On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition
aa r X i v : . [ m a t h . SP ] O c t On a waveguide with frequently alternatingboundary conditions: homogenized Neumanncondition
Denis Borisov , Renata Bunoiu , and Giuseppe Cardone
1) Bashkir State Pedagogical University, October Revolution St. 3a,450000 Ufa, Russia, e-mail: [email protected]
2) LMAM, UMR 7122, Universit´e de Metz et CNRS Ile du Saulcy,F-57045 METZ Cedex 1, France, e-mail: [email protected]
3) University of Sannio, Department of Engineering, Corso Garibaldi, 107,82100 Benevento, Italy, e-mail: [email protected]
Abstract
We consider a waveguide modeled by the Laplacian in a straight pla-nar strip. The Dirichlet boundary condition is taken on the upper boundary,while on the lower boundary we impose periodically alternating Dirichlet andNeumann condition assuming the period of alternation to be small. We studythe case when the homogenization gives the Neumann condition instead ofthe alternating ones. We establish the uniform resolvent convergence and theestimates for the rate of convergence. It is shown that the rate of the con-vergence can be improved by employing a special boundary corrector. Otherresults are the uniform resolvent convergence for the operator on the cellof periodicity obtained by the Floquet-Bloch decomposition, the two-termsasymptotics for the band functions, and the complete asymptotic expansionfor the bottom of the spectrum with an exponentially small error term.
During last decades, models of quantum waveguides attracted much attention byboth physicists and mathematicians. It was motivated by many interesting mathe-matical phenomena of these models and also by the progress in the semiconductorphysics, where they have important applications. Much efforts were exerted to studyinfluence of various perturbations on the spectral properties of the waveguides. Oneof such perturbations is a finite number of openings coupling two lateral waveguides(see, for instance, [7], [8], [9], [12], [15], [18], [19]). Such openings are usually called
This work was partially done during the visit of D.B. to the University of Sannio (Italy)and of G.C. to LMAM of University Paul Verlaine of Metz (France). They are grateful for thewarm hospitality extended to them. D.B. was partially supported by RFBR (09-01-00530), bythe grants of the President of Russia for young scientists-doctors of sciences (MD-453.2010.1)and for Leading Scientific School (NSh-6249.2010.1), by Federal Task Program “Research andeducational professional community of innovation Russia” (contract 02.740.11.0612), and by theproject “Progetto ISA: Attivit`a di Internazionalizzazione dell’Universit`a degli Studi del Sannio”. ε , wassupposed to be small, while the other, η = η ( ε ), could be either bounded or small.The main difference between the models studied in [3] and in [7], [8], [9], [12], [15],[18], [19] is the influence of the perturbation on the spectral properties: while in thelatter papers the essential spectrum remained unchanged and discrete eigenvaluesappeared below its bottom, in [3] the spectrum was purely essential and had bandstructure. Moreover, it depended on the perturbation and, for example, the bottomof the spectrum moved as ε → +0. Assuming that ε ln η ( ε ) → − ε → +0 , (1.1)it was shown in [3] that the homogenized operator is the Laplacian with the previousboundary condition on the upper boundary, while the alternation on the lowerboundary should be replaced by the Dirichlet one. More precisely, it was shownthat the uniform resolvent convergence for the perturbed operator holds true andthe rate of convergence was estimated. Other main results were the two-termsasymptotics for first band functions of the perturbed operator and the completetwo-parametric asymptotic expansion for the bottom of the spectrum.In the present paper we consider a different case: we assume that the homoge-nized operator has the Neumann condition on the lower boundary, which is guar-anteed by the condition ε ln η ( ε ) → −∞ as ε → +0 . (1.2)We observe that this condition is not new, and it was known before that it impliedthe homogenized Neumann boundary condition for the similar problems in boundeddomains, see [24], [13], [14], [16], [17], [20].We obtain the uniform resolvent convergence for the perturbed operator andwe estimate the rate of convergence. We also obtain similar convergence for theoperator appearing on the cell of periodicity after Floquet decomposition and pro-vide two-terms asymptotics for the first band function. The last main result is thecomplete asymptotic expansion for the bottom of the spectrum.Similar results were obtained [3] under the assumption (1.1), and now we wantto underline the main differences. We first observe that in [3] the estimate ofthe rate of convergence for the perturbed resolvent was obtained for the difference2f the resolvents of the perturbed and homogenized operator and this differencewas considered as an operator from L into W . In our case, in order to have asimilar good estimate, we have to consider the difference not with the resolvent ofthe homogenized operator, but with that of an additional operator depending inboundary condition on an additional parameter µ = µ ( ε ) := − ε ln η ( ε ) → +0 as ε → +0 . (1.3)Moreover, we also have to use a special boundary corrector , see Theorem 2.1. Omit-ting the corrector and estimating the difference of the same resolvents as an operatorin L , we can still preserve the mentioned good estimate. Omitting the correctoror replacing the additional operator mentioned above by the homogenized one, oneworsens the rate of convergence. At the same time, this rate can be improved par-tially by considering the difference of the resolvents as an operator in L . Suchsituation was known to happen in the case of the operators with the fast oscillatingcoefficients (see [1], [2], [6], [30], [31], [34], [35], [36], [38], [39] and the referencestherein for further results). From this point of view the results of the present paperare closer to the cited paper in contrast to the results of [3] and [29, Ch. III, Sec.4.1].One more difference to [3] is the asymptotics for the band functions and thebottom of the essential spectrum. The second term in the asymptotics for the bandfunctions is not a constant, but a holomorphic in µ function. In fact, it is a seriesin µ and this is why the mentioned two-terms asymptotics can be regarded as theasymptotics with more terms, see (2.8). Even more interesting situation occurs inthe asymptotics for the bottom of the spectrum. Here the asymptotics containsjust one first term, but the error estimate is exponential . The leading term dependson ε and µ holomorphically and can be represented as the series in ε with theholomorphic in µ coefficients. For the bounded domains the complete asymptoticexpansions for the eigenvalues in the case of the homogenized Neumann problemwere constructed in [4], [25]. These asymptotics were power in ε [25] with theholomorphic in µ coefficients [4]. At the same time, the error terms were powersin ε and the convergence of these asymptotic series was not proved. In our casethe first term in the asymptotics for the bottom of the essential spectrum is thesum of the asymptotic series analogous to those in [4], [25]. In other words, wesucceeded to prove that in our case this series converges, is holomorphic in ε and µ and gives the exponentially small error term that for singularly perturbed problemsin homogenization is regarded as a strong result.Eventually, we point out that the technique we use is different: in additionto the boundary layer method [37] used also in [3], here we also have to employthe method of matching of the asymptotic expansions [27]. Such combination wasborrowed from [4], [23], [24], [25]. We use this combination to construct the afore-mentioned corrector to obtain the uniform resolvent convergence. Similar correctorswere also constructed in [13], [20], [24], but to obtain either weak or strong resol-vent convergence. We also employ the same corrector in the combination of thetechnique developed in [21] for the analysis of the uniform resolvent convergence forthin domains. 3n conclusion, we describe briefly the structure of the paper. In the next sectionwe formulate precisely the problem and give the main results. The third section isdevoted to the study of the uniform resolvent convergence. In the fourth section wemake the similar study for the operator appearing after the Floquet decomposition,and we also establish two-terms asymptotics for the first band functions. In thelast, fifth section we construct the complete asymptotic expansion for the bottomof the spectrum. Let x = ( x , x ) be Cartesian coordinates in R , and Ω := { x : 0 < x < π } be astraight strip of width π . By ε we denote a small positive parameter, and η = η ( ε )is a function satisfying the estimate0 < η ( ε ) < π . We indicate by Γ + and Γ − the upper and lower boundary of Ω, and we partitionΓ − into two subsets (cf. fig. 1), γ ε := { x : | x − επj | < εη, x = 0 , j ∈ Z } , Γ ε := Γ − \ γ ε . The main object of our study is the Laplacian in L (Ω) subject to the Dirichletboundary condition on Γ + ∪ γ ε and to the Neumann one on Γ ε . We introducethis operator as the non-negative self-adjoint one in L (Ω) associated with thesesquilinear form h ε [ u, v ] := ( ∇ u, ∇ v ) L (Ω) on ˚ W (Ω , Γ + ∪ γ ε ) , where ˚ W ( Q, S ) indicates the subset of the functions in W ( Q ) having zero trace onthe curve S . We denote the described operator as H ε . The aim of this paper is tostudy the asymptotic behavior of the resolvent and the spectrum of H ε as ε → +0.Let H ( µ ) be the non-negative self-adjoint operator in L (Ω) associated with thesesquilinear form h ( µ ) [ u, v ] := ( ∇ u, ∇ v ) L (Ω) + µ ( u, v ) L ( ∂ Ω) on ˚ W (Ω , Γ + ) , where µ > H ( µ ) consists of the functions in W (Ω) satisfying the boundarycondition ∂u∂x − µu = 0 on Γ − , u = 0 on Γ + , (2.1)and H ( µ ) u = − ∆ u. (2.2)By k · k L (Ω) → L (Ω) and k · k L (Ω) → W (Ω) we denote the norm of an operator actingfrom L (Ω) into L (Ω) and into W (Ω), respectively.Our first main result describes the uniform resolvent convergence for H ε .4igure 1: Waveguide with frequently alternating boundary conditions Theorem 2.1.
Suppose (1.2). Then k ( H ε − i) − − ( H ( µ ) − i) − k L (Ω) → L (Ω) Cεµ | ln εµ | , (2.3) k ( H ε − i) − − ( H (0) − i) − k L (Ω) → W (Ω) Cµ / , (2.4) k ( H ε − i) − − ( H (0) − i) − k L (Ω) → L (Ω) Cµ, (2.5) where the constants C are independent of ε and µ , and µ = µ ( ε ) was defined in(1.3). There exists a corrector W = W ( x, ε, µ ) defined explicitly by (3.17) such that k ( H ε − i) − − (1 + W )( H ( µ ) − i) − k L (Ω) → W (Ω) Cεµ | ln εµ | , (2.6) where the constant C is independent of ε and µ . The spectrum of the operator H (0) is purely essential and coincides with (cid:2) , + ∞ (cid:1) .By [RS1, Ch. VIII, Sec. 7, Ths. VIII.23, VIII.24] and Theorem 2.1 we have Theorem 2.2.
