Homogenization of Spectral Problem on Small-Periodic Networks

The homogenization of a spectral problem on small-periodic networks with periodic boundary conditions is considered. Asymptotic expansions for eigenfunctions and corresponding eigenvalues on the network are con- structed. The theorem is proved which is a justi¯cation of the asymptotic expansions for...

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Datum:	2012
Hauptverfasser:	Krylova, A.S., Sandrakov, G.V.
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Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2012
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spelling	irk-123456789-1067272016-10-04T03:02:29Z Homogenization of Spectral Problem on Small-Periodic Networks Krylova, A.S. Sandrakov, G.V. The homogenization of a spectral problem on small-periodic networks with periodic boundary conditions is considered. Asymptotic expansions for eigenfunctions and corresponding eigenvalues on the network are con- structed. The theorem is proved which is a justi¯cation of the asymptotic expansions for some eigenvalues and eigenfunctions of the problem on the network. Рассматривается осреднение спектральной задачи на мелко-периодической сетке с периодическими краевыми условиями. Построены асимптотические разложения для собственных функций и соответствующих собственных значений задачи на сетке. Доказана теорема, которая является обоснованием построенной асимптотики для некоторых собственных значений и собственных функций задачи на сетке. 2012 Article Homogenization of Spectral Problem on Small-Periodic Networks / A.S. Krylova, G.V. Sandrakov // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 4. — С. 336-356. — Бібліогр.: 16 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106727 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The homogenization of a spectral problem on small-periodic networks with periodic boundary conditions is considered. Asymptotic expansions for eigenfunctions and corresponding eigenvalues on the network are con- structed. The theorem is proved which is a justi¯cation of the asymptotic expansions for some eigenvalues and eigenfunctions of the problem on the network.
format	Article
author	Krylova, A.S. Sandrakov, G.V.
spellingShingle	Krylova, A.S. Sandrakov, G.V. Homogenization of Spectral Problem on Small-Periodic Networks Журнал математической физики, анализа, геометрии
author_facet	Krylova, A.S. Sandrakov, G.V.
author_sort	Krylova, A.S.
title	Homogenization of Spectral Problem on Small-Periodic Networks
title_short	Homogenization of Spectral Problem on Small-Periodic Networks
title_full	Homogenization of Spectral Problem on Small-Periodic Networks
title_fullStr	Homogenization of Spectral Problem on Small-Periodic Networks
title_full_unstemmed	Homogenization of Spectral Problem on Small-Periodic Networks
title_sort	homogenization of spectral problem on small-periodic networks
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2012
url	http://dspace.nbuv.gov.ua/handle/123456789/106727
citation_txt	Homogenization of Spectral Problem on Small-Periodic Networks / A.S. Krylova, G.V. Sandrakov // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 4. — С. 336-356. — Бібліогр.: 16 назв. — англ.
series	Журнал математической физики, анализа, геометрии
work_keys_str_mv	AT krylovaas homogenizationofspectralproblemonsmallperiodicnetworks AT sandrakovgv homogenizationofspectralproblemonsmallperiodicnetworks
first_indexed	2025-07-07T18:54:38Z
last_indexed	2025-07-07T18:54:38Z
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fulltext	Journal of Mathematical Physics, Analysis, Geometry 2012, vol. 8, No. 4, pp. 336–356 Homogenization of Spectral Problem on Small-Periodic Networks A.S. Krylova and G.V. Sandrakov Taras Shevchenko National University of Kyiv 64 Volodymyrska Str., Kyiv, 01601 Ukraine E-mail: krylovaas@univ.kiev.ua sandrako@mail.ru Received June 13, 2012, revised August 13, 2012 The homogenization of a spectral problem on small-periodic networks with periodic boundary conditions is considered. Asymptotic expansions for eigenfunctions and corresponding eigenvalues on the network are con- structed. The theorem is proved which is a justification of the asymptotic expansions for some eigenvalues and eigenfunctions of the problem on the network. Key words: homogenization, spectral problem, small-periodic network, string cross. Mathematics Subject Classification 2000: 35B27, 34L20, 35B40. Introduction The development of modern technology leads to the research of various prob- lems for differential equations with small parameters. Some problems analysis is based on the homogenization theory and asymptotic methods. In the pa- per, the homogenization of spectral problems for a second order operator on small-periodic networks is studied. The homogenization of the equation on a small-periodic network was studied in [1], whereas in [2] it was studied on a small-periodic framework with thin bars. The research of differential equations on the networks, which in the more complicated form can be transferred to strat- ified sets or geometric graphs [3, 4], is a new enough direction. The analysis of the processes in these complex systems leads to the consideration of ordinary second order differential operators on segment systems [5]. The homogenization of a spectral problem on domains was studied in [2, 6–9]. The homogenization of the problems on domains with a high-frequency spectrum was studied in [9]. c© A.S. Krylova and G.V. Sandrakov, 2012 Homogenization of Spectral Problem on Small-Periodic Networks The spectral problems on networks were considered in [3, 5], and the spectral problems on graphs, in [10]. In the paper, in Secs. 1, 2, we consider the spectral cell problem solved in [11], and the spectral problem on the ε-periodic networks in R2, where ε is a small positive parameter. The network, spanned on a rectangle, one side of which is defined by a real l and the other is equal to the unit, is constructed as a union of the same εY -periodic string crosses. According to [3], we can define the function spaces on the string cross by identifying it with a stratified set or a geometrical graph. In Sec. 2, the main theorem of the paper is formulated. In Sec. 3, we define a derivative for the network according to [1] and construct the asymp- totic expansions by following the principle of homogenization from [12]. The expansions are approximate solutions for the spectral problem on the ε-periodic network. In the last section we provide the asymptotic expansion justification which is the proof of the main theorem. 1. Statement of the Cell Problem Let Y be a union of four closed stretched strings that are in the rectangle Q = [0, 1]× [0, l], where Q is a subset of R2 and l is a fixed positive real number. Suppose that the network of such rectangles with the same set Y covers R2. The totality of all Y is called the network G. The set Y is called the periodical recurring cell. We denote by σ1, σ2, σ3 and σ4 the corresponding cell strings, whereas strings ends are called nodes. We define T∂Y as nodes that are disposed on the rectangle boundary ∂Q intersecting with Y . For example, the points σ01 = (0, l/2), σ02 = (1, l/2), σ03 = (1/2, 0) and σ04 = (1/2, l) are the nodes for the string cross Y (Fig. 1). We define TY as the set of all nodes for the cell Y . Let T be the set of all nodes for the network G. One of the cases of the union of strings in the periodically recurring cell is the string cross described in [5] and [11]. The strings of the cross under consideration are supposed to be homogeneous segments, which are right angle arranged and have a unit tension and mass distribution density [5]. Following [11], the two- dimensional string cross, which is represented as a pencil of four strings, can be considered as a string cross formed by two strings σ̃1 and σ̃2 connected in the complementary midpoint. For the statement of the problem on Y , we define the function spaces for the string cross with string lengths, which equal 1 and l (Fig. 1), by using the simple case of the problem for stratified sets that are considered, for example, in [3]. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 337 A.S. Krylova and G.V. Sandrakov p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p p p q p p p p p p p p Á À ε !!Yε LL Qε Ω r r 0 σ01 1 σ02σ̃1 r r r σ05 0 σ03 lσ04 σ̃2 Fig. 1. The Y -periodic string cross. Fig. 2. The small-periodic network. Let C(Y ) be the set of the functions u : Y → R, which are restrictions on Y of continuous functions defined on the rectangle Q. These functions define continuous restrictions to each string σ̃i denoted as ui for i = 1, 2. An integral of the function is defined by the following equality: ∫ Y u dy = ∑ i=1,2 ∫ σ̃i ui dyi, where the right-hand side is the sum of the Riemann integrals over standard measures on the strings with the standard parametrizations y1 ∈ [0, 1] and y2 ∈ [0, l]. Thus, the function u on the string cross can be considered as a pair of contin- uous restrictions of the function to each string of the cross denoted by u1 and u2. Therefore, the integral is defined as the sum of the string integrals. However, it is useful to connect the cell integral with the ”volume” of this cell. For example, we define the cross integral of the unit function as l for the Y -periodic string cross (Fig. 1). Thus, we introduce a normalizing factor l/(l + 1) and denote ∫ Y u(y) dy = l l + 1   1∫ 0 u1(y1) dy1 + l∫ 0 u2(y2) dy2   . (1) Let C1(Y ) be the set of functions u : Y → R, which are the restrictions to Y of the functions from C1(Q) that are defined on the rectangle Q. These functions define the C1(σ̃i) restrictions ui to each string σ̃i for i = 1, 2. Let L2(Y ) be the completion of the space C(Y ) with respect to the norm induced by the inner product (u, v)L2(Y ) = ∫ Y uv dy. 338 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 Homogenization of Spectral Problem on Small-Periodic Networks The function space H1(Y ) is the completion of C1(Y ) with respect to the norm ‖ · ‖H1(Y ) induced by the inner product 〈u, v〉H1(Y ) = ∫ Y uv dy + ∫ Y (∂yu)(∂yv) dy. (2) Taking into account (1), we can rewrite the last equality in details in the form 〈u, v〉H1(Y ) = l l + 1   1∫ 0 u1v1 dy1 + l∫ 0 u2v2 dy2+ + 1∫ 0 (∂y1u1)(∂y1v1) dy1 + l∫ 0 (∂y2u2)(∂y2v2) dy2   . To be precise, the last equality defines the integral in (2), since the given function u ∈ C1(Y ) can be considered as a pair of functions u1(y1) and u2(y2) defined on the segments [0, 1] and [0, l], respectively, because the coordinates on Y can be defined with more complexity (further details are to be found in [1, 3–5]. We denote by C1 per(Y ) the space of functions u ∈ C1(Y ) satisfying the peri- odicity condition u(yi) = u(yi + li), ∂yiu(yi) = ∂yiu(yi + li), where y ∈ T∂Y , i = 1, 2, l1 = 1 and l2 = l. We mean that the given equalities hold for u1(y1) and u2(y2), which are defined by u ∈ C1(Y ). We denote by H1 per(Y ) the completion of the periodic function space C1 per(Y ) with respect to the norm ‖u‖H1(Y ) = ∫ Y u2dy + ∫ Y (∂yu)2 dy. Consider the Y -periodic spectral cell problem: find u ∈ H1 per(Y ) such that −∂2 yu(y) = λu(y) for y ∈ Y ; u(yi) = u(yi + li), ∂yiu(yi) = ∂yiu(yi + li) for y ∈ T∂Y , (3) where ‖u‖L2(Y ) = 1, i = 1, 2, l1 = 1 and l2 = l, with the conditions of function and flux continuity in string nodes for u1 and u2, which are defined by u. The conditions have the following form: u1 ( 1 2 ) = u2 ( l 2 ) , ∂y1u1 ( 1 2 + 0 ) − ∂y1u1 ( 1 2 − 0 ) + ∂y2u2 ( l 2 + 0 ) − ∂y2u2 ( l 2 − 0 ) = 0. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 339 A.S. Krylova and G.V. Sandrakov To be definite, the first equation of (3) can be written as two equations −∂ 2 y1 u1(y1) = λu1(y1) for y1∈ [0, 1], −∂ 2 y2 u2(y2) = λu2(y2) for y2∈ [0, l], for u1 and u2, which are defined by u. It agrees with the definition of the derivative ∂y by equality (2). We remark that the continuity and periodicity conditions are fulfilled automatically for u ∈ C1 per(Y ) (and, in the known sense [1], for u ∈ H1 per(Y )). Moreover, the solutions of problem (3) are smooth according to [11], where this problem was solved and the eigenvalues λm = 4π2m2 with m ∈ Z+ = {0, 1, 2 . . . } and the corresponding eigenfunctions N0(y), N1(y), . . . , which are combinations of cos 2πy1m and sin 2πy2m on each string, were obtained. In this paper, we consider only the eigenvalue λ0 = 0 of problem (3) with the eigenfunction N0(y) = l−1/2 that means N0 1 (y1) = l−1/2 and N0 2 (y2) = l−1/2. 2. Problem on Networks We shrink N times the rectangle Q and the string cross Y , where N is a given natural number. As a result, we obtain the sets Qε and Yε with coordinates x′= ε y for ε = 1/N . If we repeat the rectangles Qε by periodicity, then we obtain a closed domain Ω = [0, 1] × [0, l] ⊂ R2 with the Lipschitz boundary ∂Ω. The small-periodic network Gε spans this domain in R2 (Fig. 2) and it is the union of N2 string cross Yε. This Yε we call periodically recurring cell with edge lengths that are equal to ε and lε. The parameter x′ denotes the position of a point on the network Gε. We denote by Tε the whole node set of the network Gε. Similar networks with arbitrary arcs instead of stretched strings, which have the unit span and mass distribution density, was considered in [1]. Therefore, we will use some notation and calculations from this paper. According to [1], on the network Gε, we define H1(Gε) as the set of functions, which are continuous in nodes and absolutely continuous on each string, with the norm ‖u‖2 H1(Gε) = ε ∫ Gε ( u2 + (∂x′u)2 ) dx′. (4) The norm definition is equivalent to the norm definition induced by inner product (2), which is defined for Gε. The space H1 per(Gε) ⊂ H1(Gε) of periodic functions on Gε ∩ ∂Ω is defined similarly to the space on Y . Consider the following spectral boundary problem for the network Gε: find uε ∈ H1 per(Gε) such that ‖uε‖L2(Gε) = 1 and −ε2∂2 x′uε(x′) = λεuε(x′) for x′ ∈ Gε ∩ Ω, (5) uε(x′1, x ′ 2) = uε(x′1 + 1, x′2), uε(x′1, x ′ 2) = uε(x′1, x ′ 2 + l), ∂x′uε(x′1, x ′ 2) = ∂x′uε(x′1 + 1, x′2), ∂x′uε(x′1, x ′ 2) = ∂x′uε(x′1, x ′ 2 + l) for x′ ∈ Gε ∩ ∂Ω. (6) 340 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 Homogenization of Spectral Problem on Small-Periodic Networks This means that the coordinates x′1 and x′2 can change along the parallel lines, which are obtained by periodic extensions of the strings εσ̃1 and εσ̃2. To be precise, it is helpful to consider the function uε ∈ H1 per(Gε) as a set of 2N func- tions uε 1j(x ′ 1j) and uε 2j(x ′ 2j) that are defined on the lines. The lines are periodic extensions of the strings εσ̃1 and εσ̃2, which are parametrized by coordinates x′1j ∈ [0, 1] and x′2j ∈ [0, l] for j = 1, . . . , N . In this case, periodicity condi- tion (6) and the function and flux continuity conditions in nodes of the network Gε, which are defined for the functions uε 1j(x ′ 1j) and uε 2j(x ′ 2j), hold automatically, since uε ∈ H1 per(Gε). Thus, problem (5)–(6) reduces to the 2N elliptical equations for the functions uε 1j(x ′ 1j) and uε 2j(x ′ 2j) with j = 1, . . . , N , −ε2∂ 2 x′1j (uε 1j) = λuε 1j for x′1j ∈ [0, 1], −ε2∂ 2 x′2j (uε 2j) = λ uε 2j for x′2j ∈ [0, l]. Besides, the functions uε 1j and uε 2j are smooth enough due to the ellipticity of these equations. Problem (5)–(6) has the trivial solution uε = l−1/2 for the eigenvalue λε = 0. In order to exclude this solution from further consideration, we define the space H1 per∗(Gε) = { u ∈ H1 per(Gε) : (u, 1)L2(Gε) = 0 } with the norm ‖u‖2 H1∗(Gε) = ε ∫ Gε (∂x′u)2 dx′ that is equivalent to norm (4) on H1 per(Gε) by virtue of Poincare’s inequality [3]. We define L2∗(Gε) = { u ∈ L2(Gε) : (u, 1)L2(Gε) = 0 } in the same way. Thus, the problem (5)–(6) can be considered as variational in the sense of integral identity: find u ∈ H1 per∗(Gε) such that ε2 ∫ Gε (∂x′u)(∂x′v) dx′ = λε ∫ Gε uv dx′ ∀v ∈ H1 per∗(Gε). By the definition, the operator of problem (5)–(6) is negative definite and has a compact inverse operator on L2∗(Gε) for a fixed ε (with 0 <ε ≤ ε0) which follows from the embedding compactness H1 per∗(Gε) ⊂ L2∗(Gε) [4]. Thus, there are countable sets of eigenvalues λ1 ε, λ2 ε, . . . and orthonormalized eigenfunc- tions u1 ε, u2 ε, . . . for the problem such that αε2 ≤ λ1 ε ≤ . . . ≤ λs ε ≤ . . . with the multiplicity being taken into account, where α is some positive constant and lims→∞ λs ε = ∞. The main aim of the paper is to construct and justify the asymptotic ex- pansions for eigenvalues and eigenfunctions of problem (5)–(6) with insufficiently large numbers (s ¿ ε−2). The main statement of the paper is the following theorem. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 341 A.S. Krylova and G.V. Sandrakov Theorem 1. For eigenvalues λs ε and eigenfunctions us ε of problem (5)–(6) there exists a constant C independent of ε and s such that \|λs ε − ε2λs\| ≤ Cε3(λs)3/2 and ‖us ε − vs‖L2(Gε) ≤ Cε(λs)1/2 for λs ¿ ε−2 and 0 < ε ≤ ε0, where λs and vs are some eigenvalue and eigen- function of the relevant homogenization problem which is to be determined later. Theorem 1 will be proved in Sec. 4. The estimate of this theorem is valid for all λs and us ε such that λs ≤ c ε−2+σ with a positive constant c, where 0 < σ ≤ 2 and 0 < ε ≤ ε0 (this means that λs ¿ ε−2 precisely). However, the proof of the theorem may be incorrect when λs = c ε−2. This situation is natural, since there is a high-frequency spectrum for problem (5)–(6). The spectrum will be introduced and justified in further researches. The initial component construction of asymptotic expansions for λs ε and us ε as well as the homogenized problem will be considered in the next section. In order to construct the initial component of the asymptotic expansions for the solutions to problem (5)–(6), we will follow the principles of homogenization introduced in [12]. 3. Construction of Asymptotic Expansions We define the derivative of functions on the network in accordance with [1]. If a function v(x, y) is an element of C1 ( Ω, H1(Y \TY ) ) (where Ck ( Ω,W ) denotes the space of abstract functions on the domain Ω with values in some Hilbert space W ), then the following relation is valid: ∂x′v(x′, ε−1x′) = (∇v(x, y) + ε−1∂yv(x, y) )∣∣ x=x′, y=x′/ε , (7) where ∇ = (∂/∂x1, ∂/∂x2) and ∂y = (∂/∂y1, ∂/∂y2). Notice that sometimes it is more convenient to consider the function on the string cross Y as two functions on the corresponding strings. We will construct an expansion of the eigenfunction uε for problem (5)–(6) in the form of asymptotic sum (see [12] for more details). The components of the asymptotic sum are functions of the form u0(x, y), u1(x, y), . . . , where (x, y) ∈ Q×Y and x = x′, y = x′/ε. The functions have separated variables and are Y -periodic in second variable. The asymptotic sum is in the following form: ua(x′, x′ ε ) = u0(x′, x′ ε ) + εu1(x′, x′ ε ) + ε2u2(x′, x′ ε ). (8) Consequently, the expansion for eigenvalue λε of problem (5)–(6) should be of the form λa = λ0 + ελ1 + ε2λ2. (9) 342 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 Homogenization of Spectral Problem on Small-Periodic Networks The function ua(x, y) is Y -periodic by the definition. Therefore, we can con- sider this function as the function on Y for fixed x ∈ Ω, if necessary. Moreover, it is helpful to consider the function as two functions u1 a(x, y1) and u2 a(x, y2) on the corresponding strings. Using (7), we can rewrite equation (5) in the following form: −{ ∂2 y + 2ε1∇∂y + ε2∇2 } ua(x, y) = λaua(x, y), (10) where y = x′/ε, x = x′, and the notation ∂2 y = ( ∂2 ∂y2 1 , ∂2 ∂y2 2 ) , ∇∂y = ( ∂ ∂x1 ∂ ∂y1 , ∂ ∂x2 ∂ ∂y2 ) , ∇2 = ( ∂2 ∂x2 1 , ∂2 ∂x2 2 ) is used for convenience. Substituting asymptotic sum (8), (9) to (10) and setting equal coefficients with the same powers of ε, we obtain the equations for the functions u0, u1 and u2. Taking into account [12], it is useful to choose the asymptotic expansion components in the form ui = Ni (y) vi(x), where Ni ∈ H1 per(Y ). As stated before, we consider only the eigenvalue λ0 = 0 for cell problem (3) with