varpi^ \varepsilon}+ ( U^ \varepsilon, V^ \varepsilon)_{\varpi^ \varepsilon},$ and the positive, symmetric and compact operator $T^ \varepsilon(\eta)$,

The integral identity (17) for problem (13)-(16) can be rewritten as the abstract equation

with the new spectral parameter

Since $T^ \varepsilon(\eta)$ is compact (cf. e.g, Section I.4 in [31] and Ⅲ.9 in [1]), its spectrum consists of the point $\tau = 0$, the essential spectrum, and of the discrete spectrum $\{\tau^{ \varepsilon}_{m}(\eta) \}_{m\in {\mathbb N }}$ which, in view of (18) and (77), constitutes the infinitesimal sequence of positive eigenvalues

For the point

cf. (52) and (61), we construct a function $\mathcal{U}^{ \varepsilon}_{m} \in H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)$ such that

where $c_m$ and $C_m$ are some positive constants independent of $\varepsilon\in(0, \varepsilon_m]$, with $\varepsilon_m>0$. These inequalities imply the estimate for the norm of the resolvent operator $(T^ \varepsilon(\eta) -t^{ \varepsilon}_{ m}(\eta))^{-1}$

with $\mathbf{c}_m = c_m^{-1} C_m>0$. According to the well-known formula for self-adjoint operators

supported by the spectral decomposition of the resolvent (cf., e.g., Section V.5 in [9] and Ch. 6 in [1]), we deduce that the closed segment

contains at least one eigenvalue $\tau^{ \varepsilon}_{p}(\eta)$ of the operator $T^ \varepsilon(\eta)$. Since the eigenvalues of $T^ \varepsilon(\eta)$ satisfy (77) and we get the definition (78), we derive that

Then, simple algebraic calculations (cf. (81) and (25)) show that, for a $\varepsilon_m>0$, the estimate

is satisfied with a constant $\mathcal{C}_m$ independent of $\varepsilon\in(0, \varepsilon_m]$. Due to the convergence with conservation of the multiplicity (22), $p = m$ in (82) and this estimate becomes (70).

To conclude with the proof of Theorem 5.1, there remains to present a function $\mathcal{U}^{ \varepsilon}_{m}\in H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)$ enjoying restrictions (79) and (80). In what follows, we construct $\mathcal{U}^{ \varepsilon}_{ m}$ using (63) suitably modified with the help of cut-off functions with "overlapping supports", cf. [19], Ch. 2 in [17] and others. We define

where $U^0_m$ satisfies (55) and $U^1_m$ is the solution of (60) satisfying the boundary conditions (15)-(16) and (59). Similarly, we define

and

with $w^m_\pm$ defined in (58), and $W^\pm$ and matrix $p(\Xi)$ in Proposition 1 (cf. (38)). Finally, we set

where $X^ \varepsilon$ and $\mathcal{X}$ are two cut-off functions, both smoothly dependent on the $x_1$ variable, such that

Note that (85) takes into account components in both expressions (83) and (84), but the last subtrahend in $\mathcal{U}^{ \varepsilon}_{m}$ compensates for this duplication. In further estimations, term (85) will be joined to either $\mathcal{V}^{ \varepsilon m}_{out}$ or $\mathcal{V}^{ \varepsilon m}_{in}$ in order to obtain suitable bounds.

First, let us show that $\mathcal{U}^{ \varepsilon}_{m} \in H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)$. Indeed, the function defined in (86) enjoys the conditions (15)-(16) and (14). This is due to the fact that $\mathcal{U}^{ \varepsilon}_{m} = \mathcal{V}^{ \varepsilon m}_{out}$ near the sides $\{x_1 = \pm1/2, \, x_2\in (0, H)\}$ and the quasi-periodicity conditions (15)-(16) are verified by both terms in (83). Also, $\mathcal{U}^{ \varepsilon}_{m} = \mathcal{V}^{ \varepsilon m}_{in}$ near the perforation string (21) so that the Dirichlet conditions are fulfilled on the boundary of the perforation string $\Gamma^ \varepsilon \cap \varpi^0$ because $W^\pm$ satisfy (33). Finally, formulas (58) and (29) assure that $w^m_\pm(H;\eta) = w^m_\pm(0;\eta) = 0$ and hence the Dirichlet condition is met on $\Gamma^ \varepsilon\cap\partial\varpi^0$ as well.