The spectrum of H ε converges to that of H (0) . Namely, if λ (cid:2) , + ∞ (cid:1) , then λ σ ( H ε ) for ε small enough. If λ ∈ (cid:2) , + ∞ (cid:1) , then there exists λ ε ∈ σ ( H ε ) so that λ ε → λ as ε → +0 . The convergence of the spectral projectorsassociated with H ε and H (0) kP ( a,b ) ( H ε ) − P ( a,b ) ( H (0) ) k → , ε → , is valid for a < b . The operator H ε is periodic since the sets γ ε and Γ ε are periodic, and we employthe Floquet decomposition to study its spectrum. We denoteΩ ε := n x : | x | < επ , < x < π o , ˚ γ ε := ∂ Ω ε ∩ γ ε , ˚Γ ε := ∂ Ω ε ∩ Γ ε , ˚Γ ± := ∂ Ω ε ∩ Γ ± . By ˚ H ε ( τ ) we indicate the self-adjoint non-negative operator in L (Ω ε ) associatedwith the sesquilinear form˚ h ε ( τ )[ u, v ] := (cid:18)(cid:18) i ∂∂x − τε (cid:19) u, (cid:18) i ∂∂x − τε (cid:19) v (cid:19) L (Ω ε ) + (cid:18) ∂u∂x , ∂v∂x (cid:19) L (Ω ε ) on ˚ W ,per (Ω ε , ˚Γ + ∪ ˚ γ ε ), where τ ∈ [ − , W ,per (Ω ε , ˚Γ + ∪ ˚ γ ε ) is the set of thefunctions in ˚ W (Ω ε , ˚Γ + ∪ ˚ γ ε ) satisfying periodic boundary conditions on the lateral5oundaries of Ω ε . The operator ˚ H ε ( τ ) has a compact resolvent, since it is boundedas that from L (Ω ε ) into W (Ω ε ), and the space W (Ω ε ) is compactly embeddedinto L (Ω ε ). Hence, the spectrum of ˚ H ε ( τ ) consists of its discrete part only. Wedenote the eigenvalues of ˚ H ε ( τ ) by λ n ( τ, ε ) and arrange them in the ascending orderwith the multiplicities taking into account λ ( τ, ε ) λ ( τ, ε ) . . . λ n ( τ, ε ) . . . By [3, Lm. 4.1] we know that σ ( H ε ) = σ e ( H ε ) = ∞ [ n =1 { λ n ( τ, ε ) : τ ∈ [ − , } , where σ ( · ) and σ e ( · ) indicate the spectrum and the essential spectrum of an opera-tor.By L ε we denote the subspace of L (Ω ε ) consisting of the functions independentof x , and we shall make use the decomposition L (Ω ε ) = L ε ⊕ L ⊥ ε , where L ⊥ ε is the orthogonal complement to L ε in L (Ω ε ). Let Q µ be the self-adjointnon-negative operator in L ε associated with the sesquilinear form q [ u, v ] := (cid:18) dudx , dvdx (cid:19) L (0 ,π ) + µu (0) v (0) on ˚ W ((0 , π ) , { π } ) , i.e., Q µ is the operator − d dx in L (0 , π ) with the domain consisting of the functionsin W (0 , π ) satisfying the boundary conditions u ( π ) = 0 , u ′ (0) − µu (0) = 0 . Our next results are on the uniform resolvent convergence for ˚ H ε ( τ ) and two-terms asymptotics for the first band functions. Theorem 2.3.
Let | τ | < − κ , where < κ < is a fixed constant and suppose(1.2). Then for sufficiently small ε the estimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ˚ H ε ( τ ) − τ ε (cid:19) − − Q − µ ⊕ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω ε ) → L (Ω ε ) C κ − / ( ε / µ + ε ) (2.7) holds true, where the constant C is independent of ε , µ , and κ . Theorem 2.4.
Let the hypothesis of Theorem 2.3 holds true. Then given any N ,for ε < κ / N − the eigenvalues λ n ( τ, ε ) , n = 1 , . . . , N , satisfy the relations λ n ( τ, ε ) = τ ε + Λ n ( µ ) + R n ( τ, ε, µ ) , | R n ( τ, ε, µ ) | C κ − / n ε / µ, (2.8)6 here Λ n ( µ ) , n = 1 , . . . , N , are first N eigenvalues of Q µ , and the constant C isthe same as in (2.7). The eigenvalues Λ n ( µ ) solve the equation √ Λ cos √ Λ π + µ sin √ Λ π = 0 , (2.9) are holomorphic w.r.t. µ , and Λ n ( µ ) = (cid:18) n − (cid:19) + µπ (cid:0) n − (cid:1) + O ( µ ) . (2.10)Let θ ( β ) := − + ∞ X j =1 n p j − β (2 j + p j − β ) . (2.11)It will be shown in Lemma 5.2 that the function θ ( β ) is holomorphic in β and itsTaylor series is θ ( β ) = − + ∞ X j =1 (2 j − ζ (2 j + 1)8 j j ! β j − , (2.12)where ζ is the Riemann zeta-function.Our last main result provides the asymptotic expansion for the bottom of theessential spectrum of H ε . Theorem 2.5.
For ε small enough, the first eigenvalue λ ( τ, ε ) attains its minimumat τ = 0 , inf τ ∈ [ − , λ ( τ, ε ) = λ (0 , ε ) . (2.13) The asymptotics λ (0 , ε ) = Λ( ε, µ ) + O ( µε − / e − ε − + ε / η / ) (2.14) holds true, where Λ( ε, µ ) is the real solution to the equation √ Λ cos √ Λ π + µ sin √ Λ π − ε µ Λ / θ ( ε Λ) cos √ Λ π = 0 (2.15) satisfying the restriction Λ( ε, µ ) = Λ ( µ ) + o (1) , ε → . (2.16) The function Λ( ε, µ ) is jointly holomorphic w.r.t. ε and µ and can be representedas the series Λ( ε, µ ) = Λ ( µ ) + µ ∞ X j =1 ε j +1 K j +1 ( µ ) + µ ∞ X j =2 ε j K j ( µ ) , (2.17)7 here the functions K j ( µ ) are holomoprhic w.r.t. µ , and, in particular, K ( µ ) = − ζ (3)4 Λ ( µ ) π Λ ( µ ) + µ + πµ ,K ( µ ) = 0 ,K ( µ ) = − ζ (5)64 Λ ( µ ) π Λ ( µ ) + µ + πµ ,K ( µ ) = ζ (3)
64 Λ ( µ )(2 π Λ ( µ ) + 7 πµ Λ ( µ ) + 2 π µ Λ ( µ ) + 7 µ + 7 πµ )( π Λ ( µ ) + µ + πµ ) K ( µ ) = − ζ (7)512 Λ ( µ ) π Λ ( µ ) + µ + πµ ,K ( µ ) = 3 ζ (3) ζ (5)512 Λ ( µ )(2 π Λ ( µ ) + 9 πµ Λ ( µ ) + 2 π µ Λ ( µ ) + 9 µ + 9 µ π )( π Λ ( µ ) + µ + πµ ) . (2.18) The asymptotic expansion for the associated eigenfunction of ˚ H ε (0) reads as follows, k ˚ ψ ( · , ε ) − ˚Ψ ε k W (Ω ε ) = O ( µ e − ε − + εη / ) , (2.19) where the function ˚Ψ ε is defined in (5.27). Remark 2.6.
All other coefficients of (2.17) can be determined recursively by sub-stituting this series and (2.12) into (2.15), expanding then (2.15) in powers of ε ,and solving the obtained equations w.r.t. K i . H ε In this section we prove Theorem 2.1. Given a function f ∈ L (Ω), we denote u ε := ( H ε − i) − f, u ( µ ) := ( H ( µ ) − i) − f. The main idea of the proof is to construct a special corrector W = W ( x, ε, µ ) withcertain properties and to estimate the norms of v ε := u ε − (1 + W ) u ( µ ) and u ( µ ) W .In fact, the function W reflects the geometry of the alternation of the boundaryconditions for H ε , and this is why it is much simpler to estimate independently v ε and u ( µ ) W than trying to get directly the estimate for u ε − u ( µ ) and u ε − u (0) . Nextlemma is the first main ingredient in the proof of Theorem 2.1 and it shows how W is employed. Lemma 3.1.
Let W = W ( x, ε, µ ) be an επ -periodic in x function belonging to C (Ω) ∩ C ∞ (Ω \ { x : x = 0 , x = ± εη + επn, n ∈ Z } ) satisfying boundary conditions W = − on γ ε , ∂W∂x = − µ on Γ ε , (3.1) and having differentiable asymptotics W ( x, ε, µ ) = c ± ( ε, µ ) r / ± sin θ ± O ( ρ ± ) , r ± → +0 . (3.2)8 ere ( r ± , θ ± ) are polar coordinates centered at ( ± εη, such that the values θ ± = 0 correspond to the points of γ ε . Assume also that ∆ W ∈ C (Ω) . Then (1 + W ) u ( µ ) belongs to ˚ W (Ω , Γ + ∪ γ ε ) , and k∇ v ε k L (Ω) + i k v ε k L (Ω) = ( f, v ε W ) L (Ω) + ( u ( µ ) ∆ W, v ε ) L (Ω) − u ( µ ) W, v ε ) L (Ω) − W ∇ u ( µ ) , ∇ v ε ) L (Ω) − µ ( u ( µ ) , W v ε ) L (Γ ε ) . (3.3) Proof.
