ASYMPTOTIC STRUCTURE OF THE SPECTRUM IN A DIRICHLET-STRIP WITH DOUBLE PERIODIC PERFORATIONS

. We address a spectral problem for the Dirichlet-Laplace operator in a waveguide Π ε . Π ε is obtained from an unbounded two-dimensional strip Π which is periodically perforated by a family of holes, which are also periodically distributed along a line, the so-called “perforation string”. We assume that the two periods are diﬀerent, namely, O (1) and O ( ε ) respectively, where 0 < ε (cid:28) 1. We look at the band-gap structure of the spectrum σ ε as ε → 0. We derive asymptotic formulas for the endpoints of the spectral bands and show that σ ε has a large number of short bands of length O ( ε ) which alternate with wide gaps of width O (1).


(Communicated by Benedetto Piccoli)
Abstract. We address a spectral problem for the Dirichlet-Laplace operator in a waveguide Π ε . Π ε is obtained from an unbounded two-dimensional strip Π which is periodically perforated by a family of holes, which are also periodically distributed along a line, the so-called "perforation string". We assume that the two periods are different, namely, O(1) and O(ε) respectively, where 0 < ε 1. We look at the band-gap structure of the spectrum σ ε as ε → 0. We derive asymptotic formulas for the endpoints of the spectral bands and show that σ ε has a large number of short bands of length O(ε) which alternate with wide gaps of width O(1).
1. Introduction. In this paper we consider a spectral problem for the Laplace operator in an unbounded strip Π ≡ (−∞, ∞) × (0, H) ⊂ R 2 periodically perforated by a family of holes, which are also periodically distributed along a line, the socalled "perforation string". The perforated domain Π ε is obtained by removing the double periodic family of holes ω ε from the strip Π, cf. Figure 1,a), (4)- (6). The diameter of the holes and the distance between them in the string is O(ε), while the distance between two perforation strings is 1. ε 1 is a small positive parameter. A Dirichlet condition is prescribed on the whole boundary ∂Π ε . We study the band-gap structure of the essential spectrum of the problem as ε → 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(ε) that alternate with gaps of width O(1). In fact, there is a large number of spectral gaps and their number grows indefinitely when ε → +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.
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 ε → 0, prove essential for our analysis. We use matched asymptotic expansions methods, homogenization theory and basic techniques from the spectral perturbation theory.
be a strip of width H > 0. Let ω be a domain in the plane R 2 which is bounded by a simple closed contour ∂ω which, for simplicity, we assume to be of class C ∞ , and that has the compact closure where 0 is a rectangle, the "limit periodicity" cell in Π, We also introduce the strip Π ε (see Figure 1,a) perforated by the holes ω ε (j, k) = x : ε −1 (x 1 − j, x 2 − εkH) ∈ ω with j ∈ Z, k ∈ {0, . . . , N − 1}, (4) where ε = 1/N is a small positive parameter, and N ∈ N is a big natural number that we will send to ∞. 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 εH in the x 2 -direction. The periodicity cell in Π ε takes the form Figure 1). For brevity, we shall denote by ω ε the union of all the holes in (4), namely, while ω 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 (·, ·) Π ε is the scalar product in the space L 2 (Π ε ), and H 1 0 (Π ε ) denotes the completion, in the topology of H 1 (Π ε ), of the space of the infinitely differentiable functions which vanish on ∂Π ε and have a compact support in Π ε . Since the bi-linear form on the left of (8) is positive, symmetric and closed in H 1 0 (Π ε ), the problem (8) is associated with a positive self-adjoint unbounded operator ). Problem (7) gets a positive cutoff value λ ε † and, therefore, its spectrum σ ε ⊂ [λ ε † , ∞) (cf. (20) and Remark 5). It is known, see e.g. [30], [33], [11] and [26], that σ ε has the band-gap structure where B ε n are closed connected bounded segments in the real positive axis. The segments B ε n and B ε 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 ε , with quasi-periodicity conditions (15)-(16) on the lateral sides of ε . Obviously, each problem of the parametric family (13)-(16) depends on the Floquet-parameter η, cf. (11), (19) and (20). For a fixed η ∈ [−π, π], the problem has the discrete spectrum Λ ε i (η), i = 1, 2, · · · , 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 Λ 0 i , i = 1, 2, · · · do not depend on η, since they constitute the spectrum of the Dirichlet problem in υ = (0, 1) × (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 η and it is posed in an unbounded strip Ξ which contains the unit hole ω. Its two solutions, with a polynomial growth at the infinity, play an important role when determining correctors for the eigenvalues Λ ε i (η), i = 1, 2, · · · . Further specifying, the definition and the properties of the so-called polarization matrix p(Ξ), which depend on the "Dirichlet hole" ω, 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 εΛ 1 i (η) depend on the polarization matrix and the eigenfunctions of the homogenized problem, and we prove that for sufficiently small ε, with some c i > 0 independent of η. These bounds are obtained in Section 5, see Theorems 5.1 and 5.2 depending on the multiplicity of the eigenvalues of (24). Λ 1 i (η) is a well determined function of η (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 i + are also well determined values for each eigenvalue Λ 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 Λ 0 i < Λ 0 i+1 , cf. (23), the spectrum σ ε opens a gap of width O(1) between the corresponding spectral bands B ε i and B ε 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 ω (see, e.g., (47) and (51)). Also note, that for simplicity, we have considered that ω has a smooth boundary but most of the results hold in the case where ω has a Lipschitz boundary or even when ω 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.