First of all, we recall (83) and (87) to derive

for a small $\varepsilon>0$. Thus, (79) is fulfilled.

Using (76) and (78), we have

where the supremum is computed over all $\mathcal{W}^{ \varepsilon}\in H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)$ such that

Taking into account the Dirichlet conditions on $\partial\omega^ \varepsilon$ we use the Poincaré and Hardy inequalities, namely, for a fixed $T$ such that $\omega\subset \Pi_T\equiv\Pi\cap \{y_1<T \}$,

where $C_{T}$ is a constant independent of $U$, and

Then, we have

Clearly, from (71), $(1+ \Lambda^{0}_{m}+ \varepsilon\Lambda^{1}_{m}(\eta))^{-1}\leq 1$ for a small $\varepsilon>0$ independent of $\eta$, and there remains to estimate the last supremum in (88). We integrate by parts, take the Dirichlet and quasi-periodic conditions into account and observe that

On the basis of (83)-(86) we write

Here, $[\Delta, \chi] = 2\nabla\chi \cdot\nabla+ \Delta \chi$ is the commutator of the Laplace operator with a function $\chi$, and the equality $[\Delta, X^ \varepsilon \mathcal{X}] = [\Delta, \mathcal{X}]+[\Delta, X^ \varepsilon],$ which is valid due to the position of supports of functions in (87), is used when distributing terms originated by the last subtrahend in (86). Let us estimate the scalar products $(S^ \varepsilon_k , \mathcal{W}^ \varepsilon)_{\varpi^ \varepsilon}$ for $S_k^ \varepsilon$ in (90).

Considering $S^ \varepsilon_1$, because of (27), (60), (87) and (71), we have that in fact $S^ \varepsilon_1 = \varepsilon^2 X^ \varepsilon\Lambda^{1}_{m} U^{1}_{m}$, hence

As regards $S^ \varepsilon_2$, we take into account that the supports of the functions $\partial_{x_1} X^ \varepsilon$ and $\Delta X^ \varepsilon$ belong to the adherence of the thin domain $\varpi^ \varepsilon_{ \varepsilon R} = \{x\in\overline{\varpi^ \varepsilon}:\: |x_1|\in ( \varepsilon R, 2 \varepsilon R)\}$, cf. (87). Thus, the error in the Taylor formula up to the second term, and relations (58), (59) and (85) provide

Above, we have also used the smoothness of the function $U_m^1$ which holds on account that $V = U_m^1e^{-{\rm i} \eta y_1}$ is a periodic function in the $y_1$ variable, solution of an elliptic problem with constant coefficients (cf. (60), (15)-(16) and (91)), and therefore it is smooth. Then, we make use of the weighted inequality (89) and write

Dealing with $S^ \varepsilon_3$, we match the definitions of the cut-off functions $\chi_\pm$ and $X^ \varepsilon$ such that $X^ \varepsilon(x_1) = \chi_\pm(x_1/ \varepsilon)$ for $\pm x_1>0$ (cf. (36)). Recalling formulas (37), (84) and (85), we write

when $\pm x_1>0$, respectively. Note that $W^\pm$ are harmonics and both, $\partial W^\pm/\partial \xi_2$ and $\widetilde{W}^\pm$ decay exponentially as $|\xi_1|\to\infty$, see Proposition 3.1. Thus,

Above, obviously, we take the positive constant $\delta$ to be $2\pi /H$, cf. Proposition 3.1, and we note that the last integral has been computed to obtain the bound. With the same argument on the exponential decay of $\mathcal{V}^{ \varepsilon m}_{in}- X^ \varepsilon \mathcal{V}^{ \varepsilon m}_{mat}$, one derives that

Moreover, the supports of the coefficients $\partial_{ x_1} \mathcal{X}$ and $\Delta \mathcal{X}$ in the commutator $[\Delta, \mathcal{X}]$ are contained in the set $\overline{\varpi^ \varepsilon } \cap \{1/6<|x_1|<1/3\}$ while the above-mentioned decay brings the estimate

Revisiting the obtained estimates we find the worst bound $c \varepsilon^{3/2}$, and this shows (80).