We write the integral identities for u ε and u ( µ ) ,( ∇ u ε , ∇ φ ) L (Ω) + i( u ε , φ ) L (Ω) = ( f, φ ) L (Ω) (3.4)for all φ ∈ ˚ W (Ω , Γ + ∪ γ ε ), and( ∇ u ( µ ) , ∇ φ ) L (Ω) + µ ( u ( µ ) , φ ) L (Γ − ) + i( u ( µ ) , φ ) L (Ω) = ( f, φ ) L (Ω) (3.5)for all φ ∈ ˚ W (Ω , Γ + ). Employing the smoothness of W , (3.1), (3.2), and proceedingas in the proof of Lemma 3.2 in [3], we check that (1 + W ) φ ∈ ˚ W (Ω , Γ + ∪ γ ε ), if φ belongs to the domain of H ε or H ( µ ) . Hence, (1 + W ) u ( µ ) ∈ ˚ W (Ω , Γ + ∪ γ ε ). Thus,(1 + W ) v ε ∈ ˚ W (Ω , Γ + ∪ γ ε ) . (3.6)We take φ = (1 + W ) v ε in (3.5),( ∇ u ( µ ) , ∇ (1 + W ) v ε ) L (Ω) + µ ( u ( µ ) , (1 + W ) v ε ) L (Γ − ) + i( u ( µ ) , (1 + W ) v ε ) L (Ω) = ( f, (1 + W ) v ε ) L (Ω) , ( ∇ u ( µ ) , (1 + W ) ∇ v ε ) L (Ω) + i( u ( µ ) , (1 + W ) v ε ) L (Ω) =( f, (1 + W ) v ε ) L (Ω) − ( ∇ u ( µ ) , v ε ∇ W ) L (Ω) − µ ( u ( µ ) , (1 + W ) v ε ) L (Γ − ) , ( ∇ (1+ W ) u ( µ ) , ∇ v ε ) L (Ω) + i((1 + W ) u ( µ ) , v ε ) L (Ω) =( f, (1 + W ) v ε ) L (Ω) − ( ∇ u ( µ ) , v ε ∇ W ) L (Ω) + ( u ( µ ) ∇ W, ∇ v ε ) L (Ω) − µ ( u ( µ ) , (1 + W ) v ε ) L (Γ − ) . We deduct (3.4) with φ = v ε from the last identity, k∇ v ε k L (Ω) + i k v ε k L (Ω) = − ( f, W v ε ) L (Ω) + ( ∇ u ( µ ) , v ε ∇ W ) L (Ω) − ( u ( µ ) ∇ W, ∇ v ε ) L (Ω) + µ ( u ( µ ) , (1 + W ) v ε ) L (Γ − ) . (3.7)We integrate by parts taking into account (3.1), (3.5), and (3.6),( ∇ u ( µ ) ,v ε ∇ W ) L (Ω) − ( u ( µ ) ∇ W, ∇ v ε ) L (Ω) = ( ∇ u ( µ ) , v ε ∇ W ) L (Ω) + Z Γ ε u ( µ ) ∂W∂x v ε d x + (div u ( µ ) ∇ W, v ε ) L (Ω) = 2( ∇ u ( µ ) , v ε ∇ W ) L (Ω) − µ ( u ( µ ) , v ε ) L (Γ ε ) + ( u ( µ ) ∆ W, v ε ) L (Ω) , and ( ∇ u ( µ ) ,v ε ∇ W ) L (Ω) = ( ∇ u ( µ ) , ∇ W v ε ) L (Ω) − ( ∇ u ( µ ) , W ∇ v ε ) L (Ω) = ( f, W v ε ) L (Ω) − i( u ( µ ) , W v ε ) L (Ω) − µ ( u ( µ ) , W v ε ) L (˚Γ − ) − ( ∇ u ( µ ) , W ∇ v ε ) L (Ω) . We substitute the obtained identities into (3.7) and this completes the proof.9s it follows from (3.3), to prove the smallness of v ε in W (Ω)-norm, it issufficient to construct a function W satisfying the hypothesis of Lemma 3.1 so thatthe quantities W and ∆ W are small in certain sense. This is why we introduce W as a formal asymptotic solution to the equation∆ W = 0 in Ω , (3.8)satisfying (3.1), (3.2) and other assumptions of Lemma 3.1. To construct suchsolution, we shall employ the asymptotic constructions from [4], [25] based on themethod of matching of asymptotic expansions [27] and the boundary layer method[37]. We also mention that similar approach was used in [24, Lm. 1] for constructinga different corrector.First we construct W formally, and after that we shall prove rigourously allthe required properties of the constructed corrector. Denote ξ = ( ξ , ξ ) = xε − , ς ( j ) = ( ς ( j )1 , ς ( j )2 ), ς ( j )1 = ( ξ − πj ) η − , ς ( j )2 = ξ η − . Outside a small neighborhood of γ ε we construct W as a boundary layer W ( x, ε, µ ) = εµX ( ξ ) . We pass to ξ in (3.8) and let η = 0 in the boundary conditions. It yields a boundaryvalue problem for X ,∆ ξ X = 0 , ξ > , ∂X∂ξ = − , ξ ∈ Γ := { ξ : ξ = 0 } \ + ∞ [ j = −∞ { ( πj, } , (3.9)where the function X should be π -periodic in ξ and decay exponentially as ξ → + ∞ . It was shown in [23] that the required solution to (3.9) is X ( ξ ) := Re ln sin( ξ + i ξ ) + ln 2 − ξ . It was also shown that X ∈ C ∞ ( { ξ : ξ > , ξ = ( πj, , j ∈ Z } ) , and this function satisfies the differentiable asymptotics X ( ξ ) = ln | ξ − ( πj, | + ln 2 − ξ + O ( ξ − ( πj, | ) , ξ → ( πj, , j ∈ Z . (3.10)In view of the last identity we rewrite the asymptotics for X as ξ → ( πj,
0) in termsof ς ( j ) , εµX ( ξ ) = εµ (cid:0) ln | ξ − ( πj, | + ln 2 − ξ (cid:1) + O ( εµ | ξ − ( πj, | )= − εµ (cid:0) ln | ς ( j ) | + ln 2 (cid:1) − εµης ( j )2 + O ( εµη | ς ( j ) | ) . (3.11)In accordance with the method of matching of asymptotic expansions it followsfrom the obtained identities that in a small neighborhood of each interval of γ ε weshould construct W as an internal layer, W ( x, ε, µ ) = − εµW ( j ) in ( ς ( j ) ) , (3.12)10here W ( j ) in ( ς ( j ) ) = ln | ς ( j ) | + ln 2 + o (1) , ς ( j ) → + ∞ . (3.13)We substitute (3.12) into (3.8), (3.1), which leads us to the boundary value problemfor W ( j ) in , ∆ ς ( j ) W ( j ) in = 0 , ς ( j )2 > ,W ( j ) in = 0 , ς ( j ) ∈ γ , ∂W ( j ) in ∂ς ( j )2 = 0 , ς ( j ) ∈ Γ ,γ := { ς : | ς | < , ς = 0 } , Γ := Oς \ γ . (3.14)It was shown in [23] that the problem (3.13), (3.14) is solvable and W ( j ) in ( ς ( j ) ) = Y ( ς ( j ) ) , Y ( ς ) := Re ln( z + √ z − , z = ς + i ς , (3.15)where the branch of the root is fixed by the requirement √ Y ( ς ) = ln | ς | + ln 2 + O ( | ς | − ) , ς → ∞ . (3.16)As it follows from the last asymptotics, the term − εµς ( j )2 in (3.11) is not matchedwith any term in the boundary layer. At the same time, it was found in [4], [24], [25]that such terms should be either matched or cancelled out to obtain a reasonableestimate for the error terms. This is also the case in our problem. In contrast to[4], [24], [25], to solve this issue we shall not construct additional terms in W , butemploy a different trick to solve this issue. Namely, we add the function εµξ to theboundary layer and add also − µx as the external expansion. It changes neitherequations nor boundary conditions for W but allows us to cancel out the mentionedterm in (3.11). The final form of W is as follows, W ( x, ε, µ ) = − µx + εµ ( X ( ξ ) + ξ ) + ∞ Y j = −∞ (cid:16) − χ (cid:0) | ς ( j ) | η α (cid:1)(cid:17) + + ∞ X j = −∞ χ (cid:0) | ς ( j ) | η α (cid:1)(cid:0) − εµY ( ς ( j ) ) (cid:1) , (3.17)where α ∈ (0 ,
1) is a constant, which will be chosen later, and χ = χ ( t ) is aninfinitely differentiable cut-off function taking values in [0 , t < t > /
2. It can be easily seen that the sum and the product inthe definition of (3.17) are always finite.Let us check that the function W satisfies the hypothesis of Lemma 3.1. Bydirect calculations we check that the function W is επ -periodic w.r.t. x , belongsto C (Ω) ∩ C ∞ (Ω \ { x : x = 0 , x = ± εη + επn, n ∈ Z } ), and satisfies (3.2). Theboundary condition on γ ε in (3.1) is obviously satisfied. Taking into account theboundary conditions (3.9), (3.13), we check ∂W∂x (cid:12)(cid:12)(cid:12) x ∈ Γ ε = − µ + εµ (cid:18) ∂X∂ξ (cid:12)(cid:12)(cid:12) ξ ∈ Γ + 1 (cid:19) + ∞ Y j = −∞ (cid:0) − χ ( | ς ( j ) | η α ) (cid:1) εµ + ∞ X j = −∞ χ (cid:0) | ς ( j ) | η α (cid:1) ∂Y∂ς ( j )2 (cid:12)(cid:12)(cid:12) ς ( j ) ∈ Γ = − µ, i.e., the boundary condition on Γ ε in (3.1) is satisfied, too.Let us calculate ∆ W . In order to do it, we employ the equations in (3.9), (3.13),∆ W ( x ) = 2 + ∞ X j = −∞ ∇ x χ (cid:0) | ς ( j ) | η α (cid:1) · ∇ x W ( j ) mat ( x, ε, µ )+ + ∞ X j = −∞ W ( j ) mat ( x, ε, µ )∆ x χ (cid:0) | ς ( j ) | η α (cid:1) ,W ( j ) mat ( x, ε, µ ) = − εµ (cid:0) Y ( ς ( j ) ) − X ( ξ ) − ξ (cid:1) . (3.18)It follows from the definition of ξ , ς ( j ) , χ , X , Y , and the last formula that ∆ W ∈ C ∞ (Ω). Thus, we can apply Lemma 3.1. To estimate the right hand side of (3.3)we need two auxiliary lemmas.Given any δ ∈ (0 , π/ δ := + ∞ [ j = −∞ Ω δj , Ω δj := { x : | x − ( πj, | < εδ } ∩ Ω . Lemma 3.2.
For any u ∈ W (Ω) and any δ ∈ (0 , π/ the inequality k u k L (Ω δ ) Cδ (cid:0) | ln δ | / + 1 (cid:1) k u k W (Ω) (3.19) holds true, where the constant C is independent of δ and u .Proof. We begin with the formulas k u k L (Ω δ ) = + ∞ X j = −∞ k u k L (Ω δj ) , (3.20) k u k L (Ω δj ) = Z Ω δj | u ( x ) | d x = ε Z | ξ − ( πj, | <δ, ξ > | u ( εξ ) | d ξ = ε Z | ξ − ( πj, | <δ, ξ > | χ ( ξ − ( πj, u ( εξ ) | d ξ, where χ = χ ( ξ ) is an infinitely differentiable function being one as | ξ | < δ andvanishing as | ξ | > π/