The function η ∈ [−π, π] → Λ ε m (η) (19) is continuous and 2π-periodic (see, e.g., Ch. 7 of [9]). Consequently, the sets are closed, connected and bounded intervals of the real positive axis R + . Results (9) and (20) for the spectrum of the operator A ε (η) 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)).

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 0 which coincides with ε at ε = 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 η ∈ [−π, π], 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 (Λ ε , U ε ) with Λ ε ≡ Λ ε 1 (η) and apply the the Poincaré inequality in H 1 ( 0 ) once that U ε is extended by zero in ω ε . To get C m in (25) we use the minimax principle, where the minimum is computed over the set of subspaces E ε m of H 1,η per ( ε ; Γ ε ) with dimension m. Indeed, let us take a particular E ε m 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) × (0, H), with Neumann condition on the part of the boundary {1/2} × (0, H), and Dirichlet condition on the rest of the boundary. Extend these eigenfunctions by zero for x ∈ [0, 1/4] × (0, H), and by symmetry for x ∈ [−1/2, 0] × (0, H). Finally, multiplying these eigenfunctions by e iηx1 gives E ε m and the rigth hand side of (25). Hence, for each η and m, we can extract a subsequence, still denoted by ε such that for a certain positive Λ 0 m (η) and a certain function U 0 m (·; η) ∈ H 1,η per ( 0 ), both of which, in principle, can depend on η. Obviously, U 0 m (·; η) vanish on the lower and upper bases of 0 . Also, we use the Poincaré inequality in 0 ⊃ ω, cf. (3), , e.g., [16] and (25)). Hence, we identify (Λ 0 m (η), U 0 m (·; η)) with an eigenpair of the following problem: where the differential equation has been obtained by taking limits in the variational formulation (17) for . 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 0 m (·; η)} ∞ m=1 forms a basis of L 2 ( 0 ).
Remark 1. Note that the eigenpairs of (24) can be computed explicitly The eigenvalues Λ 0 np are numerated with two indexes and must be reordered in the sequence (23); the corresponding eigenfunctions U 0 np are normalized in L 2 (υ). 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 Ξ, 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 ε , 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 ω ε (0, k) into the unit opening ω. 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 ξ 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 ω Regarding (31)-(33), it should be noted that, for any Λ ε ≤ C, we have and ε 2 Λ ε ≤ Cε 2 while the main part ∆ ξ 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 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 ξ 1 → ±∞. Here, we search for these two solutions W ± (ξ) by setting ±1 for the constants accompanying ξ 1 (cf. Proposition 3.1). In order to do it, let us consider a fixed positive R such that and define the cut-off functions χ ± ∈ C ∞ (R) as follows where the subindex ± represent the support in ±ξ 1 ∈ [0, ∞).
Proposition 3.1. There are two normalized solutions of (31)- (33) in the form where the remainder W ± (ξ) gets the exponential decay rate O(e −|ξ1|2π/H ), and the coefficients p τ ± ≡ p τ ± (Ξ), with τ = ±, which are independent of R and compose a 2 × 2-polarization matrix Proof. The existence of two linearly independent normalized solutions W ± of (31)-(33) with a linear polynomial behavior ±ξ 1 + p ±± , as ±ξ 1 → ∞, 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 ∓± 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 (Ξ)) 2 . Let us consider the functions which, obviously, satisfy (32), (33) and Let C ∞ c per (Ξ) be the space of the infinitely differentiable H-periodic functions, vanishing on ∂ω, with compact support in Ξ. Let us denote by H the completion of C ∞ c per (Ξ) in the norm The variational formulation of (40), (32) and (33) reads: to find W ± ∈ H satisfying the integral identity Since supp(F ± ) is compact, we can apply the Poincaré inequality to the elements of {V ∈ H 1 ([−2R, 2R]) × (0, H)) : V | ∂ω = 0}, to derive that the right hand side of (41) defines a linear continuous functional on H. In addition, the left-hand side of the integral identity (41) implies a norm in the Hilbert space H, and consequently, the Riesz representation theorem assures that the problem (41) has a unique solution W ∈ H satisfying (41).