The fact that the constants $\varepsilon_m$ and $c_m$ of the statement of the theorem are independent of $\eta$ follows from the independence of $\eta$ of the above inequalities throughout the proof. Indeed, we use formulas (55) and (71) for the boundedness of $U_m^0$ and $\Lambda_m^1(\eta)$, while we note that the fact that $\|U^{1}_{m} ; H^1(\varpi^ \varepsilon)\|$ is bounded by a constant independent of $\eta$ follows from the definition of the solution of (60) with the quasi-periodic boundary conditions (15)-(16). Further specifying, the change $V = U_m^1e^{-{\rm i} \eta y_1}$ converts the Laplacian into the differential operator

and therefore, performing this change in (60), gives the solution $V\in H^{1 }_{per} (\varpi^0)$. Then, as a consequence of the variational formulation of the problem for $V$ in the set of spaces $L^2 (\varpi_0)\subset H^{1 }_{per} (\varpi^0)$, the bound of $\|U^{1}_{m} ; H^1(\varpi^ \varepsilon)\|$ independently of $\eta\in [-\pi, \pi]$ holds true. Hence, the proof of Theorem 5.1 is completed.

Proof of Corollary 5.1. Due to the continuity of the function (19), the maximum and minimum of $\Lambda_m^ \varepsilon(\eta)$ for $\eta\in [-\pi, \pi]$ are achieved at two points $\eta^\pm_{ \varepsilon, m}\in [-\pi, \pi]$. Thus, the endpoints $B^{ \varepsilon\pm}(m)$ of the spectral band (20) are given by $B^{ \varepsilon\pm}(m) = \Lambda_m^ \varepsilon(\eta^\pm_{ \varepsilon, m})$.

In order to show (73) for the maximum $B^{ \varepsilon+}(m)$, we consider $\eta = \eta^+$ to be $\pi$ or $-\pi$ in such a way that $\Lambda^1(\eta^+) = B_0(m) +|B_1(m)|$. Since (70) is satisfied for $\eta = \eta^\pm_{ \varepsilon, m}$ and for $\eta = \pm \pi$, we write

and

Consequently, from (71), we derive

which gives (73) for $B^{ \varepsilon+}(m)$.

We proceed in a similar way for the minimum $B^{ \varepsilon-}(m)$ and $\eta = \eta^-$ such that $\Lambda^1(\eta^-) = B_0(m) -|B_1(m)|$ and we obtain (73). Obviously, this implies that $B^{ \varepsilon\pm}(m)$ belong to the interval

Therefore, the whole band $B_m^ \varepsilon$ is contained in the interval above whose length is $2 \varepsilon |B_1(m)| + O( \varepsilon^{3/2})$ and the corollary is proved.

Proof of Theorem 5.2. This proof holds exactly the same scheme of Theorem 5.1. Indeed, for each $l = 0, \cdots, \kappa_m-1$, we follow the reasoning in (76)-(82) and we deduce (cf. (81) and (25)) that for each $l$, and for a $\varepsilon_{m, l}>0$, the estimate

is satisfied for a certain natural $p{ \equiv p(l)}$ and $\mathcal{C}_{m, l}$ independent of $\varepsilon\in(0, \varepsilon_{m, l}]$. Due to the convergence with conservation of the multiplicity (64), the only possible eigenvalues $\Lambda^{ \varepsilon}_{p}(\eta)$ of problem (13)-(16) satisfying (92) are the set $\{ \Lambda^{ \varepsilon}_{m+l}(\eta) \}_{l = 0, \cdots, \kappa_m-1}$. Then, it suffices to show that there are $\kappa_m$ linearly independent eigenfunctions associated with the eigenvalues $\{ \Lambda^{ \varepsilon}_{p(l)}(\eta) \}_{l = 0, \cdots, \kappa_m-1}$ in (92), to deduce the result of the theorem.