3. We also suppose that the functions χ , χ ′ are boundeduniformly in ξ and δ . Hence, χ ( · − ( πj, u ∈ ˚ W (Π j , ∂ Π j ) , Π j := n ξ : | ξ − πj | < π , < ξ < o . By [28, Lm. 3.2], we obtain ε Z | ξ − ( πj, | <δ, ξ > | χ u | d ξ Cε δ ( | ln δ | + 1) Z Π j (cid:0) |∇ ξ χ u | + | χ u | (cid:1) d ξ Cε δ ( | ln δ | + 1) (cid:0) k∇ ξ u k L (Π j ) + k u k L (Π j ) (cid:1) Cδ ( | ln δ | + 1) k u k W ( { x : | x − επj | <επ/ , For any u ∈ W (Ω) and any δ ∈ (0 , π/ the inequality k u k L ( γ δε ) Cδ / k u k W (Ω) , γ δε := { x : | x − επj | < εδ, x = 0 } , holds true, where the constant C is independent of ε , δ , and u .Proof. It is clear that k u k L ( γ δε ) = + ∞ X j = −∞ k u k L ( γ δε,j ) , γ δε,j := { x : | x − επj | < εδ, x = 0 } . (3.21)It follows from the definition of χ (see the proof of Lemma 3.2) that k u k L ( γ δε,j ) = Z γ δε,j (cid:12)(cid:12)(cid:12) χ (cid:16) x ε − πj (cid:17) u ( x , (cid:12)(cid:12)(cid:12) d x . (3.22)Since χ (cid:16) x ε − πj (cid:17) u ( x , 0) = x Z επj − επ ∂∂x (cid:16) χ (cid:16) x ε − πj (cid:17) u ( x , (cid:17) d x , by the Cauchy-Schwartz inequality we get ∂∂x (cid:16) χ (cid:16) x ε − πj (cid:17) u ( x , (cid:17) = χ (cid:16) x ε − πj (cid:17) ∂u∂x ( x , 0) + ε − χ ′ (cid:16) x ε − πj (cid:17) u ( x ) , (cid:12)(cid:12)(cid:12) χ (cid:16) x ε − πj, (cid:17) u ( x , (cid:12)(cid:12)(cid:12) C ε Z γ ε,j (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂x ( x , (cid:12)(cid:12)(cid:12)(cid:12) d x + ε − Z γ ε,j | u ( x , | d x ,γ ε,j := n x : | x − επj | < επ , x = 0 o , where the constants C are independent of j , ε , δ , and u . The last estimate and(3.22) imply k u k L ( γ δε,j ) Cδ (cid:18)(cid:13)(cid:13)(cid:13) ∂u∂x (cid:13)(cid:13)(cid:13) L ( γ ε,j ) + k u k L ( γ ε,j ) (cid:19) , where the constant C is independent of j , ε , δ , and u . We substitute the obtainedinequality into (3.21) and employ the standard embedding of W (Ω) into W (Γ − )that completes the proof. 13 emma 3.4. The estimates | ∆ W | Cε − µ (1 + η α − ) , x ∈ Ω , (3.23) | W | Cεµ ( | ln δ | + 1) , x ∈ Ω \ Ω δ , η α < δ < π , (3.24) | W | C, x ∈ Ω δ , η α < δ < π , (3.25) are valid, where the constants C are independent of ε , µ , η , δ , and x .Proof. Since W is επ -periodic w.r.t. x , it is sufficient to prove the estimates onlyfor | x | < επ/ 2, 0 < x < π . It follows directly from the definition of X , Y , and(3.13), (3.16) that for any δ ∈ (0 , π/ | X ( ξ ) | C (cid:0) | ln δ | + 1 (cid:1) , | ξ | < π , ξ > , | ξ | > δ, | Y ( ς ) | C (cid:0) | ln δη − | + 1 (cid:1) C (cid:0) | ln δ | + ε − µ − (cid:1) , | ς | δη − , where the constants C are independent of ε , µ , η , δ , and x . These estimates and(3.17) imply (3.24), (3.25).It follows from the definition of χ that ∆ W is non-zero only as η − α < | ς (1) | < η − α . For the corresponding values of x due to (3.13), (3.15) the differentiable asymptotics W (1) mat ( x, ε, µ ) = O (cid:0) εµ ( | ς (1) | − + | ξ | ) (cid:1) , η − α < | ς (1) | < η − α , η − α < | ξ | < η − α , holds true. Hence, for the same values of ξ and ς (1) W (1) mat = O (cid:0) εµ ( η α + η − α ) (cid:1) , ∇ x W (1) mat = O (cid:0) µ ( η − | ς (1) | − + | ξ | ) (cid:1) = O (cid:0) µ ( η − α + η α − ) (cid:1) . Substituting the identities obtained into (3.18) and taking into account the relations ∇ x χ (cid:0) | ς ( j ) | η α (cid:1) = O ( ε − η α − ) , ∆ x χ (cid:0) | ς ( j ) | η α (cid:1) = O ( ε − η α − ) , we arrive at (3.23).Let us estimate the right hand side of (3.3). We have | ( f, W v ε ) L (Ω) | k f k L (Ω) k W v ε k L (Ω) , k W v ε k L (Ω) = k W v ε k L (Ω \ Ω δ ) + k W v ε k L (Ω δ ) . (3.26)Let δ ∈ (cid:0) η α , π (cid:1) . Applying Lemma 3.2 and using (3.24), (3.25), we have k v ε W k L (Ω \ Ω δ ) Cε µ ( | ln δ | + 1) k v ε k L (Ω \ Ω δ ) , k v ε W k L (Ω δ ) Cδ ( | ln δ | + 1) k v ε k W (Ω) . (3.27)14ere and till the end of this section we indicate by C various non-essential constantsindependent of ε , µ , η , δ , x , v ε , u ( µ ) , and f . The inequalities (3.27) yield | ( f, v ε W ) L (Ω) | C (cid:0) εµ | ln δ | + δ | ln δ | / + δ (cid:1) k v ε k W (Ω) k f k L (Ω) . (3.28)It follows from the definition of u ( µ ) that k u ( µ ) k W (Ω) C k f k L (Ω) . (3.29)Taking into account this inequality, we proceed in the same way as in (3.26), (3.27),(3.28), k u ( µ ) W k L (Ω) C ( εµ | ln δ | + δ | ln δ | / + δ ) k u ( µ ) k W (Ω) C ( εµ | ln δ | + δ | ln δ | / + δ ) k f k L (Ω) , (3.30) k W ∇ u ( µ ) k L (Ω) C ( εµ | ln δ | + δ | ln δ | / + δ ) k u ( µ ) k W (Ω) C ( εµ | ln δ | + δ | ln δ | / + δ ) k f k L (Ω) , (3.31) (cid:12)(cid:12) ( u ( µ ) , W v ε ) L (Ω) +( ∇ u ( µ ) , W ∇ v ε ) L (Ω) (cid:12)(cid:12) k u ( µ ) W k L (Ω) k v ε k L (Ω) + k W ∇ u ( µ ) k L (Ω) k∇ v ε k L (Ω) C ( εµ | ln δ | + δ | ln δ | / + δ ) k f k L (Ω) k v ε k W (Ω) . (3.32)Employing (3.23) instead of (3.24), (3.25), and applying then Lemma 3.2 with δ = η α , we get k u ( µ ) ∆ W k L (Ω) = k u ( µ ) ∆ W k L (Ω ηα ) Cη α ε − / µ / (1 + η α − ) k u ( µ ) k W (Ω) Cη α ε − / µ / (1 + η α − ) k f k L (Ω) . (3.33)Using (3.24), (3.25), (3.28), Lemma 3.3 with δ = e δ ∈ ( η α , π/ W (Ω) in W (Γ − ), and proceeding as in (3.26), (3.27), (3.28), we obtain (cid:12)(cid:12) ( u ( µ ) , W v ε ) L (Γ ε ) (cid:12)(cid:12) k u ( µ ) W k L (Γ ε ) k v ε k L (Γ − ) C k u ( µ ) W k L (Γ ε ) k v ε k W (Ω) , k u ( µ ) W k L (Γ ε ) = k u ( µ ) W k L (Γ ε \ γ e δε ) + k u ( µ ) W k L ( γ e δε ) Cε µ ( | ln e δ | + 1) k u ( µ ) k L (Γ ε ) + C e δ k u ( µ ) k W (Ω) C (cid:0)e δ + ε µ ( | ln e δ | + 1) (cid:1) k f k L (Ω) , (3.34) (cid:12)(cid:12) ( u ( µ ) , W v ε ) L (Γ ε ) (cid:12)(cid:12) C (cid:0)e δ / + εµ ( | ln e δ | + 1) (cid:1) k f k L (Ω) , Let α ∈ (1 / , k v ε k W (Ω) C ( δ | ln δ | / + εµ | ln δ | + εµ | ln e δ | + µ e δ / ) k f k L (Ω) k v ε k W (Ω) , and it is assumed here that η α < δ < π/ , η α < e δ < π/ , δ = δ ( ε ) → +0 , e δ = e δ ( ε ) → +0 as ε → +0 . Thus, taking δ = εµ , e δ = ε µ , we get k v ε k W (Ω) Cεµ | ln εµ |k f k L (Ω) , δ = εµ in (3.30) and employ (2.6), k ( H ε − i) − f − ( H ( µ ) − i) − f k L (Ω) = k u ε − u ( µ ) k L (Ω) k u ε − (1 + W ) u ( µ ) k L (Ω) + k u ( µ ) W k L (Ω) Cεµ | ln εµ |k f k L (Ω) , which proves (2.3). Lemma 3.5. The estimate k∇ ( u ( µ ) W ) k L (Ω) Cµ / k f k L (Ω) (3.35) holds true.Proof. We integrate by parts employing (3.1), (3.2), (2.1), (2.2), k∇ ( u ( µ ) W ) k L (Ω) = − (cid:18) ∂∂x u ( µ ) W, u ( µ ) W (cid:19) L (Γ − ) − (cid:0) ∆( u ( µ ) W ) , u ( µ ) W (cid:1) L (Ω) = − µ k u ( µ ) W k L (Γ − ) + Z γ ε | u ( µ ) | ∂W∂x d x + µ ( u ( µ ) , u ( µ ) W ) L (Γ ε ) − ( W ∆ u ( µ ) , W u ( µ ) ) L (Ω) − (cid:0) W ∇ u ( µ ) , u ( µ ) ∇ W (cid:1) L (Ω) − (cid:0) u ( µ ) ∆ W, u ( µ ) W (cid:1) L (Ω) . We take the real part of this identity, k∇ ( u ( µ ) W ) k L (Ω) = µ ( u ( µ ) , u ( µ ) W ) L (Γ ε ) + Z γ ε | u ( µ ) | ∂W∂x d x − µ k u ( µ ) W k L (Γ − ) − Re( W ∆ u ( µ ) , W u ( µ ) ) L (Ω) − (cid:0) W ∇ u ( µ ) , u ( µ ) ∇ W (cid:1) L (Ω) − (cid:0) u ( µ ) ∆ W, u ( µ ) W (cid:1) L (Ω) . (3.36)Let us calculate the fifth term in the right hand side of the last equation. Weintegrate by parts employing (2.1),2 Re (cid:0) W ∇ u ( µ ) , u ( µ ) ∇ W (cid:1) L (Ω) = 12 Z Ω ∇ W · ∇| u ( µ ) | d x = − Z Γ − W ∂∂x | u ( µ ) | d x − Z Ω W ∆ | u ( µ ) | d x = − µ k u ( µ ) W k L (Γ − ) − Re( W u ( µ ) , W ∆ u ( µ ) ) L (Ω) − k W ∇ u ( µ ) k L (Ω) . We substitute the last identity into (3.36), k∇ ( u ( µ ) W ) k L (Ω) = µ ( u ( µ ) , u ( µ ) W ) L (Γ ε ) + Z γ ε | u ( µ ) | ∂W∂x d x + k W ∇ u ( µ ) k L (Ω) − (cid:0) u ( µ ) ∆ W, u ( µ ) W (cid:1) L (Ω) . (3.37)16aking δ = εµ in (3.31), we get k W ∇ u ( µ ) k L (Ω) Cεµ | ln εµ |k f k L (Ω) . (3.38)It follows from (3.30) with δ = εµ and (3.33) that (cid:12)(cid:12) ( u ( µ ) ∆ W, u ( µ ) W ) L (Ω) (cid:12)(cid:12) Cη α ε − / µ / | ln εµ |k f k L (Ω) , α ∈ (1 / , . (3.39)Employing (3.17), (3.15), by direct calculations we check that Z γ ε | u ( µ ) | ∂W∂x d x = + ∞ X j = −∞ Z γ ε,j | u ( µ ) | ∂W∂x d x = εµ + ∞ X j = −∞ Z γ ε,j | u ( µ ) | ∂∂x arcsin x − επjεη d x , and Z γ ε,j | u ( µ ) | ∂∂x arcsin x − επjεη d x = επj Z επj − εη | u ( µ ) | ∂∂x (cid:18) arcsin x − επjεη + π (cid:19) d x + επj + εη Z επj | u ( µ ) | ∂∂x (cid:18) arcsin x − επjεη − π (cid:19) d x , = π | u ( µ ) ( επj, | + Z γ ε,j (cid:18) arcsin x − επjεη − π x − επj ) (cid:19) ∂∂x | u ( µ ) | d x , where π | u ( µ ) ( επj, | = 1 ε επj Z επ ( j − ∂∂x (cid:0) ( x − επ ( j − | u ( µ ) | (cid:1) d x . Thus, in view of the embedding of W (Ω) into W (Γ − ) and (3.29) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z γ ε | u ( µ ) | ∂W∂x d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ + ∞ X j = −∞ επj Z επ ( j − (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x ( x − επ ( j − | u ( µ ) | (cid:12)(cid:12)(cid:12)(cid:12) d x + εµπ + ∞ X j = −∞ Z γ jε (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x | u ( µ ) | (cid:12)(cid:12)(cid:12)(cid:12) d x Cµ k f k L (Ω) . We substitute the obtained estimate, (3.34) with e δ = ε µ , (3.38), (3.39) into (3.37)and arrive at (3.35).The proven lemma and (2.6), (3.30) with δ = εµ imply k ( H ε − i) − − ( H ( µ ) − i) − k L (Ω) → W (Ω) C µ / . (3.40)The resolvent ( H ( µ ) − i) − is obviously analytic in µ and thus k ( H ( µ ) − i) − − ( H (0) − i) − k L (Ω) → W (Ω) Cµ. This inequality, (3.40), and (2.3) yield (2.4), (2.5).17 Uniform resolvent convergence for ˚ H ε ( τ ) This section is devoted to the proof of Theorems 2.3, 2.4. The proof of the firsttheorem is close in spirit to that of Theorem 2.3 in [3]. The difference is that here weemploy the corrector W as we did in the previous section. This is why an essentialmodification of the proof of Theorem 2.3 in [3] is needed.We begin with several auxiliary lemmas. The first one was proved in [3], seeLemma 4.2 in this paper. Lemma 4.1. Let | τ | < − κ , where < κ < , and U ε = (cid:18) ˚ H ε ( τ ) − τ ε (cid:19) − f, f ∈ L (Ω ε ) . Then k U ε k L (Ω ε ) k f k L (Ω ε ) , (4.1) (cid:13)(cid:13)(cid:13) ∂U ε ∂x (cid:13)(cid:13)(cid:13) L (Ω ε ) k f k L (Ω ε ) , (cid:13)(cid:13)(cid:13) ∂U ε ∂x (cid:13)(cid:13)(cid:13) L (Ω ε ) κ / k f k L (Ω ε ) . If, in addition, f ∈ L ⊥ ε , then k U ε k L (Ω ε ) ε κ / k f k L (Ω ε ) , k∇ U ε k L (Ω ε ) ε κ k f k L (Ω ε ) . (4.2)It was also shown in [3] in the proof of the last lemma that for any u ∈ ˚ W ,per (Ω ε , ˚Γ + ) and | τ | − κ (cid:13)(cid:13)(cid:13) (cid:18) i ∂∂x − τε (cid:19) u (cid:13)(cid:13)(cid:13) L (Ω ε ) − τ ε k u k L (Ω ε ) > κ (cid:13)(cid:13)(cid:13) ∂u∂x (cid:13)(cid:13)(cid:13) L (Ω ε ) , (cid:13)(cid:13)(cid:13) ∂u∂x (cid:13)(cid:13)(cid:13) L (Ω ε ) > k u k L (Ω) . (4.3) Lemma 4.2. Let F ∈ L (0 , π ) . Then | ( Q − µ F )(0) | k F k L (0 ,π ) . Proof. We can find Q − µ F explicitly( Q − µ F )( x ) = − π Z (cid:18) | x − t | − π + x − π πµ (1 + µ ( t − π )) (cid:19) F ( t ) d t. Hence, by the Cauchy-Schwartz inequality | ( Q − µ F )(0) | πµ ) π Z (2 π − t ) | F ( t ) | d t k F k L (0 ,π ) , that completes the proof. 18 roof of Theorem 2.3. Let f ∈ L (Ω ε ), f = F ε + f ⊥ ε , where F ε ∈ L ε , f ⊥ ε ∈ L ⊥ ε , F ε ( x ) = 1 επ επ Z − επ f ε ( x ) d x ,επ k F ε k L (0 ,π ) + k f ⊥ ε k L (Ω ε ) = k f k L (Ω ε ) . (4.4)Then (cid:18) ˚ H ε ( τ ) − τ ε (cid:19) − f = (cid:18) ˚ H ε ( τ ) − τ ε (cid:19) − F ε + (cid:18) ˚ H ε ( τ ) − τ ε (cid:19) − f ⊥ ε . By (4.2), (4.4) we obtain (cid:13)(cid:13)(cid:13)(cid:13) (cid:18) ˚ H ε ( τ ) − τ ε (cid:19) − f ⊥ ε (cid:13)(cid:13)(cid:13)(cid:13) L (Ω ε ) ε κ / k f ⊥ ε k L (Ω ε ) ε κ / k f k L (Ω ε ) . (4.5)We denote U ε := (cid:18) ˚ H ε ( τ ) − τ ε (cid:19) − F ε , U ( µ ) ε := Q − µ F ε ,V ε ( x ) := U ε ( x ) − U ( µ ) ε ( x ) − U ( µ ) ε (0) W ( x, ε, µ ) χ ( x ) , where, we remind, the function χ was introduced in the third section. In view of(3.1) and the definition of U ε the function V ε belongs to ˚ W ,per (Ω ε , ˚Γ + ∪ ˚ γ ε ).We write the integral identities for U ε and U ( µ ) ε ,˚ h ε ( τ )[ U ε , φ ] − τ ε ( U ε , φ ) L (Ω ε ) = ( F ε , φ ) L (Ω ε ) (4.6)for all φ ∈ ˚ W ,per (Ω ε , ˚Γ + ∪ ˚ γ ε ), and dU ( µ ) ε dx , dφdx ! L (0 ,π ) + µU ( µ ) ε (0) φ (0) = ( F ε , φ ) L (0 ,π ) (4.7)for all φ ∈ ˚ W ((0 , π ) , { π } ). Given any φ ∈ ˚ W ,per (Ω ε , ˚Γ + ), for a.e. x ∈ ( − επ/ , επ/ φ ( x , · ) ∈ ˚ W ((0 , π ) , { π } ). We take such φ in (4.7) and integrate it over x ∈ ( − επ/ , επ/ dU ( µ ) ε dx , ∂φ∂x ! L (Ω ε ) + µ ( U ( µ ) ε , φ ) L (˚Γ − ) = ( F ε , φ ) L (Ω ε ) . The function U ( µ ) ε is independent of x , and hence (cid:18)(cid:18) i ∂∂x − τε (cid:19) U ( µ ) ε , (cid:18) i ∂∂x − τε (cid:19) φ (cid:19) L (Ω ε ) = − τε (cid:18) U ( µ ) ε , (cid:18) i ∂∂x − τε (cid:19) φ (cid:19) L (Ω ε ) τ ε ( U ( µ ) ε , φ ) L (Ω ε ) . The sum of two last equations is as follows, (cid:18)(cid:18) i ∂∂x − τε (cid:19) U ( µ ) ε , (cid:18) i ∂∂x − τε (cid:19) φ (cid:19) L (Ω ε ) + ∂U ( µ ) ε ∂x , ∂φ∂x ! L (Ω ε ) − τ ε ( U ( µ ) ε , φ ) L (Ω ε ) + µ ( U ( µ ) ε , φ ) L (˚Γ − ) = ( F ε , φ ) L (Ω ε ) (4.8)We let φ = V ε in (4.6), (4.8) and take the difference of these two equations, (cid:18)(cid:18) i ∂∂x − τε (cid:19) ( U ε − U ( µ ) ε ) , (cid:18) i ∂∂x − τε (cid:19) V ε (cid:19) L (Ω ε ) + (cid:18) ∂∂x ( U ε − U ( µ ) ε ) , ∂V ε ∂x (cid:19) L (Ω ε ) − τ ε ( U ε − U ( µ ) ε , V ε ) L (Ω ε ) = µ ( U ( µ ) ε , V ε ) L (˚Γ − ) . We represent U ε − U ( µ ) ε as V ε + U ( µ ) ε (0) W χ and substitute it into the last equation, (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) i ∂∂x − τε (cid:19) V ε (cid:13)(cid:13)(cid:13)(cid:13) L (Ω ε ) + (cid:13)(cid:13)(cid:13)(cid:13) ∂V ε ∂x (cid:13)(cid:13)(cid:13)(cid:13) L (Ω ε ) − τ ε k V ε k L (Ω ε ) = µ ( U ( µ ) ε , V ε ) L (˚Γ ε ) − U ( µ ) ε (0) (cid:18)(cid:16) i ∂∂x − τε (cid:17) W χ , (cid:18) i ∂∂x − τε (cid:19) V ε (cid:19) L (Ω ε ) − U ( µ ) ε (0) (cid:18) ∂W χ ∂x , ∂V ε ∂x (cid:19) L (Ω ε ) − τ ε U ( µ ) ε (0)( W χ , V ε ) L (Ω ε ) = U ( µ ) ε (0) µ ( W, V ε ) L (˚Γ ε ) − ( ∇ W χ , ∇ V ε ) L (Ω ε ) − τε (cid:18) ∂W χ ∂x , V ε (cid:19) L (Ω ε ) ! . (4.9)We integrate by parts employing (3.1), − τε (cid:18) ∂W χ ∂x , V ε (cid:19) L (Ω ε ) = 2i τε (cid:18) W, χ ∂V ε ∂x (cid:19) L (Ω ε ) , and µ ( W, V ε ) L (˚Γ ε ) − ( ∇ ( W χ ) , ∇ V ε ) L (Ω ε ) = µ ( W, V ε ) L (˚Γ ε ) + (cid:18) ∂W∂x , V ε (cid:19) L (˚Γ ε ) + (∆( W χ ) , V ε ) L (Ω ε ) = (∆ W χ , V ε ) L (Ω ε ) . Together with (4.9) it yields (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) i ∂∂x − τε (cid:19) V ε (cid:13)(cid:13)(cid:13)(cid:13) L (Ω ε ) + (cid:13)(cid:13)(cid:13)(cid:13) ∂V ε ∂x (cid:13)(cid:13)(cid:13)(cid:13) L (Ω ε ) − τ ε k V ε k L (Ω ε ) = U ( µ ) ε (0) (cid:0) ∆( W χ ) , V ε (cid:1) L (Ω ε ) + 2i τε (cid:18) W χ , ∂V ε ∂x (cid:19) L (Ω ε ) ! . (4.10)20t follows from Lemma 4.2 and (4.4) that | U ( µ ) ε (0) | πε − / k f k L (Ω ε ) . Hence, we can estimate the right hand side of (4.10) as follows, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U ( µ ) ε (0) (cid:0) ∆( W χ ) , V ε (cid:1) L (Ω ε ) + 2i τε (cid:18) W χ , ∂V ε ∂x (cid:19) L (Ω ε ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) πε − / k f k L (Ω ε ) k ∆( W χ ) k L (Ω ε ) k V ε k L (Ω ε ) + 2 ε − k W χ k L (Ω ε ) (cid:13)(cid:13)(cid:13)(cid:13) ∂V ε ∂x (cid:13)(cid:13)(cid:13)(cid:13) L (Ω ε ) ! π ε − k ∆( W χ ) k L (Ω) k f k L (Ω ε ) + 18 k V ε k L (Ω ε ) + 25 π κ − ε − k W k L (Ω ε ) k f k L (Ω ε ) + κ (cid:13)(cid:13)(cid:13)(cid:13) ∂V ε ∂x (cid:13)(cid:13)(cid:13)(cid:13) L (Ω ε ) . We substitute this inequality and (4.3) into (4.10), κ (cid:13)(cid:13)(cid:13)(cid:13) ∂V ε ∂x (cid:13)(cid:13)(cid:13)(cid:13) L (Ω ε ) + 14 k V ε k L (Ω ε ) π ε − k f k L (Ω ε ) k ∆( W χ ) k L (Ω ε ) + 25 π κ − ε − k W k L (Ω ε ) k f k L (Ω ε ) + 18 k V ε k L (Ω ε ) + κ (cid:13)(cid:13)(cid:13)(cid:13) ∂V ε ∂x (cid:13)(cid:13)(cid:13)(cid:13) L (Ω ε ) , k V ε k L (Ω ε ) C (cid:0) ε − k f k L (Ω ε ) k ∆( W χ ) k L (Ω ε ) + κ − ε − k f k L (Ω ε ) k W k L (Ω ε ) (cid:1) , k V ε k L (Ω ε ) C (cid:0) ε − / k ∆( W χ ) k L (Ω ε ) + κ − / ε − / k W k L (Ω ε ) (cid:1) k f k L (Ω ε ) , where the constants C are independent of ε , µ , κ , and f . Combining the lastinequality, (4.4) and Lemma 4.2, we arrive at k U ε − U ( µ ) ε k L (Ω ε ) k V ε k L (Ω ε ) + | U ( µ ) (0) |k W k L (Ω ε ) k V ε k L (Ω ε ) + Cε − / k f k L (Ω ε ) k W k L (Ω ε ) C (cid:0) ε − / k ∆ W χ k L (Ω ε ) + κ − / ε − / k W k L (Ω ε ) (cid:1) k f k L (Ω ε ) , (4.11)where the constants C are independent of ε , µ , κ , and f .Let us estimate k W k L (Ω ε ) and k ∆( W χ ) k L (Ω ε ) . We have k W k L (Ω ε ) = k W k L (Ω ε \ Ω δ ) + k W k L (Ω ε ∩ Ω δ ) . We take δ = η α and in view of the definition (3.17) of W we obtain k W k L (Ω ε \ Ω δ ) = ε µ k X k L (Ω ε \ Ω δ ) ε