3.1. Properties of the polarization matrix. In this section, we detect certain properties of the matrix p(Ξ). This matrix represents an integral characteristics of the "Dirichlet hole" ω in the strip Π. 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(Ξ) as well as for examples on its dependence on the shape and dimensions of the hole. Proof. We represent (37) in the form The function W ± 0 still satisfies the periodicity condition of (32) and the homogeneous Dirichlet condition (33) but remains harmonic in Ξ \ Υ ± (R), Υ ± (R) = {ξ ∈ Ξ : ±ξ 1 = R}, and its derivative has a jump on the segment Υ ± (R), namely In what follows, we write the equations for τ = ±. Since ∆W ± 0 = 0, we multiply it with W τ 0 and apply the Green formula in (Ξ \ Υ ± (R)) ∩ {|ξ 1 | < T }. Finally, we send T to +∞ and get On the other hand, on account of (43) and the definition of W τ , we have W τ 0 (±R, ξ 2 ) = W τ (±R, ξ 2 ) and Consequently, we can write and using again the Green formula for W τ and W ± 0 , in a similar way to (44) we get Here, we have used the following facts: ∂/∂|ξ 1 | is the outward normal derivative at the end of the truncated domain {ξ ∈ Ξ : |ξ 1 | < R}, the function W τ 0 is smooth near Υ ± (R), the derivative ∂W ± 0 /∂|ξ 1 | decays exponentially and, according to (37) and (43), the function W ± 0 admits the representation when ±ξ 1 > 2R (cf. (37)) (44) and (45) we have shown the equality for the Gram matrix , which gives the symmetry and the positiveness of the matrix p(Ξ) + R 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 ω is an open domain in the plane with a positive measure mes 2 (ω), or it is a crack with mes 2 (ω) = 0.
From Proposition 3.4, note that when ω is a vertical crack, the inequality in Proposition 3.3 must be replaced by H (2p +− − p ++ − p −− ) = mes 2 (ω) = 0. Also, we observe that in order to get property (47) for a domain ω 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 δ > 0.
4. Asymptotic analysis in the periodicity cell ε . In this section we construct asymptotic expansions for the eigenpairs (Λ ε m (η), U ε m (·; η)) of problem (13)-(16) on the periodicity cell ε . The parameters m ∈ N and η ∈ [−π, π] are fixed in this analysis. In Sections 4.1-4.2 we consider the case in which the eigenvalue Λ 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 Λ 0 m is an eigenvalue of (24) of multiplicity κ m ≥ 2.
To construct asymptotics of the corresponding eigenfunctions U ε m (x; η), 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; η) is built from the eigenfunction U 0 m of (24) by formula W ± are the solutions (37) to problem (31)-(33), while the functions U 1 m , w m ± and the number Λ 1 m (η) 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; η) implies that the term accompanying ε 0 in the inner expansion vanishes (see, e.g., [24]); this is why the first order function in (54) is ε. Also, above and in what follows, the ellipses stand for higher-order terms, inessential in our formal analysis.

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.
and there are no other different eigenvalues in the sequence (18) satisfying (70).
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(Ξ) is diagonal and the solutions of (37), W ± , decay exponentially when ξ 1 → ∓∞, respectively. However, we have given examples of cases where p +− = 0 (cf. (47) and (51)).
Note that for the holes such that the polarization matrix (38) satisfies p +− = 0, asymptotically, the bands B ε m have the precise length 2ε|B 1 (m)| + O(ε 3/2 ) and they cannot reduce to a point, namely to the point Λ 0 m + εB 0 (m) (cf. (71) and Remark 3). Also note that if p +− = 0, Theorem 5.1 still provides a correction term for Λ ε m (η) which however does not depend on η (cf. (70), (71) and Remark 3), the width of the band being O(ε 3/2 ). Although the length of the band is shorter than in the cases where p +− = 0, bounds in Corollary 5.1 may not be optimal (cf. Remark 3) and further information on the corrector depending on η can be obtained by constructing higher-order terms in the asymptotic ansatz (53).
Corollary 5.2. Under the hypothesis in Theorem 5.2, the spectral bands B ε m+l associated with Λ ε m+l (η), for l = 0, · · · , κ m − 1, cf. (20), are contained in the interval (75) Hence, the length of the the bands B ε m+l are O(ε) but they may not be disjoint.

Remark 4.