We use a classical argument of contradiction (cf. [15] and [25]). We consider the set of functions $\{ \mathcal{U}^{ \varepsilon}_{m+l}(x;\eta)\}_{l = 0, \cdots, \kappa_m-1}$, constructed in (86), and we verify that they satisfy the almost orthogonality conditions

for certain constants $\widetilde c_m$ and $\widetilde C_m$. Indeed, the first inequality above is a consequence of (79), for $l = 0, \cdots, \kappa_m-1$, while the second one follows from the orthogonality of the set of eigenfunctions $\{U^0_{m+l}(x, \eta)\}_{l = 0, \cdots, \kappa_m-1}$ and the definitions (83)-(87).

Then, we define $\widetilde{\mathcal{U }}^ \varepsilon_{m+l} = \mathcal{U}^ \varepsilon_{m+l} \, \| \mathcal{U}^{ \varepsilon}_{m+l} ; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)\|^{-1}$ and consider $\mathcal{W}^ \varepsilon_{m+ l }$ the projection of $T^ \varepsilon(\eta)\widetilde{\mathcal{U }}^{ \varepsilon}_{m+l} -t^{ \varepsilon}_{m}(\eta)\widetilde{\mathcal{U }}^{ \varepsilon}_{m+l}$ in the space of the eigenfunctions of $T^ \varepsilon(\eta)$ associated with all the eigenvalues $\{ \Lambda^{ \varepsilon}_{p(l)}(\eta)\}_{l = 0, \cdots, \kappa_m-1}$ in (92) for certain constants $\mathcal{C}_{m, l}$, and more precisely, satisfying

for a constant $\widetilde{\mathbf{c }}_m$ that we shall set later in the proof, cf. (81) and definitions (76)-(78) for the operator $T^ \varepsilon(\eta)$ and the "almost eigenvalue" $t^{ \varepsilon}_{m}(\eta)$. Then, we show

where $\widetilde{\mathcal{W }}^ \varepsilon_{m+l} = { \mathcal{W}^ \varepsilon_{m+l }}\| \mathcal{W}^{ \varepsilon}_{m+l} ; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)\|^{-1}$, and $\widetilde{\mathcal{C }}_m = 2\widetilde{\mathbf{c }}_m^{-1} \max_{0\leq l\leq \kappa_m-1} \mathcal{C}_{l, m}$. This is due to the fact that

and some straightforward computation (cf., eg., Lemma 1 in Ch. 3 of [27]). Now, from (93) and (95) and straightforward computations we obtain

and this allows us to assert that set $\{ \widetilde{\mathcal{W }}^ \varepsilon_{m+l}\}_{l = 0, \cdots, \kappa_m-1}$ defines $\kappa_m$ linearly independent functions. Indeed, to prove it, we proceed by contradiction, by assuming that there are constants $\alpha_l^ \varepsilon$ different from zero such that

Let us consider $\alpha^{*, \varepsilon} = \max_{0\leq l\leq \kappa_m-1} \vert \alpha_l^ \varepsilon \vert$ and assume, without any restriction that $\alpha^{*, \varepsilon} = \alpha_0^ \varepsilon$. Then, we write

Now, setting $(\kappa_m-1)5\widetilde{\mathcal{C }}_m <1$ gives a contradiction, since the left hand side takes the value $1$, cf. (96). In this way, we also have fixed $\widetilde{\mathbf{c}}_m$ in (94).

Thus, $\{ \widetilde{\mathcal{W }}^ \varepsilon_{m+l}\}_{l = 0, \cdots, \kappa_m-1}$ define $\kappa_m$ linearly independent functions, which obviously implies that they are associated with $\kappa_m$ eigenvalues; hence the set of eigenvalues $\{ \Lambda^{ \varepsilon}_{p(l)}(\eta) \}_{l = 0, \cdots, \kappa_m-1}$ coincides with $\{ \Lambda^{ \varepsilon}_{m+l}(\eta) \}_{l = 0, \cdots, \kappa_m-1}$ and this concludes the proof of the theorem.