Asymptotic approximation for the solution to a boundary-value problem with varying type of boundary conditions in a thick two-level junction

We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ωε, which is the union of a domain Ω₀ and a large number 3N of thin rods with thickness of order ε = O(N⁻¹). The thin rods are divided into two levels depending on their length. In addition, the thin ro...

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Datum:	2006
Hauptverfasser:	Durante, T., Mel'nyk, T.A., Vashchuk, P.S.
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Sprache:	English
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spelling	irk-123456789-1783752021-02-19T01:27:57Z Asymptotic approximation for the solution to a boundary-value problem with varying type of boundary conditions in a thick two-level junction Durante, T. Mel'nyk, T.A. Vashchuk, P.S. We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ωε, which is the union of a domain Ω₀ and a large number 3N of thin rods with thickness of order ε = O(N⁻¹). The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. The uniform Dirichlet conditions and the nonuniform Neumann conditions are given respectively on the sides of the thin rods from the first level and the second level. Using the method of matched asymptotic expansions and special junction-layer solutions, we construct the asymptotic approximation for the solution and prove the corresponding estimates in the Sobolev space H¹ (Ωε) as ε → 0 (N → +∞). Розглядається мiшана крайова задача для рiвняння Пуассона у плоскому дворiвневому з’єднаннi Ωε, яке є об’єднанням деякої областi Ω₀ та великої кiлькостi 3N тонких стержнiв з товщиною порядку ε = O(N⁻¹). Тонкi стержнi роздiлено на два рiвнi в залежностi вiд їх довжини, i стержнi з кожного рiвня ε-перiодично чергуються. На сторонах тонких стержнiв з першого рiвня задано однорiднi крайовi умови Дiрiхле, а на сторонах стержнiв другого рiвня — неоднорiднi крайовi умови Неймана. З допомогою методу узгодження асимптотичних розвинень та спецiальних розв’язкiв типу примежового шару в зонi з’єднання побудовано асимптотичне наближення для розв’язку даної задачi та доведено вiдповiднi асимптотичнi оцiнки у просторi Соболєва H¹ (Ωε) при ε → 0 (N → +∞). 2006 Article Asymptotic approximation for the solution to a boundary-value problem with varying type of boundary conditions in a thick two-level junction / T. Durante, T.A. Mel'nyk, P.S. Vashchuk // Нелінійні коливання. — 2006. — Т. 9, № 3. — С. 336-355. — Бібліогр.: 22 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/178375 517.956 en Нелінійні коливання Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ωε, which is the union of a domain Ω₀ and a large number 3N of thin rods with thickness of order ε = O(N⁻¹). The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. The uniform Dirichlet conditions and the nonuniform Neumann conditions are given respectively on the sides of the thin rods from the first level and the second level. Using the method of matched asymptotic expansions and special junction-layer solutions, we construct the asymptotic approximation for the solution and prove the corresponding estimates in the Sobolev space H¹ (Ωε) as ε → 0 (N → +∞).
format	Article
author	Durante, T. Mel'nyk, T.A. Vashchuk, P.S.
spellingShingle	Durante, T. Mel'nyk, T.A. Vashchuk, P.S. Asymptotic approximation for the solution to a boundary-value problem with varying type of boundary conditions in a thick two-level junction Нелінійні коливання
author_facet	Durante, T. Mel'nyk, T.A. Vashchuk, P.S.
author_sort	Durante, T.
title	Asymptotic approximation for the solution to a boundary-value problem with varying type of boundary conditions in a thick two-level junction
title_short	Asymptotic approximation for the solution to a boundary-value problem with varying type of boundary conditions in a thick two-level junction
title_full	Asymptotic approximation for the solution to a boundary-value problem with varying type of boundary conditions in a thick two-level junction
title_fullStr	Asymptotic approximation for the solution to a boundary-value problem with varying type of boundary conditions in a thick two-level junction
title_full_unstemmed	Asymptotic approximation for the solution to a boundary-value problem with varying type of boundary conditions in a thick two-level junction
title_sort	asymptotic approximation for the solution to a boundary-value problem with varying type of boundary conditions in a thick two-level junction
publisher	Інститут математики НАН України
publishDate	2006
url	http://dspace.nbuv.gov.ua/handle/123456789/178375
citation_txt	Asymptotic approximation for the solution to a boundary-value problem with varying type of boundary conditions in a thick two-level junction / T. Durante, T.A. Mel'nyk, P.S. Vashchuk // Нелінійні коливання. — 2006. — Т. 9, № 3. — С. 336-355. — Бібліогр.: 22 назв. — англ.
series	Нелінійні коливання