µ Z | ξ | < π , ξ > | X ( ξ ) | d ξ Cε µ , where the constant C is independent of ε , µ , κ , and f . It follows from (3.25) that k W k L (Ω ε ∩ Ω ηα ) Cε η α , α ∈ (0 , , C is independent of ε and η . Hence, k W k L (Ω ε ) Cε µ, (4.12)where the constant C is independent of ε and µ .The definition (3.17) of W , the equations in (3.9), (3.14), the estimate (3.23),and the exponential decay of X , X ( ξ ) = O (e − ξ ) , ξ → + ∞ yield that k ∆( W χ ) k L (Ω ε ) k ∆ W k L (Ω ε ) + 2 (cid:13)(cid:13)(cid:13)(cid:13) ∂W∂x χ ′ + W χ ′′ (cid:13)(cid:13)(cid:13)(cid:13) L (Ω ε ) , k ∆ W k L (Ω ε ) Cµ η − α , α ∈ (1 / , , (cid:13)(cid:13)(cid:13)(cid:13) ∂W∂x χ ′ + W χ ′′ (cid:13)(cid:13)(cid:13)(cid:13) L (Ω ε ) Cµ e − ε − , where C are positive constants independent of ε , η , and µ . We substitute the lastestimates and (4.12) into (4.11), k U ε − U ( µ ) ε k L (Ω ε ) C κ − / µε / k f k L (Ω ε ) , where the constant C is independent of ε , µ , and κ . Together with (4.5) it completesthe proof. Proof of Theorem 2.4. First we obtain the upper bound for the eigenvalues λ n . Todo this, we employ standard bracketing arguments (see, for instance, [33, Ch. XIII,Sec. 15, Prop. 4]), and estimate the eigenvalues of ˚ H ε ( τ ) by those of the sameoperator but with η = π/ 2, i.e., with Dirichlet boundary condition on ˚Γ − . Thelowest eigenvalues of the latter operator are τ ε + n , (2 + τ ) − τ ε + n , (2 − τ ) − τ ε + n , n = 1 , , . . . Hence, for n < κ ε − the lowest eigenvalues among mentioned are τ ε − + n , andthus 14 λ n ( τ, ε ) − τ ε n , n < κ / ε − . (4.13)The lower estimate was obtained by replacing the boundary conditions on ˚Γ − bythe Neumann one. In the same way we can estimate the eigenvalues of Q µ replacingthe boundary condition at x = 0 by the Dirichlet and Neumann one,0 Λ n ( µ ) n (4.14)uniformly in µ for all n ∈ Z .By [29, Ch. III, Sec. 1, Th. 1.4], Theorem 2.3, and (4.13), (4.14) we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ n ( τ, ε ) − τ ε − n ( µ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C κ − / ε / µ, (cid:12)(cid:12)(cid:12) λ n ( τ, ε ) − τ ε − Λ n ( µ ) (cid:12)(cid:12)(cid:12)(cid:12) C κ − / ( µε / + ε ) | Λ n ( µ ) | (cid:12)(cid:12)(cid:12)(cid:12) λ n ( τ, ε ) − τ ε (cid:12)(cid:12)(cid:12)(cid:12) Cn κ − / ( µε / + ε ) , which proves (2.8).The eigenvalues Λ n ( µ ) are solutions to the equation (2.9), and the associatedeigenfunctions are sin √ Λ n ( x − π ). Hence, these eigenvalues are holomorphic withrespect to µ by the inverse function theorem. The formula (2.10) can be checkedby expanding the equation (2.15) and Λ n ( µ ) w.r.t. µ . In this section we prove Theorem 2.5. The proof of (2.13) reproduces word by wordthe proof of similar equation (2.5) in [3] with one minor change, namely, one shoulduse here identity λ (0 , ε ) = 14 + o (1) , ε → +0 , (5.1)instead of similar identity in [3]. The identity (5.1) follows from (2.8), (2.10).In order to construct the asymptotic expansion for λ (0 , ε ), we employ the ap-proach suggested in [4], [23], [24], [25] for studying similar problems in boundeddomains.The eigenvalue λ (0 , ε ) and the associated eigenfunction ˚ ψ ( x, ε ) of ˚ H ε (0) satisfythe problem − ∆˚ ψ ( x, ε ) = λ (0 , ε )˚ ψ ( x, ε ) in Ω ε , ˚ ψ ( x, ε ) = 0 on ˚Γ + ∪ ˚ γ ε , ∂ ˚ ψ∂x ( x, ε ) = 0 on ˚Γ ε . (5.2)and periodic boundary conditions on the lateral boundaries of Ω ε . We constructthe asymptotics for λ (0 , ε ) as λ (0 , ε ) = Λ( ε, µ ) , where Λ = Λ( ε, µ ) is a function to be determined. It view of (2.8) with τ = 0 thefunction Λ should satisfy (2.16).The asymptotics of the associated eigenfunction ˚ ψ ε is constructed as the sumof three expansion, namely, the external expansion, the boundary layer, and theinternal expansion. The external expansion has a closed form, ψ exε ( x, Λ) = sin √ Λ( x − π ) . (5.3)It is clear that for any choice of Λ( ε, µ ) this function solves the equation in (5.2),and satisfies the periodic boundary conditions on the lateral boundaries of Ω ε .The boundary layer is constructed in terms of the variables ξ , i.e., ψ blε = ψ blε ( ξ, µ ).The main aim of introducing the boundary layer is to satisfy the boundary conditionon ˚Γ ε . We construct ψ blε by the boundary layer method. In accordance with this23ethod, the series ψ blε should satisfy the equation in (5.2), the periodic boundarycondition on the lateral boundaries of Ω ε , the boundary condition ∂ψ exε ∂x + ∂ψ blε ∂x = 0 on ˚Γ ε , (5.4)and it should decay exponentially as ξ → + ∞ .It follows from (5.3) and the definition of ξ that ψ blε should satisfy the boundarycondition ∂ψ blε ∂ξ = −√ Λ cos √ Λ π on ˚Γ , (5.5)˚Γ := n ξ : 0 < | ξ | < π , ξ > o . Here we passed to the limit η → +0 in the definition of ˚Γ ε .We substitute ψ blε into the equation in (5.2) and rewrite it in the variables ξ , − ∆ ξ ψ blε = ε Λ ψ blε , ξ ∈ Π , Π := n ξ : | ξ | < π , ξ > o . (5.6)To construct ψ blε , in [4], [23], [24], [25] the authors used the standard way. Namely,they sought ψ blε and Λ( ε, µ ) as asymptotic series power in ε . Then these series weresubstituted into (5.5), (5.6), and equating the coefficients at like powers of ε impliedthe boundary value problems for the coefficients of the mentioned series. In our casewe do not employ this way. Instead of this we study the existence of the requiredsolution to the problem (5.5), (5.6) and describe some of its properties needed inwhat follows.By V we denote the space of π -periodic even in ξ functions belonging to C ∞ (Π \{ } ) and exponentially decaying as ξ → + ∞ together with all their derivativesuniformly in ξ . We observe that X ∈ V . Lemma 5.1. The function X can be represented as the series X ( ξ ) = − + ∞ X n =1 n e − nξ cos 2 nξ , (5.7) which converges in L (Π) and in C k (Π ∩ { ξ : ξ > R } ) for each k > , R > .Proof. Since X ∈ V , for each ξ > k > C k [ − π/ , π/ X ( ξ ) = + ∞ X n =1 X n ( ξ ) cos 2 nξ , k X ( · , ξ ) k L ( − π , π ) = π + ∞ X n =1 X n ( ξ ) , (5.8) X n ( ξ ) = 2 π π Z − π X ( ξ ) cos 2 nξ d ξ . ξ , we obtain the Parseval identity k X k L (Π) = π + ∞ X n =1 k X n k L (0 , + ∞ ) . It yields that the first series in (5.8) converges also in L (Π), since (cid:13)(cid:13)(cid:13) X − N X n =1 X n cos 2 nξ (cid:13)(cid:13)(cid:13) L (Π) = k X k L (Π) − π N X n =1 k X n k L (0 , + ∞ ) . The harmonicity of X and the exponential decay as ξ → + ∞ yield X ′′ n ( ξ ) = − π Z − π ∂ X∂ξ cos 2 nξ d ξ = − n X n ( ξ ) ,X n ( ξ ) = k n e − nξ , k n = 2 π Z ˚Γ X n cos 2 nξ d ξ . Denote Π δ := Π \ { ξ : | ξ | < δ } . Employing (3.9) and the harmonicity of X , weintegrate by parts,0 = − lim δ → +0 Z Π e − nξ cos 2 nξ ∆ ξ X d ξ = Z ˚Γ (cid:18) cos 2 nξ ∂X∂ξ + 2 nX cos 2 nξ (cid:19) d ξ + lim δ → +0 Z | ξ | <δ, ξ > (cid:18) e − nξ cos 2 nξ ∂X∂ | ξ | − X ∂∂ | ξ | e − nξ cos 2 nξ (cid:19) d s = − Z ˚Γ cos 2 nξ d ξ + πnk n + π. (5.9)Thus, k n = − /n , which implies (5.7). The convergence of this series in C k (Π ∩ { ξ : ξ > R } ) follows from the exponential decay of its terms in (5.6) as n → + ∞ . Lemma 5.2. For small real β the problem − ∆ ξ Z − β Z = β X, ξ ∈ Π , ∂Z∂ξ = 0 , ξ ∈ ˚Γ , (5.10) has a solution in W (Π) ∩ V . This solution and all its derivatives w.r.t. ξ decayexponentially as ξ → + ∞ uniformly in ξ and β . The differentiable asymptotics Z ( ξ, β ) = Z (0 , β ) + O ( | ξ | ln | ξ | ) , ξ → , (5.11) holds true uniformly in β . The function ( X + Z ) is bounded in L (Π) uniformly in β . The identity Z (0 , β ) = β θ ( β ) (5.12) is valid, where the function θ is defined in (2.11). The function θ is holomorphicand its Taylor series is (2.12). roof. Let W be the subspace of W (Π) consisting of the functions satisfying peri-odic boundary conditions on the lateral boundaries of Π, the Neumann boundarycondition on ˚Γ , and being orthogonal in L (Π) to all functions φ = φ ( ξ ) belongingto L (Π). The space W is the Hilbert one.By B we denote the operator in L (Π) acting as − ∆ ξ on W . This operator issymmetric and closed. It follows from the definition of W that each v ∈ W satisfiesthe equation π Z − π v ( ξ ) d ξ = 0 for a.e. ξ ∈ (0 , + ∞ ) . Using this fact, one can check easily