Under the hypothesis of Theorem 5.2, it may happen that, for l = 0, · · · , κ m − 1, only the eigenvalue Λ ε m+l0 (η) in the sequence (18) satisfies (74). This depends on the polarization matrix p(Ξ). As a matter of fact, it can be shown by contradiction under the assumption that for two different l the functions Λ 1 m+l (η) do not intersect at any point η, cf. (71) and (72). For instance, this follows for ω with p +− (Ξ) = 0.
To conclude with the proof of Theorem 5.1, there remains to present a function U ε m ∈ H 1,η per ( ε ; Γ ε ) enjoying restrictions (79) and (80). In what follows, we construct U ε 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 ± defined in (58), and W ± and matrix p(Ξ) in Proposition 3.1 (cf. (38)). Finally, we set where X ε and 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 U ε m compensates for this duplication. In further estimations, term (85) will be joined to either V εm out or V εm in in order to obtain suitable bounds. First, let us show that U ε m ∈ H 1,η per ( ε ; Γ ε ). Indeed, the function defined in (86) enjoys the conditions (15)- (16) and (14). This is due to the fact that U ε m = V εm out near the sides {x 1 = ±1/2, x 2 ∈ (0, H)} and the quasi-periodicity conditions (15)- (16) are verified by both terms in (83). Also, U ε m = V εm in near the perforation string (21) so that the Dirichlet conditions are fulfilled on the boundary of the perforation string Γ ε ∩ 0 because W ± satisfy (33). Finally, formulas (58) and (29) assure that w m ± (H; η) = w m ± (0; η) = 0 and hence the Dirichlet condition is met on Γ ε ∩ ∂ 0 as well.
Using (76) and (78), we have where the supremum is computed over all W ε ∈ H 1,η per ( ε ; Γ ε ) such that Taking into account the Dirichlet conditions on ∂ω ε we use the Poincaré and Hardy inequalities, namely, for a fixed T such that ω ⊂ Π T ≡ Π ∩ {y 1 < T }, where C T is a constant independent of U , and Then, we have Clearly, from (71), (1 + Λ 0 m + εΛ 1 m (η)) −1 ≤ 1 for a small ε > 0 independent of η, 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, [∆, χ] = 2∇χ · ∇ + ∆χ is the commutator of the Laplace operator with a function χ, and the equality [∆, X ε X ] = [∆, X ] + [∆, X ε ], 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 ε k , W ε ) ε for S ε k in (90). Considering S ε 1 , because of (27), (60), (87) and (71), we have that in fact S ε As regards S ε 2 , we take into account that the supports of the functions ∂ x1 X ε and ∆X ε belong to the adherence of the thin domain ε εR = {x ∈ ε : |x 1 | ∈ (εR, 2εR)}, 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 1 m which holds on account that V = U 1 m e −iηy1 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 ε 3 , we match the definitions of the cut-off functions χ ± and X ε such that X ε (x 1 ) = χ ± (x 1 /ε) for ±x 1 > 0 (cf. (36)). Recalling formulas (37), (84) and (85), we write when ±x 1 > 0, respectively. Note that W ± are harmonics and both, ∂W ± /∂ξ 2 and W ± decay exponentially as |ξ 1 | → ∞, see Proposition 3.1. Thus, Above, obviously, we take the positive constant δ to be 2π/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 V εm in − X ε V εm mat , one derives that |(S ε 5 , W ε ) ε | ≤ cε Moreover, the supports of the coefficients ∂ x1 X and ∆X in the commutator [∆, X ] are contained in the set ε ∩ {1/6 < |x 1 | < 1/3} while the above-mentioned decay brings the estimate |(S ε 4 , W ε ) ε | ≤ ce −2δ/(3ε) . Revisiting the obtained estimates we find the worst bound cε 3/2 , and this shows (80).
The fact that the constants ε m and c m of the statement of the theorem are independent of η follows from the independence of η of the above inequalities throughout the proof. Indeed, we use formulas (55) and (71) for the boundedness of U 0 m and Λ 1 m (η), while we note that the fact that U 1 m ; H 1 ( ε ) is bounded by a constant independent of η follows from the definition of the solution of (60) with the quasi-periodic boundary conditions (15)- (16). Further specifying, the change V = U 1 m e −iηy1 converts the Laplacian into the differential operator and therefore, performing this change in (60), gives the solution V ∈ H 1 per ( 0 ). Then, as a consequence of the variational formulation of the problem for V in the set of spaces L 2 ( 0 ) ⊂ H 1 per ( 0 ), the bound of U 1 m ; H 1 ( ε ) independently of η ∈ [−π, π] holds true. Hence, the proof of Theorem 5.1 is completed.