work_keys_str_mv	AT durantet asymptoticapproximationforthesolutiontoaboundaryvalueproblemwithvaryingtypeofboundaryconditionsinathicktwoleveljunction AT melnykta asymptoticapproximationforthesolutiontoaboundaryvalueproblemwithvaryingtypeofboundaryconditionsinathicktwoleveljunction AT vashchukps asymptoticapproximationforthesolutiontoaboundaryvalueproblemwithvaryingtypeofboundaryconditionsinathicktwoleveljunction
first_indexed	2025-07-15T16:43:09Z
last_indexed	2025-07-15T16:43:09Z
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fulltext	UDC 517 . 956 ASYMPTOTIC APPROXIMATION FOR THE SOLUTION TO A BOUNDARY-VALUE PROBLEM WITH VARYING TYPE OF BOUNDARY CONDITIONS IN A THICK TWO-LEVEL JUNCTION АСИМПТОТИЧНЕ НАБЛИЖЕННЯ РОЗВ’ЯЗКУ КРАЙОВОЇ ЗАДАЧI ЗI ЗМIНОЮ ТИПУ КРАЙОВИХ УМОВ У ГУСТОМУ ДВОРIВНЕВОМУ З’ЄДНАННI T. Durante Univ. Salerno via Ponte don Melillo, 84084 Fisciano (SA), Italy e-mail: durante@diima.unisa T. A. Mel’nyk, P. S. Vashchuk Kyiv Nat. Taras Shevchenko Univ. Volodymyrska Str. 64, 01033, Kyiv, Ukraine e-mail: melnyk@imath.kiev.ua pavel vaschuk@ukr.net We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ωε, which is the union of a domain Ω0 and a large number 3N of thin rods with thickness of order ε = = O(N−1). The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. The uniform Dirichlet conditions and the nonuni- form Neumann conditions are given respectively on the sides of the thin rods from the first level and the second level. Using the method of matched asymptotic expansions and special junction-layer solutions, we construct the asymptotic approximation for the solution and prove the corresponding estimates in the Sobolev space H1(Ωε) as ε → 0 (N → +∞). Розглядається мiшана крайова задача для рiвняння Пуассона у плоскому дворiвневому з’єднаннi Ωε, яке є об’єднанням деякої областi Ω0 та великої кiлькостi 3N тонких стержнiв з товщиною порядку ε = O(N−1). Тонкi стержнi роздiлено на два рiвнi в залежностi вiд їх довжини, i стержнi з кожного рiвня ε-перiодично чергуються. На сторонах тонких стержнiв з першого рiвня зада- но однорiднi крайовi умови Дiрiхле, а на сторонах стержнiв другого рiвня — неоднорiднi крайо- вi умови Неймана. З допомогою методу узгодження асимптотичних розвинень та спецiальних розв’язкiв типу примежового шару в зонi з’єднання побудовано асимптотичне наближення для розв’язку даної задачi та доведено вiдповiднi асимптотичнi оцiнки у просторi СоболєваH1(Ωε) при ε → 0 (N → +∞). 1. Introduction and statement of the problem. There are two ways for investigation of boundary- value problems in perturbed domains. The first one is the proof of the corresponding convergen- ce theorem. The second way consists in construction of the asymptotic approximation for the solution and in proving the corresponding asymptotic estimate. For these two ways we need different assumptions for data of the investigated problem. The last way is more suitable for applied problems. Boundary-value problems in thick one-level junctions (thick junctions) are very intensively c© T. Durante, T. A. Mel’nyk, P. S. Vashchuk, 2006 336 ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 ASYMPTOTIC APPROXIMATION FOR THE SOLUTION TO A BOUNDARY-VALUE PROBLEM . . . 337 investigated in recent time. As was shown in the papers [1, 2], such problems lose the coerci- tivity and compactness as ε → 0. This creates special difficulties in the asymptotic investigati- on. In addition, thick junctions are non-convex domains with Lipschitz boundaries, therefore the solutions to boundary-value problems in such domains have only minimal H1-smoothness. In [3 – 9], classification of thick one-level junctions was given and basic results (convergence theorems and asymptotic approximations) were obtained both for boundary-value and spectral problems in thick junctions of different types. It was shown that qualitative properties of soluti- ons essentially depend on the junction type and on the conditions given on the boundaries of the attached thin domains. A survey of results obtained in this direction is presented in the papers [3 – 9]. Here we mention only the pioneer papers [10 – 12], where the asymptotic behavi- our of Green’s function of the homogeneous Neumann problem for the Helmholtz equation in unbounded thick junctions was studied. In the present paper we continue the asymptotic research of boundary-value problems in thick multilevel junctions. Those thick junctions have a more complex structure and because of this the asymptotic study has own particularities and qualitative new results (see [13 – 18]). Here we construct the asymptotic approximation for a solution to a mixed boundary-value problem in a thick two-level junction and investigate influence of varying type of boundary conditi- ons on the asymptotic behavior. In particular, the homogeneous Dirichlet boundary conditi- ons are given on the lateral sides of the thin rods from the first level and the inhomogeneous Neumann boundary conditions ∂νuε = εgε are given on the lateral sides of the thin rods from the second level. At first sight it seems that there is no difference between this inhomogeneous Neumann condition and the homogeneous Neumann condition since the term gε is multiplied by the factor ε. But, as we will see in this paper, this is quite false. The Fourier conditions or the inhomogeneous Neumann conditions make the process of homogenization and approxi- mation more complicated and, to homogenize boundary-value problems in thick junctions with those conditions, the method of the integral identities was proposed in [7 – 9]. The convergence theorem for the solution of the investigated problem was proved in [16]. 