that B > 4, and therefore the bounded inverseoperator exists, and kB − k / 4. Hence,( B − β ) − = B − (I − β B − ) − , i.e., the inverse operator ( B − β ) − exists and is bounded uniformly in β .We let Z := β ( B − β ) − X . It is clear that the function Z ∈ W (Π) solves(5.10) and satisfies the periodic boundary conditions on the lateral boundaries ofΠ. By the standard smoothness improving theorems and the smoothness of X weconclude that Z ∈ C ∞ (Π \ { } ).Using Lemma 5.1, for ξ > Z by the separation ofvariables, Z ( ξ, β ) = + ∞ X n =1 n e − nξ − n p n − β e − √ n − β ξ ! cos 2 nξ . (5.13)In the same way as in the proof of Lemma 5.1 one can check that this series convergesin L (Π) and C k (Π ∩ { ξ : ξ > R } ) for each k > R > 0. Thus, this function andall its derivatives w.r.t. ξ decay exponentially as ξ → + ∞ uniformly in ξ and β ,and Z ∈ V .By (5.7), (5.13) we have X + Z = − + ∞ X n =1 p n − β e − √ n − β ξ cos 2 nξ , k X + Z k L (Π) = + ∞ X n =1 π n − β ∞ Z n =1 e − √ n − β ξ d ξ = + ∞ X n =1 π n − β ) / . Hence, the function ( X + Z ) is bounded in L (Π) uniformly in β .Reproducing the proof of Lemma 3.2 in [22], one can show easily that the func-tion Z satisfies differentiable asymptotics (5.11) uniformly in β . Let us calculate Z (0 , β ). The function e Z ( ξ, β ) := X ( ξ ) + Z ( ξ, β ) + β − sin βξ (5.14)26olves the boundary value problem(∆ ξ + β ) e Z = 0 , ξ ∈ Π , ∂ e Z∂ξ = 0 , ξ ∈ ˚Γ , is bounded, satisfies periodic boundary condition on the lateral boundaries of Π,and has the asymptotics e Z ( ξ, β ) = ln | ξ | + O (1) , ξ → . Using these properties and (5.10), we integrate by parts in the same way as in (5.9), β Z Π X e Z d ξ = − lim δ → +0 Z Π δ e Z (∆ ξ + β ) Z d ξ = lim δ → +0 Z | ξ | = δ, ξ > e Z ∂Z∂ | ξ | − Z ∂ e Z∂ | ξ | ! d s = − πZ (0 , β ) , and hence Z (0 , β ) = − β π Z Π X e Z d ξ. We substitute (5.7), (5.13), (5.14) into the last identity, Z (0 , β ) = − β ∞ X n =1 n p n − β ∞ Z e − (2 n + √ n − β ) ξ d ξ = − β ∞ X n =1 n p n − β (2 n + p n − β )that proves (5.12).The series in the definition of θ converges uniformly in β , and by the firstWeierstrass theorem this function is holomorphic in small β . It is easy to see that1 n p n − β (2 n + p n − β ) = 2 n − p n − ββn p n − β = 1 β p n − β − n ! = 1 β n q − β n − n = + ∞ X j =1 (2 j − β j − j n j +1 j ! . We substitute this identity into the definition of θ ( β ), θ ( β ) = − + ∞ X n =1 + ∞ X j =1 (2 j − β j − j n j +1 j ! = − + ∞ X j =1 (2 j − ζ (2 j + 1) β j − j j ! , which yields (2.12). The proof is complete.27e choose the boundary layer as ψ blε ( ξ, Λ) = ε √ Λ cos √ Λ π (cid:0) X ( ξ ) + Z ( ξ, ε √ Λ) (cid:1) . (5.15)It is clear that this function satisfies all the aforementioned requirements for theboundary layer.In accordance with Lemma 5.2, the boundary layer has a logarithmic singularityat ξ = 0, and the sum of the external expansion and the boundary layer does notsatisfy the boundary condition on ˚ γ ε in (5.2). This is the reason of introducingthe internal expansion. We construct it as depending on ς := ς (1) and employ themethod of matching of the asymptotic expansions. It follows from (5.3), (2.9) that ψ exε ( x, µ ) = ψ exε (0 , µ ) + ∂ψ exε ∂x (0 , µ ) x + O ( | x | ) , x → , (5.16) ψ exε (0 , µ ) = − sin p Λ( ε, µ ) π, (5.17)where the asymptotics is uniform in Λ( ε, µ ). Using the definition of ς = ξη − and(1.3), by (5.15), (5.11), (3.10) we obtain ψ blε ( ξ, Λ) = √ Λ cos √ Λ π (cid:18) − µ + ε (ln | ς | + ln 2) − x (cid:19) + ε Λ / θ ( ε Λ) cos √ Λ π + O ( ε | ξ | ln | ξ | ) , ξ → , uniformly in ε and Λ. In view of (5.5), (5.16), (5.17) we have ψ exε ( x, Λ) + ψ blε ( ξ, Λ) = − √ Λ µ cos √ Λ π − sin √ Λ π + ε Λ / θ ( ε Λ) cos √ Λ π + ε √ Λ cos √ Λ π (ln | ζ | + ln 2) + O (cid:0) εη | ζ | ( | ln | ζ || + | ln η | ) (cid:1) , as x → 0. Hence, in accordance with the method of matching of asymptotic expan-sions we conclude that the internal expansion should be as follows, ψ inε ( ς, Λ) = ψ in ( ζ , Λ , ε ) + εψ in ( ζ , Λ , ε ) , (5.18)where the coefficients should satisfy the asymptotics ψ in ( ς, Λ , ε ) = − √ Λ µ cos √ Λ π − sin √ Λ π + ε Λ / θ ( ε Λ) cos √ Λ π + o (1) , ς → ∞ , (5.19) ψ in ( ς, Λ) = ε √ Λ cos √ Λ π (ln | ζ | + ln 2) + o (1) , ς → ∞ . We substitute (5.18) into (5.2) and pass to the variables ς . It yields the boundaryvalue problems for ψ ini ,∆ ς ψ ini = 0 , ς > , ψ ini = 0 , ς ∈ ˚ γ , ∂ψ ini ∂ς = 0 , ς ∈ ˚Γ . (5.20)28or i = 0 this problem has the only bounded solution which is trivial, ψ in = 0 . (5.21)Thus, by (5.19) we obtain the equation (2.15) for Λ( ε, µ ).In view of the properties of the function Y described in the third section thefunction ψ in should be chosen as ψ in ( ζ , Λ , ε ) = ε √ Λ cos √ Λ πY ( ζ ) . (5.22)The formal constructing of λ (0 , ε ) and ˚ ψ ε is complete.We proceed to the studying of the equation (2.15). Since the function θ isholomorphic by Lemma 5.2, the function T ( ε, µ, Λ) := √ Λ cos √ Λ π + µ sin √ Λ π − ε µ Λ / θ ( ε Λ) cos √ Λ π is jointly holomorphic w.r.t. small ε , µ , and Λ close to 1 / 4. Employing the formula(2.12), we continue T analytically to complex values of ε , µ , and Λ.As ε = µ = 0, the equation (2.15) becomes √ Λ cos √ Λ π = 0 , and it has the root Λ = 1 / 4. It is clear that ∂T∂ Λ (cid:18) , , (cid:19) = 0 . Hence, by the inverse function theorem there exists the unique root of the equation(2.15). This root is jointly holomorphic in ε and µ and satisfies (2.16). We representthis root as Λ( ε, µ ) = Λ ( µ ) + + ∞ X j =1 ε j e K j ( µ ) , (5.23)where e K j ( µ ) are holomorphic in µ functions. We choose the leading term in thisseries as Λ ( µ ), since as ε = 0 the equation (2.15) coincides with (2.9).We substitute (5.23) and (2.12) into (2.15) and equate the coefficients at ε i , i = 1 , . . . , 8. It implies the equations for e K i , i = 1 , . . . , 8. Solving these equations,we obtain e K = e K = 0 and (2.18).Let us prove that e K j +1 ( µ ) = µ K j +1 ( µ ), e K j ( µ ) = µ K j ( µ ), where K j ( µ ) areholomorphic in µ functions. It is sufficient to prove that e K j (0) = e K ′ j (0) = 0 , e K ′′ j (0) = 0 . We take µ = 0 in (2.15) and (5.23), p Λ(0 , ε ) cos p Λ(0 , ε ) π = 0 , (5.24)Λ(0 , ε ) = 14 . (5.25)29y (2.10), (5.23) it implies e K j (0) = 0. We differentiate the equation (2.15) w.r.t. µ and then we let µ = 0. It implies the equation − π p Λ( ε, 0) sin p Λ( ε, π − cos p Λ( ε, π p Λ( ε, ∂ Λ ∂µ ( ε, − ε Λ / ( ε, θ ( ε Λ( ε, p Λ( ε, π + sin p Λ( ε, π = 0 . We substitute here the identity (5.25) and arrive at the equation − π ∂ Λ ∂µ ( ε, 0) + 1 = 0 , which by (2.10) implies ∂ Λ ∂µ ( ε, 0) = 2 π = ∂ Λ ∂µ (0) . (5.26)These identities and (5.23) yield e K ′ j (0) = 0.We differentiate the equation (2.15) twice w.r.t. µ and then we let µ = 0 takinginto account the identities (5.25), (5.26), and (2.12), − π + ε θ (cid:16) ε (cid:17) − π ∂ Λ ∂µ ( ε, 0) = 0 ,∂ Λ ∂µ ( ε, 0) = 1 π (cid:18) − ε πθ (cid:16) ε (cid:17)(cid:19) = − π π + ∞ X j =1 (2 j − ζ (2 j + 1)32 j − j ! ε j +1 ! . Hence, e K ′′ j (0) = 0, j > ε . Itimplies certain equations, which can be solved w.r.t. K i . Since all the coefficientsin the expansion in ε of θ and other terms in the equation (2.15) are real, thefunctions K i are real, too. Hence, by (2.17) the function Λ is real-valued for real ε and µ .We proceed to the justification of the asymptotics. Denote˚Ψ ε ( x ) := (cid:0) ψ exε ( x, Λ( ε, µ )) + χ ( x ) ψ blε ( ξ, Λ( ε, µ )) (cid:1)(cid:0) − χ ( | ς | η / ) (cid:1) + χ (cid:0) | ς | η / (cid:1) ψ inε ( ς, Λ( ε, µ )) . (5.27)where, we remind, χ is the cut-off function introduced in the third section. Lemma 5.3. The function ˚Ψ ε ∈ C ∞ (Ω ε \ { x : x = ± εη, x = 0 } ) belongs to thedomain of ˚ H ε (0) , satisfies the convergence (cid:13)(cid:13)(cid:13)(cid:13) ˚Ψ ε − sin x − π (cid:13)(cid:13)(cid:13)(cid:13) L (Π) = O ( ε / µ ) , ε → +0 , (5.28) and solves the equation (cid:0) ˚ H ε (0) − Λ( ε, µ ) (cid:1) ˚Ψ ε = h ε , (5.29) where for the function h ε ∈ L (Ω ε ) an uniform in ε , µ , and η estimate k h ε k L (Ω ε ) C ( µ e − ε − + εη / ) (5.30) holds true. roof. It follows from the definition of ˚Ψ ε that˚Ψ ε ∈ C ∞ (Ω