1.1. The statement of the problem. Let a, d1, d2, b1, h1, h2 be positive real numbers and let d1 ≤ d2, 0 < b1 < 1 2 , 0 < b1 − h1 2 , b1 + h1 2 < 1 2 − h2 2 . The last restrictions mean that the intervals Ih2(s2) := ( s2 − h2 2 , s2 + h2 2 ) , Ih1(sn) := ( sn − h1 2 , sn + h1 2 ) , n = 1, 3, belong to (0, 1) and don’t intersect; here s1 = b1, s2 = 1/2, s3 = 1− b1. Let us divide the segment I0 := {x : x1 ∈ [0, a], x2 = 0} into N equal segments [εj , ε(j + 1)], j = 0, . . . , N − 1 ( ε = a N ) . Here N is a large integer, therefore, the value ε is a small discrete parameter. A model thick two-level junction Ωε consists of the junction’s body Ω0 = {x ∈ R2 : 0 < x1 < a, 0 < x2 < γ(x1) }, ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 338 T. DURANTE, T. A. MEL’NYK, P. S. VASHCHUK where γ ∈ C1([0, a]), min[0,a] γ =: γ0 > 0, and a large number of the thin rods G (1) j (sn, ε) = { x ∈ R2 : \|x1 − ε (j + sk)\| < εh1 2 , x2 ∈ (−d1, 0] } , n = 1, 3, G (2) j (s2, ε) = { x ∈ R2 : \|x1 − ε (j + s2)\| < εh2 2 , x2 ∈ (−d2, 0] } , j = 0, 1, . . . , N − 1. Thus Ωε = Ω0 ∪G(1) ε ∪G(2) ε , where G(1) ε = N−1⋃ j=0 ( G (1) j (s1, ε) ∪G(1) j (s3, ε) ) , G(2) ε = N−1⋃ j=0 G (2) j (s2, ε), see Fig. 1. Fig. 1 We see that the number of the thin rods is equal to 3N and they are divided into two levels G (1) ε and G(2) ε depending on their length. The length of the rods from the first level is equal to d1 and it is equal to d2 for the rods from the second one. The parameter ε characterizes the distance between the neighboring thin rods and their thickness. These thin rods from each level are ε-periodically alternated along the segment I0. In Ωε we consider the problem −∆uε(x) = fε(x), x ∈ Ωε, uε(x) = 0, x ∈ S (1) ε = ∂G (1) ε \ I0, ∂νuε(x) = εgε(x), x ∈ S (2) ε = ∂G (2) ε \ ( I0 ∪ {x : x2 = −d2} ) , ∂νuε(x) = 0, x ∈ Γε = ∂Ωε \ ( S (1) ε ∪ S(2) ε ) . (1) Here ∂ν = ∂/∂ν is the outward normal derivative. ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 ASYMPTOTIC APPROXIMATION FOR THE SOLUTION TO A BOUNDARY-VALUE PROBLEM . . . 339 Without loss of generality, we can assume that fε ∈ L2(Ω2), where Ω2 = Ω0 ∪ D2; D2 = = I0 × (−d2, 0) is the rectangle that is filled up by the thin rods from the second level in the limit passage as ε → 0. Analogously, we define D1 = I0 × (−d1, 0) and Ω1 = Ω0 ∪D1. We also suppose that the function gε and its generalized derivative with respect to x1 belong to L2(D2) and ∃ C0 > 0 ∀ ε > 0 : ‖∂x1gε‖L2(D2) ≤ C0. (2) A function uε ∈ Hε = {u ∈ H1(Ωε) : u = 0 on S (1) ε } is called a weak solution of problem (1), if it satisfies the integral identity∫ Ωε ∇uε · ∇ϕ dx = ∫ Ωε fεϕ dx+ ε ∫ S (2) ε gεϕ dx2 ∀ϕ ∈ Hε. (3) It follows from the fundamental statements of the theory of boundary-value problems that for every fixed value ε > 0 there exists a unique generalized solution to problem (1). The aim of our research is to construct the asymptotic approximation of the solution to problem (1) as ε → 0, i.e., when the number of the attached thin rods from each level infinitely increases and their thickness tends to 0, and to prove the corresponding asymptotic estimates. 2. Formal asymptotics for the solution. In this section to construct the leading terms of formal asymptotic expansions we assume that the right-hand sides in (1) are independent of ε, i.e., fε = f0, gε = g0 and f0, g0 are smooth. 2.1. Outer expansions. We seek the leading terms for the solution uε, restricted to Ω0, in the form u(x, ε) ≈ v+ 0 (x) + ∞∑ k=1 εkv+ k (x, ε), (4) and, restricted to each of the thin rods G(1) j (s1, ε), G (1) j (s3, ε), G (2) j (s2, ε), j = 0, . . . , N − 1, respectively in the form u(x, ε) ≈ v (i,−) 0 (x) + ∞∑ k=1 εkv (i,−) k (x, ξ1 − j), ξ1 = x1 ε , i = 1, 2. (5) In (5) we don’t indicate the dependence of s1 = b1 and s3 = 1−b1 at i = 1, since the asymptotic expansions on these rods are the same. Substituting the series (4) in the equation of problem (1) and in the boundary conditions on Γ0 = ∂Ω0\I0, and collecting coefficients of the same powers of ε,we get the following relations for the function v+ 0 : −∆v+ 0 (x) = f0(x), x ∈ Ω0, ∂νv + 0 (x) = 0, x ∈ Γ0. ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 340 T. DURANTE, T. A. MEL’NYK, P. S. VASHCHUK Now let us find limit relations in the rectangles Di, i = 1, 2. We write the Taylor series of v (i,−) k with respect to the x1 at the point x1 = ε(j+sn) and pass to the "fast"variable ξ1 = x1/ε, where s1 = b1, s2 = 1/2, s3 = 1− b1. Then (5) takes the form u(x, ε) ≈ v (i,−) 0 (ε(j + sn), x2) + ∞∑ k=1 εkV i,j k (sn, ξ1, x2), x ∈ G (i) j (sn, ε), (6) where V i,j k = v (i,−) k (ε(j + sn), x2, ξ1 − j) + k∑ m=1 (ξ1 − j − sn)m m! ∂mv (i,−) k−m ∂xm 1 ( ε(j + sn), x2, ξ1 − j ) . (7) Substituting the series (6) in the differential equation of problem (1) on the rod G(2) j (s2, ε) from the second level and into the Neumann condition and collecting the coefficients of the same power of ε, we obtain one dimensional boundary-value problems with respect to ξ1. The first problem is the following: ∂2 ξ2 1 V 2,j 1 (s2, ξ1, x2) = 0, ξ1 ∈ Ih2(s2), ∂ξ1V 2,j 1 ( s2, s2 ± h2 2 , x2 ) = 0, (8) where ∂ξ1 = ∂/∂ξ1, ∂ 2 ξ1ξ1 = ∂2/∂ξ21 ; the variable x2 is regarded as a parameter in this problem. From (8) it follows that the function V 2,j 1 doesn’t depend on ξ1. We restrict ourselves to the leading terms of the asymptotics and set V 2,j 1 ≡ 0. Then, due to (7), we have v (2,−) 1 (ε(j + s2), x2, ξ − j) = (−ξ1 + j + s2)∂x1v (2,−) 0 (ε(j + s2), x2). The problem for the function V 2,j 2 is as follows: −∂2 ξ2 1 V 2,j 2 (s2, ξ1, x2) = ∂2 x2 2 v (2,−) 0 (ε(j + s2), x2) + f0(ε(j + s2), x2), ξ1 ∈ Ih2(s2), ∂ξ1V 2,j 2 ( s2, s2 ± h2 2 , x2 ) = ±g0(ε(j + s2), x2). (9) The solvability condition for (9) is given by the differential equation −h2 ∂ 2 x2 v (2,−) 0 (ε(j + s2), x2) = h2 f0(ε(j + s2), x2) + 2g0(ε(j + s2), x2). (10) If we substitute (6) into the Neumann conditions on the lower base Q (2) j (s2, ε) of rod G (2) j (s2, ε), we get ∂x2v (2,−) 0 (ε(j + s2),−d2) = 0. (11) ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 ASYMPTOTIC APPROXIMATION FOR THE SOLUTION TO A BOUNDARY-VALUE PROBLEM . . . 341 Since the segments {x : x1 = ε(j + s2), x2 ∈ [−d2, 0]}, j = 0, 1, . . . , N − 1, fill out the rectangle D2 in the limit as ε → 0 (N → +∞), we can continue the differential equation (10) into the rectangle D2 and the relations (11) into the segment Id2 = {x : x1 ∈ [0, a], x2 = = −d2}. Substitute now the series (6) in the differential equation of problem (1) on the rodsG(1) j (s1, ε) andG(1) j (s3, ε) from the first level. Since these asymptotic expansions are the same, we consider the rod G(1) j (s1, ε). Due to the Dirichlet conditions on the vertical sides, we have v(1,−) 0 ≡ 0. The problem for the function V 1,j 1 (s1, ξ1, x2) is as follows: ∂2 ξ2 1 V 1,j 1 (s1, ξ1, x2) = 0, ξ1 ∈ Ih1(s1), V 1,j 1 ( s1, s1 ± h1 2 , x2 ) = 0, from which V 1,j 1 ≡ 0. Then, due to (7), we have v(1,−) 1 (ε(j + s1), x2, ξ − j) = 0. The problem for V 1,j 2 is the following: −∂2 ξ2 1 V 1,j 2 (s1, ξ1, x2) = f0(ε(j + s1), x2), ξ1 ∈ Ih1(s1), V 1,j 2 ( s1 ± h1 2 , x2 ) = 0. It is easy to verify that V (1,j) 2 (s1, ξ1, x2) = h−1 1 ( ξ1 − s1 − h1 2 ) ξ1∫ s1−h1/2 ( s1 − h1 2 − t ) f0(t, x2) dt+ + h−1 1 ( s1 − h1 2 − ξ1 ) s1+h1/2∫ ξ1 ( t− s1 − h1 2 ) f0(t, x2)dt. According to (7), we obtain v (1,−) 2 (ε(j + s1), x2, ξ1 − j) = V (1,j) 2 (s1, ξ1, x2). Thus, the asymptotic expansions on the rods from the first level have the form ε2v (1,−) 2 (ε(j + sn), x2) + ∞∑ k=3 εkV 1,j k (sn, ξ1, x2), x ∈ G (1) j (sn, ε), n = 1, 3. (12) It is evident that the first terms of the asymptotic expansions must coincide on the joint zone I0. Therefore, from (12) and (4) it follows that v+ 0 (ε(j + s1), 0) = v+ 0 (ε(j + s3), 0) = 0, ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 342 T. DURANTE, T. A. MEL’NYK, P. S. VASHCHUK j = 0, 1, . . . , N − 1. On the other hand, v+ 0 (ε(j + s2), 0) must be equal to v(2,−) 0 (ε(j + s2), 0) at j = 0, 1, . . . , N − 1. This is possible if v+ 0 (x1, 0) = v (2,−) 0 (x1, 0) = 0, x1 ∈ I0. As a result, the first terms of the asymptotic expansions (4) and (5) on G (2) ε have to be solutions of the following problems: −∆v+ 0 (x) = f0(x), x ∈ Ω0, ∂νv + 0 (x) = 0, x ∈ Γ0, v+ 0 (x1, 0) = 0, x1 ∈ I0; (13) −h2 ∂ 2 x2 v (2,−) 0 (x) = h2 f0(x) + 2g0(x), x ∈ D2, v (2,−) 0 (x1, 0) = 0, x1 ∈ I0, ∂x2v (2,−) 0 (x1,−d2) = 0, x1 ∈ I0. (14) Obviously, there is a unique weak solution to problem (13) that belongs to the Sobolev space H1(Ω0, I0) = {u ∈ H1(Ω0) : u = 0 on I0}. It is easy to calculate that v (2,−) 0 (x) = −x2 x2∫ −d2 ( f0(x1, t) + 2h−1 2 g0(x1, t) ) dt− 0∫ x2 t ( f0(x1, t) + 2h−1 2 g0(x1, t) ) dt. (15) Thus, the asymptotic expansion on the rod G(2) j (s2, ε) has the following form: v (2,−) 0 (ε(j + s2), x2) + ε(−ξ1 + j + s2)∂x1v (2,−) 0 (ε(j + s2), x2) + ∞∑ k=2 εkV 2,j k (s2, ξ1, x2). If we consider the function R(x) =  v+ 0 (x), x ∈ Ω0, v (2,−) 0 (x) + εY (x1 ε ) ∂x1v (2,−) 0 (x), x ∈ G (2) ε , 0, x ∈ G (1) ε , (16) as a first approximation for the solution to problem (1), then it leaves the remainder ∫ I0∩G (2) ε ( ∂x2v + 0 (x1, 0)− h2∂x2v (2,−) 0 (x1, 0) ) ϕ(x1, 0) dx1, ϕ ∈ Hε, (17) ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 ASYMPTOTIC APPROXIMATION FOR THE SOLUTION TO A BOUNDARY-VALUE PROBLEM . . . 343 in the corresponding integral identity. In (16), Y (t) = −t+1 2 +[t], where [t] is the integral part of t. To neutralize this remainder, we should construct a special inner expansion in a neighborhood of the joint zone I0. 