ε \ { x : x = ± εη, x = 0 } ) ∩ ˚ W ,per (Ω ε , ˚Γ + ) . (5.31)The boundary condition (5.4), (5.17), and (3.14) for Y yield those for ˚Ψ ε ,˚Ψ ε = 0 on ˚Γ + ∪ ˚ γ ε , ∂ ˚Ψ ε ∂x = 0 on ˚Γ ε . (5.32)Let us show that − (∆ ξ + Λ( ε, µ ))˚Ψ ε = h ε , x ∈ Ω ε , (5.33)where h ε ∈ L (Ω ε ) satisfies (5.30). Employing the equations (5.6), (5.20), we obtain − (∆ ξ + Λ)˚Ψ ε = h ε , h ε = − ( h (1) ε + h (2) ε + h (3) ε ) , (5.34) h (1) ε ( x ) = 2 χ ′ ( x ) ∂∂x ψ blε ( ξ, Λ( ε, µ )) + χ ′′ ( x ) ψ blε ( ξ, Λ( ε, µ )) ,h (2) ε ( x ) = Λ( ε, µ ) χ ( | ς | η / ) ψ inε ( ς, Λ( ε, µ )) ,h (3) ε ( x ) = 2 ∇ x χ ( | ς | η / ) · ∇ x ˚Ψ matε ( x ) + ˚Ψ ( mat ) ε ( x )∆ x χ ( | ς | η / ) , ˚Ψ ( mat ) ε ( x ) := ψ inε ( ς, Λ( ε, µ )) − ψ exε ( x, Λ( ε, µ )) − ψ blε ( ξ, Λ( ε, µ )) . (5.35)It is clear that h ( i ) ε ∈ L (Ω ε ) that implies the same for h ε .Due to (2.15) the function ψ blε can be rewritten as follows, ψ blε ( ξ, Λ( ε, µ )) = µ (cid:0) ε Λ / ( ε, µ ) θ ( ε Λ( ε, µ )) cos p Λ( ε, µ ) π − sin p Λ( ε, µ ) π (cid:1)(cid:0) X ( ξ ) + Z ( ξ, ε p Λ( ε, µ )) (cid:1) . Thus, h (1) ε ( x ) = µ (cid:0) ε Λ / ( ε, µ ) θ ( ε Λ( ε, µ )) cos p Λ( ε, µ ) π − sin p Λ( ε, µ ) π (cid:1)(cid:16) χ ′ ( x ) ∂∂x + χ ′′ ( x ) (cid:17)(cid:0) X ( ξ ) + Z ( ξ, ε p Λ( ε, µ )) (cid:1) . The functions χ ′ ( x ), χ ′′ ( x ) are non-zero only for 1 < x < that corresponds to ε − < ξ < ε − . For such values of ξ we can use the series (5.7), (5.13) for X and Z which converge in C k (cid:0) (cid:8) ξ : ε − ξ ε − , | ξ | π (cid:9) (cid:1) . It yields the exponentialestimate for h (1) ε , k h (1) ε k L (Ω ε ) Cµ e − ε − , (5.36)where the constant C is independent of ε and µ .Taking into account (5.21), and replacing in (5.22) the factor √ Λ cos √ Λ π by µ (cid:0) ε Λ / ( ε, µ ) θ ( ε Λ( ε, µ )) cos p Λ( ε, µ ) π − sin p Λ( ε, µ ) π (cid:1) as we did it in (5.34), weestimate h (2) ε , k h (2) ε k L (Ω ε ) Cε µ η Z | ς | <η − / , ς > | Y ( ς ) | d ς Cε µ η | ln η | Cε η, (5.37)31here the constants C are independent of ε , µ , and η .The asymptotics (3.10), (5.11), (3.16), the equation (2.15), and the identities(5.3), (5.15), (5.18), (5.21), (5.22) imply the differentiable asymptotics for ˚Ψ matε ,˚Ψ matε ( x ) = ε √ Λ cos √ Λ π (cid:0) ln | ς | + ln 2 + O ( | ς | − ) (cid:1) − sin √ Λ( x − π ) − ε √ Λ cos √ Λ π (cid:0) ln | ξ | + ln 2 + ε Λ θ ( ε Λ) − ξ + O ( | ξ | ) (cid:1) = − sin √ Λ( x − π ) − sin √ Λ π + √ Λ x cos √ Λ π + O (cid:0) εµ ( | ξ | + | ς | − ) (cid:1) = O (cid:0) | x | + εµ ( | ξ | + | ς | − ) (cid:1) uniformly in ε , µ , and η as εη / < | x | < εη / , x ∈ Ω ε . (5.38)Thus, for such x | ˚Ψ matε ( x ) | C ( ε ( ε + µ ) η ) , |∇ x ˚Ψ matε ( x ) | C (( ε + µ ) η / ) , where the constants C are independent of x , ε , µ , and η . Since the functions ∇ x χ ( | ς | η / ), ∆ x χ ( | ς | η / ) are non-zero only for x satisfying (5.38), the last in-equalities for ˚Ψ matε and ∇ x ˚Ψ matε enable us to estimate h (3) ε , k h (3) ε k L (Ω ε ) C (( ε + µ ) η / ) , where the constant C is independent of ε , µ , and η . We sum the last estimate and(5.36), (5.37), k h ε k L (Ω ε ) C ( µ e − ε − + εη / ) , where the constant C is independent of ε , µ , and η . This estimate imply (5.30).Due to the smoothness (5.31) of ˚Ψ ε , the boundary value conditions (5.32), andthe equation (5.33), the function ˚Ψ ε is a generalized solution to the boundary valueproblem (5.33), (5.32). Hence, ˚Ψ ε belongs to the domain of ˚ H ε (0).Let us prove the estimate (5.28). Completely as in the estimating h ε , we checkthat k χ ( x ) ψ blε (cid:0) − χ ( | ς | η / ) (cid:1) + χ (cid:0) | ς | η / (cid:1) ψ inε − ψ exε χ ( | ς | η / ) k L (Ω ε ) = O ( ε µ ) . In view of (2.10) and the definition (5.3) of ψ exε the estimate (cid:13)(cid:13)(cid:13)(cid:13) ψ exε − sin x − π (cid:13)(cid:13)(cid:13)(cid:13) L (Π) = O ( ε / µ )holds true. Two last estimates and the definition (5.27) of ˚Ψ ε imply (5.28).We proceed to the estimating of the error terms. The core of these estimates areLemmas 12, 13 in [37]. We employ these results in the form they were formulatedin [29, Ch. III, Sec. 1.1, Lm. 1.1]. For the reader’s convenience we provide thislemma below. 32 emma 5.4. Let A : H → H be a continuous linear compact self-adjoint operatorin a Hilbert space H . Suppose that there exist a real M > and a vector u ∈ H ,such that k u k H = 1 and kA u − M u k H κ , α = const > . Then there exists an eigenvalue M i of operator A such that | M i − µ | κ . Moreover, for any d > κ there exists a vector u such that k u − u k H κ d − , k u k H = 1 , and u is a linear combination of the eigenvectors of the operator A correspondingto the eigenvalues of A from the segment [ M − d, M + d ] . Since the operator ˚ H ε (0) is non-negative and self-adjoint in L (Ω ε ) and satisfies(4.1), the inverse A := ˚ H − ε (0) exists, is bounded and self-adjoint, and satisfies theestimate kAk . (5.39)The operator A is also bounded as that from L (Ω ε ) into W (Ω ε ) and in view ofthe compact embedding of W (Ω ε ) in L (Ω ε ) the operator A is compact in L (Ω ε ).We rewrite the equation (5.29) as follows,Λ − ( ε, µ )˚Ψ ε = A ˚Ψ ε + e h ε , e h ε := Λ − ( ε, µ ) A h ε . By (2.16), (2.10), (5.39), (5.30) the function e h ε satisfies the estimate k e h ε k L (Ω ε ) = O ( µ e − ε − + εη / ) . Hence, by (5.28) k e h ε k L (Ω ε ) k ˚Ψ ε k − L (Ω ε ) = O ( µε − / e − ε − + ε / η / ) . Taking this estimate into account, we apply Lemma 5.4 with H = L (Ω ε ) , u = ˚Ψ ε k ˚Ψ ε k L (Ω ε ) ,M = Λ − ( ε, µ ) , κ = k e h ε k L (Ω ε ) k ˚Ψ ε k − L (Ω ε ) , (5.40)and conclude that there exists an eigenvalue f M ( ε, µ ) of A satisfying the estimate | f M ( ε, µ ) − Λ − ( ε, µ ) | = O ( µε − / e − ε − + ε / η / ) . Thus, by (2.16), (2.10) | f M ( ε, µ ) | > | Λ − ( ε, µ ) | − O ( µε − / e − ε − + ε / η / ) > , | f M − ( ε, µ ) | , f M − ( ε, µ ) − Λ( ε, µ ) | = O (cid:0) ( µε − / e − ε − + ε / η / ) | Λ( ε, µ ) || f M − ( ε, µ ) | (cid:1) = O ( µε − / e − ε − + ε / η / ) . (5.41)The number f M − ( ε, µ ) is an eigenvalue of ˚ H ε (0). Due to (2.8), (2.10) there ex-ists exactly one eigenvalue of this operator satisfying (5.41), and this eigenvalue is λ (0 , ε ). Thus, | λ (0 , ε ) − Λ( ε, µ ) | = O ( µε − / e − ε − + ε / η / ) (5.42)that proves (2.14).The asymptotics (2.8), (2.10), (2.16), (2.14) imply that for ε small enough thesegment [Λ( ε, µ ) − , Λ( ε, µ ) + 1] contains exactly one eigenvalue of ˚ H ε , which is λ (0 , ε ). Bearing in mind this fact and (5.30), we apply Lemma 5.4 with d = 1and other quantities given by (5.40) and conclude that the normalized in L (Ω ε )eigenfunction ˚ φ ( x, ε ) associated with λ (0 , ε ) satisfies the estimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˚Ψ ε k ˚Ψ ε k L (Ω ε ) − ˚ φ ( · , ε ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω ε ) k h ε k L (Ω ε ) k ˚Ψ ε k L (Ω ε ) C (cid:0) µ e − ε − + εη / (cid:1) k ˚Ψ ε k L (Ω ε ) , where the constant C is independent of ε , µ , and η . Hence, for the eigenfunction˚ ψ ( x, ε ) := k ˚Ψ ε k L (Ω ε ) ˚ φ ( x, ε ) associated with λ (0 , ε ) we have k ˚ ψ ( · , ε ) − ˚Ψ ε k L (Ω ε ) = O (cid:0) µ e − ε − + εη / (cid:1) . (5.43)Denote ˚Φ ε ( x ) := ˚Ψ ε ( x ) − ˚ ψ ( x, ε ). The equations (5.29) and the eigenvalueequation for ˚ ψ ( x, ε ) imply the equation for ˚Φ ε ,˚ H ε (0)˚Φ ε = λ (0 , ε )˚Φ ε + (cid:0) λ (0 , ε ) − Λ( ε, µ ) (cid:1) ˚Ψ ε . Hence, we can write the integral identity k∇ ˚Φ ε k L (Ω ε ) = λ (0 , ε ) k ˚Φ ε k L (Ω ε ) + (cid:0) λ (0 , ε ) − Λ( ε, µ ) (cid:1) (˚Ψ ε , ˚Φ ε ) L (Ω ε ) . Thus, by (5.43), (5.42), (5.28), (2.14), (2.16), (2.10) k∇ ˚Φ ε k L (Ω ε ) λ (0 , ε ) k ˚Φ ε k L (Ω ε ) + (cid:0) λ (0 , ε ) − Λ( ε, µ ) (cid:1) (˚Ψ ε , ˚Φ ε ) L (Ω ε ) k ˚Φ ε k L (Ω ε ) + | λ (0 , ε ) − Λ( ε, µ ) |k ˚Ψ ε k L (Ω ε ) k ˚Φ ε k L (Ω ε ) C (cid:0) µ e − ε − + ε η (cid:1) . The last estimate and (5.43) prove the asymptotics (2.19). Theorem 2.5 is proved. References [1] M.Sh. Birman, T.A. Suslina, Homogenization with corrector term for periodicelliptic differential operators . St. Petersburg Math. J. (2006), 897-973.342] M.Sh. Birman, T.A. 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