2.2. Inner expansion. Near the joint zone I0 we introduce the "rapid"coordinates ξ = = (ξ1, ξ2), where ξ1 = x1/ε, ξ2 = x2/ε. Passing to ε = 0, we see that the rods G(1) 0 (s1, ε), G (2) 0 (s2, ε), G (1) 0 (s3, ε) transform into the semiinfinite strips Π− 1 = Ih1(s1)× (−∞, 0), Π− 2 = Ih2(s2)× (−∞, 0), Π− 3 = Ih1(s3)× (−∞, 0). Fig. 2 The domain Ω0 transforms into the first quadrant {ξ : ξ1 > 0, ξ2 > 0}. Taking into account the periodicity of the thin rods we can regard the union Π = Π+ ∪ Π− 1 ∪ Π− 2 ∪ Π− 3 , where Π+ = (0, 1) × (0,+∞), as the base domain, see Fig. 2, in which junction-layer problems have to be considered. Obviously, solutions of these junction-layer problems must be 1-periodic in ξ1, i.e., Z(0, ξ2) = Z(1, ξ2), ∂ξ1Z(0, ξ2) = ∂ξ1Z(1, ξ2), ξ2 > 0. (18) We seek the first term of the inner expansion in a neighborhood of I0 in the form uε(x) ≈ ε ( Z1 (x ε ) ∂x2v + 0 (x1, 0) + Ξ1 (x ε ) ( h−1 2 ∂x2v + 0 (x1, 0)− ∂x2v (2,−) 0 (x1, 0) )) +O(ε2), (19) where the functions Z1(ξ), Ξ1(ξ), ξ ∈ Π, are 1-periodic with respect to ξ1 (see (18)). The last summand in (19) to neutralize the remainder (17). ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 344 T. DURANTE, T. A. MEL’NYK, P. S. VASHCHUK Substituting (19) in the differential equation of problem (1) and in the corresponding boun- dary conditions, taking into account that the Laplace operator takes the form ε−2∆ξ in the coor- dinates ξ and collecting the coefficients of the same power of ε, we obtain that the function Z1 must be a nontrivial solution to the following homogeneous problem: −∆Z1(ξ) = 0, ξ ∈ Π, ∂p ξ1 Z1(ξ)\|ξ1=0 = ∂p ξ1 Z1(ξ)\|ξ1=1, ξ2 > 0, p = 0, 1, ∂ξ2Z1(ξ1, 0) = 0, ξ1 ∈ (0, 1) \ (Ih1(s1) ∪ Ih2(s2) ∪ Ih1(s3)), [Z1]\|ξ2=0 = [∂ξ2Z1]\|ξ2=0 = 0, ξ1 ∈ Ih1(s1) ∪ Ih2(s2) ∪ Ih1(s3), Z1(ξ) = 0, ξ ∈ S−1 ∪ S − 3 , ∂ξ1Z1(ξ) = 0, ξ ∈ S−2 , (20) where S−n = ∂Π− n \ Ih1(sn), n = 1, 3, S−2 = ∂Π− 2 \ Ih2(s2), the brackets denote the jump of the enclosed quantities. The function Ξ1 must be a solution to the following problem: −∆Ξ1(ξ) = 0, ξ ∈ Π, ∂p ξ1 Ξ1(ξ)\|ξ1=0 = ∂p ξ1 Ξ1(ξ)\|ξ1=1, ξ2 > 0, p = 0, 1, ∂ξ2Ξ1(ξ1, 0) = 0, ξ1 ∈ (0, 1) \ (Ih1(s1) ∪ Ih2(s2) ∪ Ih1(s3)), [Ξ1]\|ξ2=0 = 0, ξ1 ∈ Ih1(s1) ∪ Ih2(s2) ∪ Ih1(s3), [∂ξ2Ξ1]\|ξ2=0 = 0, ξ1 ∈ Ih1(s1) ∪ Ih1(s3), [∂ξ2Ξ1]\|ξ2=0 = 1, ξ1 ∈ Ih2(s2), Ξ1(ξ) = 0, ξ ∈ S−1 ∪ S − 3 , ∂ξ1Ξ1(ξ) = 0, ξ ∈ S−2 . (21) The main asymptotic relations for a solution to problem (20) can be obtained from general results about the asymptotic behaviour of solutions to elliptic problems in domains with di- fferent exits to infinity [19, 20]. The proofs simplify substantially if the polynomial property of the corresponding sesquilinear forms is employed [21]. However, using symmetry of the domain Π, we can define more exactly the asymptotic relations and detect other properties of the junction-layer solution Z1 similarly as in the papers [4, 5]. Statement 1. There exits a unique solution Z1 to problem (20), which has the following differentiable asymptotics: Z1(ξ) =  ξ2 + α+ +O(exp(−δ1ξ2)), ξ2 → +∞, ξ ∈ Π+, h−1 2 ξ2 + α− +O(exp(δ1ξ2)), ξ2 → −∞, ξ ∈ Π− 2 , O(exp(δ1ξ2)), ξ2 → −∞, ξ ∈ Π− 1 ∪Π− 3 , (22) where α± are some fixed constants; δ1 is an arbitrary fixed number from the interval (0, π). In addition, the function Z1 is even in ξ2 with respect to 1/2. ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 ASYMPTOTIC APPROXIMATION FOR THE SOLUTION TO A BOUNDARY-VALUE PROBLEM . . . 345 In order to find the constant α± in (22), it is necessary to substitute the function Z1 and ξ2 into the second Green’s formula in the domains Π+ ∩ {ξ : 0 < ξ2 < R}, Π− 1 ∩ {ξ : −R < < ξ2 < 0}, Π− 2 ∩ {ξ : −R < ξ2 < 0},Π− 3 ∩ {ξ : −R < ξ2 < 0}, and to pass to the limit as R → ∞. As a result, we obtain α+ = ∫ [0,1]\(Ih1 (s1)∪Ih1 (s3)) Z1(ξ1, 0) dξ1, α− = 1 h2 ∫ Ih2 (s2) Z1(ξ1, 0) dξ1. (23) Now let us investigate the solvability of problem (21). Let Ĉ∞ 0 (Π, S−1 ∪ S−3 ) be a space of infinitely differentiable functions that are equal to zero on S−1 ∪ S−3 , satisfy the periodic conditions (18), and are finite in ξ2, i.e., ∀ϕ ∈ Ĉ∞ 0 (Π, S−1 ∪ S − 3 ) ∃R > 0 ∀ξ ∈ Π \|ξ2\| ≥ R : ϕ(ξ) = 0. Let H be the completion of the space Ĉ∞ 0 (Π, S−1 ∪ S − 3 ) with respect to the norm ‖u‖H = ‖∇ξu‖L2(Π). A function Ξ ∈ H is called a weak solution of problem (21), if it satisfies the integral identity ∫ Π ∇ξΞ · ∇ξϕdξ = − ∫ Ih2 (s2) ϕ(ξ1, 0) dξ1 ∀ ϕ ∈ H. Lemma 1. There exits a unique weak solution Ξ1 to problem (21), which has the following differentiable asymptotics: Ξ1(ξ) =  β+ +O(exp(−δ2ξ2)), ξ2 → +∞, ξ ∈ Π+, β− +O(exp(δ2ξ2)), ξ2 → −∞, ξ ∈ Π− 2 , O(exp(δ2ξ2)), ξ2 → −∞, ξ ∈ Π− 1 ∪Π− 3 , (24) where β± are some fixed constants; δ2 is an arbitrary fixed number from the interval (0, π). In addition, the function Ξ0 is even in ξ2 with respect to 1/2. Proof. To prove the first part of this lemma it is enough to show that the linear functional l(ϕ) := ∫ Ih2 (s2) ϕ(ξ1, 0) dξ1, ϕ ∈ H, is bounded. With help of the Friedrichs inequality and the cut-off function χ0 ∈ C∞(R), 0 ≤ ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 346 T. DURANTE, T. A. MEL’NYK, P. S. VASHCHUK ≤ χ0 ≤ 1, χ0(ξ2) = 1 if \|ξ2\| ≤ 1/2, and χ0(ξ2) = 0 if \|ξ2\| ≥ 1, we deduce \|l(ϕ)\|2 ≤ h2 ∫ Ih2 (s2) ϕ2(ξ1, 0) dξ1 ≤ h2 ∫ Ih2 (s2)×(0,1) ( ∂ξ2(χ0(ξ2)ϕ(ξ1, ξ2) ) )2 dξ1dξ2 ≤ ≤ 2h2  ∫ Ih2 (s2)×(0,1) (χ′0) 2ϕ2 dξ + ∫ Ih2 (s2)×(0,1) ( ∂ξ2ϕ )2 dξ  ≤ ≤ c1  ∫ Π∩{ξ:−1<ξ2<1} ϕ2 dξ + ‖ϕ‖2 H  ≤ ≤ c2  ∫ Π∩{ξ:−1<ξ2<1} \|∇ϕ\|2 dξ + ‖ϕ‖2 H  ≤ c2‖ϕ‖2 H. Taking into account the properties of solutions to elliptic problems in semiinfinite domains, we can state that the solution Ξ1 has the asymptotics (24). Due to the symmetry of the domain ω and using the substitution ξ1 = 1 − ξ1 in problem (21), we obtain that the difference Ξ1(ξ1, ξ2)− Ξ1(1− ξ1, ξ2) is a solution of the homogeneous problem (20) and it belongs to H. By virtue of the uniqueness of such a solution, it follows that this difference vanishes. By the same way as we obtained (23), we get β+ = ∫ [0,1]\(Ih1 (s1)∪Ih1 (s3)) Ξ1(ξ1, 0) dξ1, β− = 1 h2 ∫ Ih2 (s2) Ξ1(ξ1, 0) dξ1. If we apply the second Green’s formula to the functionsZ1 and Ξ1 in the domain Π∩{ξ : −R < < ξ2 < R} and then pass to the limit as R → ∞, we deduce the relation β+ − β− = h2α −. The lemma is proved. 3. Asymptotic approximations. Here we construct an approximation function Rε using the terms v+ 0 , v (2,−) 0 , Z1, Ξ1 defined in the previous section and the following cut-off function χ1 ∈ C∞(R), 0 ≤ χ1 ≤ 1, χ1(x2) = 1 if \|x2\| ≤ λ1/2, and χ1(x2) = 0 if \|x2\| ≥ λ1, where λ1 = 2−1 min{γ0, d1, d2}. It is equal to R+ ε (x) := Rε(x) = v+ 0 (x) + εχ1(x2)N+(ξ, x1)\|ξ=x ε , x ∈ Ω0, R(2,−) ε (x) := Rε = v (2,−) 0 (x) + ε ( Y (ξ1)∂x1v (2,−) 0 (x) + χ1(x2)N (2,−)(ξ, x1) ) \|ξ=x ε , x ∈ G(2) ε , (25) R(1,−) ε (x) := Rε(x) = ε χ1(x2)N (1,−)(ξ, x1)\|ξ=x ε , x ∈ G(1) ε . ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 ASYMPTOTIC APPROXIMATION FOR THE SOLUTION TO A BOUNDARY-VALUE PROBLEM . . . 347 Here Y is defined in (16); the functions N+(ξ, x1), N (2,−)(ξ, x1), N (1,−)(ξ, x1) are 1-periodic with respect to ξ1 and N+(ξ, x1) = (Z1(ξ)− ξ2)∂x2v + 0 (x1, 0) + Ξ1(ξ) ( ∂x2v (2,−) 0 (x1, 0)− h−1 2 ∂x2v + 0 (x1, 0) ) , ξ1 > 0, N (2,−) = (Z1(ξ)− h−1 2 ξ2)∂x2v + 0 (x1, 0) + Ξ1(ξ) ( ∂x2v (2,−) 0 (x1, 0)− h−1 2 ∂x2v + 0 (x1, 0) ) , ξ1 < 0, ξ ∈ Π− 2 , N (1,−) = Z1(ξ)∂x2v + 0 (x1, 0) + Ξ1(ξ) ( ∂x2v (2,−) 0 (x1, 0)− h−1 2 ∂x2v + 0 (x1, 0) ) , ξ1 < 0, ξ ∈ Π− 1 ∪Π− 3 . It is easy to verify that R+ ε (x1, 0) = R (1,−) ε (x1, 0), x1 ∈ I0 ∩ G (1) ε , and R+ ε (x1, 0) = = R (2,−) ε (x1, 0), x1 ∈ I0 ∩G(2) ε . Thus, Rε ∈ Hε. In addition, [∂x2Rε]\|x2=0 = 0, x1 ∈ I0 ∩G(1) ε ; and [∂x2Rε]\|x2=0 = −εY (ξ1)∂2 x2x1 v (2,−) 0 (x1, 0), x1 ∈ I0 ∩G(2) ε . (26) Theorem 1. Suppose that f0 ∈ C2 0 (Ω2), g0 ∈ C2 0 (D2). Then for any δ0 ∈ (0, 1) there exist positive constants C0, ε0 such that for all values ε ∈ (0, ε0) the difference between the solution uε to problem (1) and the approximation functionRε defined by (25) satisfies the following estimate: ‖uε −Rε‖H1(Ωε) ≤ C0 ( ε1−δ0 + ε+ ‖fε − f0‖L2(Ω0∪G (2) ε ) + ‖g0 − gε‖L2(G (2) ε ) ) . (27) Proof. 1. Discrepancies in the domain Ω0. Taking into account the properties of the functions Z1, Ξ1, v + 0 and f0, g0, we conclude that R+ ε satisfies all boundary conditions for problem (1) on ∂Ω0 ∩ ∂Ωε. Putting R+ ε in the equation of problem (1), we get −∆xR + ε (x)− fε(x) = −χ′1(x2) (∂ξ2N +(ξ, x1))\|ξ=x/ε− − χ1(x2) (∂2 x1ξ1N +(ξ, x1))\|ξ=x/ε − ε∂x2 ( χ′1(x2)N+ (x ε , x1 )) − − εχ1(x2) ∂x1(∂x1N +(ξ, x1)\|ξ=x/ε)− (fε(x)− f0(x)), x ∈ Ω0. (28) Further, the arguments of functions involved in calculations are indicated only if their absence may cause confusion. We multiply the identify (28) by a test function ψ ∈ Hε and integrate by parts in Ω0, ∫ I0∩Ωε ∂x2R + ε (x1, 0)ψ dx1 + ∫ Ω0 ∇xR + ε · ∇xψdx− ∫ Ω0 fεψdx = = I+ 1 (ε, ψ) + . . .+ I+ 5 (ε, ψ), (29) ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 348 T. DURANTE, T. A. MEL’NYK, P. S. VASHCHUK where I+ 1 (ε, ψ) = − ∫ Ω0 χ′1(x2) (∂ξ2N +(ξ, x1))\|ξ=x/εψ dx, I+ 2 (ε, ψ) = − ∫ Ω0 χ1(x2) (∂2 x1ξ1N +(ξ, x1))\|ξ=x/εψ dx, I+ 3 (ε, ψ) = ε ∫ Ω0 χ′1(x2)N+ (x ε , x1 ) ∂x2ψdx, I+ 4 (ε, ψ) = ε ∫ Ω0 χ1(x2) ( ∂x1N +(ξ, x1) ) \|ξ=x/ε ∂x1ψ dx, I+ 5 (ε, ψ) = − ∫ Ω0 (fε(x)− f0(x))ψ dx. 2. Discrepancies in the thin rings from the second level. It is easy to calculate that ∂x2R (2,−) ε (x1,−d1) = 0, and ∂νR 1,− ε = ±ε ( Y (ξ1)∂2 x1x1 v (2,−) 0 (x) + χ1(x2)(∂x1N (2,−)(ξ, x1))\|ξ=x/ε ) , x ∈ S(2) ε , (30) where we take ” + ” or ”− ” depending on the right or left side of the thin rod. Putting R(2,−) ε in the differential equation of problem (1), we obtain −∆xR (2,−) ε (x)− fε(x) = = −χ′1(x2) (∂ξ2N (2,−)(ξ, x1))\|ξ=x/ε − χ1(x2) (∂2 x1ξ1N (2,−)(ξ, x1))\|ξ=x/ε− − ε∂x2 ( χ′1(x2)N (2,−) (x ε , x1 )) − εχ1(x2) ∂x1 ( ∂x1N (2,−)(ξ, x1)\|ξ=x/ε ) − − (fε(x)− f0(x))− ε div ( Y1 (x1 ε ) ∇x ( ∂x1v (2,−) 0 )) + 2h−1 2 g0(x), x ∈ G(2) ε . (31) Using the following integral identity: ε2−1h2 ∫ S (2) ε v dx2 = ∫ G (2) ε v dx− ε ∫ G (2) ε Y (x1 ε ) ∂x1v dx ∀ v ∈ Hε, (32) ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 ASYMPTOTIC APPROXIMATION FOR THE SOLUTION TO A BOUNDARY-VALUE PROBLEM . . . 349 which was proved in [8], and taking into account (26) and the boundary values of ∂νR − ε (see (30)), we multiply (31) by a test function ψ ∈ Hε and integrate by parts in G(2) ε . This yields − ∫ I0∩G (2) ε ∂x2R + ε (x1, 0)ψ(x1, 0) dx1 + ∫ G (2) ε ∇xR (2,−) ε · ∇xψ dx− − ∫ G (2) ε fεψdx− ε ∫ S (2) ε gε ψ dx2 = I2,− 1 (ε, ψ) + . . .+ I2,− 7 (ε, ψ), (33) where I2,− 1 (ε, ψ) = − ∫ G (2) ε χ′1(x2) ( ∂ξ2N (2,−)(ξ, x1) ) \|ξ=x/ε ψ dx, I2,− 2 (ε, ψ) = − ∫ G (2) ε χ1(x2) ( ∂2 x1ξ1N (2,−)(ξ, x1) ) \|ξ=x/ε ψ dx, I2,− 3 (ε, ψ) = ε ∫ G (2) ε χ′1(x2)N (2,−) (x ε , x1 ) ∂x2ψ dx, I2,− 4 (ε, ψ) = ε ∫ G (2) ε χ1(x2) ( ∂x1N (2,−)(ξ, x1) ) \|ξ=x/ε ∂x1ψ dx, I2,− 5 (ε, ψ) = − ∫ G (2) ε ( fε(x)− f0(x) ) ψ dx, I2,− 6 (ε, ψ) = ε ∫ G (2) ε Y1 (x1 ε ) ∇x ( ∂x1v (2,−) 0 ) · ∇xψ dx, I2,− 7 (ε, ψ) = ε ∫ S (2) ε (g0(x)− gε(x)) ψ(x) dx2 − 2εh−1 2 ∫ G (2) ε Y (x1 ε ) ∂x2(g0 ψ) dx. 3. Discrepancies in the thin rings from the first level. It is easy to verify that ∂x2R (1,−) ε (x1, −d1) = 0 and R (1,−) ε (x) = 0 on S(1) ε . Putting R(1,−) ε in the differential equation of problem (1), we obtain ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 350 T. DURANTE, T. A. MEL’NYK, P. S. VASHCHUK −∆xR (1,−) ε (x)− fε(x) = −χ′1(x2) (∂ξ2N (1,−)(ξ, x1))\|ξ=x/ε− − χ1(x2) (∂2 x1ξ1N (1,−)(ξ, x1))\|ξ=x/ε − ε∂x2 ( χ′1(x2)N (1,−) (x ε , x1 )) − − εχ1(x2) ∂x1 ( ∂x1N (1,−)(ξ, x1)\|ξ=x/ε ) − fε(x), x ∈ G(1) ε . (34) Multiplying (34) by a test function ψ ∈ Hε and integrating by parts in G(1) ε , we get − ∫ I0∩G (1) ε ∂x2R + ε (x1, 0)ψ(x1, 0) dx1 + ∫ G (1) ε ∇xR (1,−) ε · ∇xψ dx− ∫ G (1) ε fεψdx = = I1,− 1 (ε, ψ) + . . .+ I1,− 5 (ε, ψ), (35) where I1,− 1 (ε, ψ) = − ∫ G (1) ε χ′1(x2) ( ∂ξ2N (1,−)(ξ, x1) ) \|ξ=x/ε ψ dx, I1,− 2 (ε, ψ) = − ∫ G (1) ε χ1(x2) ( ∂2 x1ξ1N (1,−)(ξ, x1) ) \|ξ=x/ε ψ dx, I1,− 3 (ε, ψ) = ε ∫ G (1) ε χ′1(x2)N (1,−) (x ε , x1 ) ∂x2ψ dx, I1,− 4 (ε, ψ) = ε ∫ G (1) ε χ1(x2) ( ∂x1N (1,−)(ξ, x1) ) \|ξ=x/ε ∂x1ψ dx, I1,− 5 (ε, ψ) = − ∫ G (1) ε fε(x)ψ dx. 4. Asymptotic estimates. Summing (29), (33) and (35), we see that the functionRε constructed by formulas (25) satisfies the following integral identity: ∫ Ωε ∇xRε · ∇xψ dx = ∫ Ωε fεψdx+ ε ∫ S (2) ε gε ψ dx2 + Fε(ψ) ∀ψ ∈ Hε, ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 ASYMPTOTIC APPROXIMATION FOR THE SOLUTION TO A BOUNDARY-VALUE PROBLEM . . . 351 where Fε(ψ) = I±1 (ε, ψ) + . . . + I±5 (ε, ψ) + I2,− 6 (ε, ψ) + I2,− 7 (ε, ψ), I±j (ε, ψ) = I+ j (ε, ψ) + +I1,− j (ε, ψ) + I2,− j (ε, ψ), j = 1, . . . , 5. Subtracting the integral identity (3) from (32), we get ∫ Ωε ∇x ( Rε − uε ) · ∇xψ dx = Fε(ψ) ∀ψ ∈ Hε. (36) Now we should estimate the value Fε(ψ). At first we note that due to Lemma 1 [16] the usual norm ‖u‖H1(Ωε) and the norm ‖∇u‖L2(Ωε) are uniformly equivalent with respect to ε. Integrals in I±1 (ε, ψ) and I±3 (ε, ψ) are taken, in fact, over supp ( χ′1(x2) ) ∩ Ωε = { x : λ1 2 < \|x2\| < λ1 } ∩ Ωε, respectively. In this domain, by virtue of Statement 1 and Lemma 1, the functions ( ∂ξ2N +(ξ, x1) ) \|ξ=x/ε, ( ∂ξ2N (1,−)(ξ, x1) ) \|ξ=x/ε, ( ∂ξ2N (2,−)(ξ, x1) ) \|ξ=x/ε are exponentially small and the functions N+, N (1,−), N (2,−) are uniformly bounded with respect to ε. Therefore, \|I±1 (ε, ψ) + I±3 (ε, ψ)\| ≤ εC1‖ψ‖H1(Ωε). Here and in what follows all constants in asymptotic inequalities are independent of ε. In order to estimate the term I±2 (ε, ψ) we use the following statement. Statement 2 [4, 5]. Assume that a function N is 1-periodic in ξ1, belongs to the space L2(Π) and is exponentially decreasing at infinity as ξ2 → ∞, i.e., there exist positive constants c,R, σ such that for any \|ξ2\| ≥ R ∣∣N (ξ) ∣∣ ≤ c exp(−σ\|ξ2\|). Then for any δ > 0 there exist positive constants c1, ε0 such that for all values ε ∈ (0, ε0) the following inequality holds:∣∣∣∣∣∣ ∫ Ωε N (x ε ) ψ(x) dx ∣∣∣∣∣∣ ≤ c1ε 1−δ‖ψ‖H1(Ωε) ∀ψ ∈ Hε. Since all functions of ξ entering in ∂2 x1ξ1 N+(ξ, x1), ∂2 x1ξ1 N (1,−)(ξ, x1), ∂2 x1ξ1 N (2,−)(ξ, x1) exponentially decrease as \|ξ2\| → +∞, on the basis of Statement 2 we get that \|I±2 (ε, ψ)\| ≤ ε1−δ0 C2(δ0)‖ψ‖H1(Ωε), where δ0 is an arbitrary fixed positive number. ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 352 T. DURANTE, T. A. MEL’NYK, P. S. VASHCHUK Integrals in I±4 (ε, ψ) are over {x : \|x2\| < λ1} ∩ Ωε and they can be estimated in the following way. Consider, for example, the integral ε−1\|I+ 4 (ε, ψ)\| = ∣∣∣∣∣∣ ∫ Ω0 χ1(x2) ∂x1ψ ( (Z1(ξ)− ξ2)∂2 x1x2 v+ 0 (x1, 0) + + Ξ1(ξ) ( ∂2 x1x2 v (2,−) 0 (x1, 0)− h−1 2 ∂2 x1x2 v+ 0 (x1, 0) ))∣∣ ξ=x/ε dx ∣∣∣∣∣∣ ≤ ≤ ∫ Ω0 χ1 \|∂x1ψ\| ( \|Z1(ξ)− ξ2 − α+\|+ \|Ξ1(ξ)− β+\| )∣∣ ξ=x/ε ( \|∂2 x1x2 v (2,−) 0 (x1, 0)\|+ + h−1 2 \|∂2 x1x2 v+ 0 (x1, 0)\| ) dx+ + ∫ Ω0 χ1 \|∂x1ψ\| ( \|α+\|+ \|β+\| )( \|∂2 x1x2 v (2,−) 0 (x1, 0)\|+ h−1 2 \|∂2 x1x2 v+ 0 (x1, 0)\| ) dx ≤ ≤ c1‖∂x1ψ‖L2(Ω0) √√√√∫ Ω0 χ1 ∣∣Z1(ξ)− ξ2 − α+ ∣∣2 ξ=x/ε dx + + √√√√∫ Ω0 χ1 ∣∣Ξ1(ξ)− β+ ∣∣2 ξ=x/ε dx+ 1  ≤ ≤ c2‖∂x1ψ‖L2(Ω0) (√ ε ‖Z1(ξ)− ξ2 − α+‖L2(Π+) + √ ε ‖Ξ1(ξ)− β+‖L2(Π+) + 1 ) , where \|Ω0\| is measure of Ω0. The values ‖Z1(ξ) − ξ2 − α+‖L2(Π+) and ‖Ξ1(ξ) − β+‖L2(Π+) are bounded because of (22) and (24). As a result, we have \|I±4 (ε, ψ)\| ≤ εC4‖ψ‖H1(Ωε). (37) Remark 1. The constant C4 in (37) depends on the quantities sup x1∈I0 ∣∣∣ ∂2v+ 0 ∂x1∂x2 ( x1, 0 )∣∣∣, sup x1∈I0 ∣∣∣∂2v (2,−) 0 ∂x1∂x2 ( x1, 0 )∣∣∣. (38) Since f0 ∈ C2 0 (Ω2) the function v+ 0 and its derivatives have no singularities at the points (0, 0) and (a, 0). Therefore, by virtue of classical results on the smoothness of solutions to boundary- ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 ASYMPTOTIC APPROXIMATION FOR THE SOLUTION TO A BOUNDARY-VALUE PROBLEM . . . 353 value problems, the first quantity in (38) is bounded. The boundedness of the second quanti- ty follows from (15) and the condition g0 ∈ C2 0 (D2). With the help of the Cauchy – Buniakovsky inequality we estimate the summand I±5 (ε, ψ), \|I±5 (ε, ψ)\| = ∣∣∣∣∣∣∣ ∫ Ω0∪G (2) ε (fε(x)− f0(x))ψ dx + ∫ G (1) ε fε(x)ψ dx ∣∣∣∣∣∣∣ ≤ ≤ ‖fε − f0‖L2(Ω0∪G (2) ε ) ‖ψ‖ L2(Ω0∪G (2) ε ) + ‖fε‖L2(G (1) ε ) ε ‖∂x1ψ‖L2(G (1) ε ) ≤ ≤ ( ‖fε − f0‖L2(Ω0∪G (2) ε ) + εC5 ) ‖ψ‖H1(Ωε). Since ∂x1v (2,−) 0 ∈ H1(D2), we have \|I2,− 6 (ε, ψ)\| ≤ εC6 ‖ψ‖H1(Ωε). Now it remains to estimate I2,− 7 . It is obvious that the last summand in I2,− 7 is not greater than c1ε‖ψ‖H1(Ωε). Using identity (32), assumptions (2) and g0 ∈ C2 0 (D2), we have∣∣∣∣∣∣∣ε ∫ S (2) ε ( g0(x)− gε(x) ) ψ(x) dx2 ∣∣∣∣∣∣∣ ≤ c2  ∫ G (2) ε ∣∣g0 − gε ∣∣ \|ψ\| dx+ ε ∫ G (2) ε \|∂x1 ( (g0 − gε)ψ ) \| dx  ≤ ≤ c3 ( ‖g0 − gε‖L2(G (2) ε ) + ε ) ‖ψ‖H1(Ωε). Thus, \|I2,− 7 (ε, ψ)\| ≤ C7 ( ‖g0 − gε‖L2(G (2) ε ) + ε ) ‖ψ‖H1(Ωε). With regard to the inequalities obtained, we conclude that for the right-hand side in (36) the following inequality holds: \|Fε(ψ)\| ≤ C0 ( ε1−δ0 + ε+ ‖fε − f0‖L2(Ω0∪G (2) ε ) + ‖g0 − gε‖L2(G (2) ε ) ) ‖ψ‖H1(Ωε) ∀ψ ∈ Hε, (39) where δ0 is a positive fixed number. From (36) and (39) it follows the inequality (27). Theorem is proved. Corollary 1. From (27) it follows that ‖uε − v+ 0 ‖L2(Ω0) + ‖uε − v (2,−) 0 ‖ L2(G (2) ε ) + ‖uε‖L2(G (1) ε ) ≤ ≤ c2 ( ε1−δ0 + ε+ ‖fε − f0‖L2(Ω0∪G (2) ε ) + ‖g0 − gε‖L2(G (2) ε ) ) . ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 354 T. DURANTE, T. A. MEL’NYK, P. S. VASHCHUK Corollary 2. Assume that fε(x) = f0(x) + εf1(x, ε), x ∈ Ωε, where ‖f1(·, ε)‖L2(Ωε) = O(1) as ε → 0, gε(x) = g0(x) + εg1(x, ε), x ∈ G(2) ε , where ‖g1(·, ε)‖L2(G (2) ε ) = O(1) as ε → 0. Then for any δ0 > 0 there exist positive constants c3, ε0 such that for all values ε ∈ (0, ε0) ‖uε−Rε‖H1(Ωε) ≤ c3 ε 1−δ0 , ‖uε−v+ 0 ‖L2(Ω0) +‖uε−v(2,−) 0 ‖ L2(G (2) ε ) +‖uε‖L2(G (1) ε ) ≤ c3 ε 1−δ0 . Conclusion. An important problem for existing multiscale methods is their stability and accuracy. The proof of the error estimate between the constructed approximation and the exact solution is a general principle that has been applied to the analysis of the efficiency of a multi- scale method. In our paper we have constructed the asymptotic approximation for the solution to problem (1) and proved such estimates in Theorem 1 and Corollaries 1 and 2. It was shown that, due to the homogeneous Dirichlet boundary conditions on the vertical sides of the rods from of first level, the original boundary-value problem (1) decomposes in the limit into two independent problems (13) and (14). Thanks to the inhomogeneous Neumann boundary conditions on the vertical sides of the rods the second level, we have obtained the corresponding term in the right-hand side of the ordinary differential equation of problem (14). This fact was noted in [22], where elliptic boundary-value problems that describe processes in strongly inhomogeneous thin perforated domains with rapidly varying thickness were studied. It follows from these results that for applied problems or for numerical calculations in thick two-level junctions we can use the corresponding limit problems, which are more simple, instead of the initial problem with a sufficient validity. Furthermore, owing to the convergence theorem for the solution to problem (1) proved in [16], we can use the limit problems (13) and (14) with minimal conditions for the right-hand sides f0 and g0. In this paper we consider the thick two-level junction Ωε, which has more dense pa- cking of the thin rods from the first level on the cell of the joining. As a result we have a more complex domain (the domain Π, see Fig. 2), where special junction-layer problems are consi- dered. The junction-layer solutions behave as powers at infinity and do not decrease exponenti- ally. Therefore, they influence directly the leading terms of the asymptotics. 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Asymptotic approximation for the solution to a boundary-value problem with varying type of boundary conditions in a thick two-level junction

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