the eigenfunc- tion N0(y) = l−1/2. Thus, setting equal coefficients for ε0, we obtain the equation −∂2 yu0 = 0, which is a cell problem. Thus, we can choose u0 = Av(x), where the constant A and the function v(x) are defined later. Next, we consider the equation for ε1. The equation has the form −∂2 yu1 = λ1u0, since λ0 = 0 and ∂y∇u0 = 0 due to the definition of u0. It is well known [1] and may be directly verified that the equation for w ∈ H1 per(Y ) such that −∂2 yw(y) = g(y) with a given function g ∈ L2(Y ) is solvable if and only if the right-hand side is orthogonal to N0(y). The equation for ε1 satisfies the condition only if λ1 = 0 for v(x) 6= 0. The function u1 = Av1(x) is a periodic solution of the equation, where the function v1(x) is chosen to be null for further simplicity. Setting equal coefficients for ε2, we obtain the cell equation which, by the definition, may be considered as the system of equations −∂2 y1 u1 2(y1, x) = A∂2 x1 v + λ2Av, −∂2 y2 u2 2(y2, x) = A∂2 x2 v + λ2Av. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 343 A.S. Krylova and G.V. Sandrakov Calculating the inner product of the right-hand side of the equation with the kernel element N0(y) of the cell equation and setting the product to be equal to zero, we get A ∫ Y (∇2v(x) + λ2v(x) ) N0(y) dy = 0. By the integration rule (1), for A 6= 0, we get the spectral homogenized problem for v(x) ∂2 x1 v(x) + l∂2 x2 v(x) + (l + 1)λ2v(x) = 0 for x ∈ Ω, v(x) = v(x + li), ∂xv(x) = ∂xv(x + li) for x ∈ ∂Ω, (11) where the periodicity conditions are conformed to (5)–(6), v(x) is normalized by the condition ‖v‖L2(Ω) = 1, and l1 = 1, l2 = l. The solutions of the spectral problem are determined by the countable sets of eigenvalues λs = 4π2(l + 1)−1(n2 + m2l−1) for m,n ∈ N = {1, 2, . . . } to be such arranged that 0 < λ1 ≤ λ2 ≤ . . . (with the multiplicity being taken into account, which may be 2, 4 or 8 depending on l) and the corresponding eigenfunctions vs 0(x) have the form 2l−1/2 cos 2πnx1 cos 2πmx2l −1, 2l−1/2 sin 2πnx1 cos 2πmx2l −1, 2l−1/2 cos 2πnx1 sin 2πmx2l −1 or 2l−1/2 sin 2πnx1 sin 2πmx2l −1 for m,n ∈ N. It is known [13, 14] that λs = 4πs l−1 + O(s1/2) for large s. Thus, fixing λs, vs 0 and substituting the homogenized equation (11) in the equation for ε2, we can define A = 1 and us 0 = vs 0(x), us 2 = N2(y) ( ∂2 x1 vs 0(x) + λsvs 0(x) ) , where N2(y) ∈ H1 per(Y ) satisfies the system of the equations −∂2 y1 N1 2 (y1) = 1, −∂2 y2 N2 2 (y2) = −l−1. There exists a solution of the system (that is determined up to the constant function AN0(y)) and the solution is normalized such that ∫ Y N0(y)N2(y)dy = 0. Thus, we get the eigenvalue and the eigenfunction asymptotics in the following form: λs a = ε2λs, us a(x, y) = vs 0(x) + ε2N2(y) ( ∂2 x1 vs 0(x) + λsvs 0(x) ) . In what follows, we assume that N2(y) is periodically extended to the whole network. Thus, the function us a(x ′, x′ ε ) is well defined on Gε. Moreover, the 344 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 Homogenization of Spectral Problem on Small-Periodic Networks function us a(x ′,x′ ε ) satisfies automatically all periodicity conditions (6) and the function and flux continuity conditions in string nodes of the network Gε, since the smooth function vs 0(x) is defined on the rectangle Ω and satisfies the periodic conditions on Ω whereas the function N2(y) is an element of H1 per(Y ) and is regular enough on Y as a solution of the elliptical equation on the cross [1, 3]. 4. Asymptotics Justification Before justifying the asymptotic expansions, we must return to the network definition. We enumerate all N2 cells Qij ε and the same number of string crosses Y ij ε by the numbers i, j = 1, . . . , N . Then, for the whole domain Ω = ⋃ ij Qij ε and the network Gε = ⋃ ij Y ij ε , we obtain the equalities ∫ Ω v(x) dx = N∑ i=1 N∑ j=1 ∫ Qij ε v(x) dx, ∫ Gε v(x′) dx′ = N∑ i=1 N∑ j=1 ∫ Y ij ε v(y) dy, where for the cell integral and the cross integral we have ∫ Qij ε v(x) dx = iε∫ ε(i−1) lεj∫ lε(j−1) v(x1, x2) dx1dx2 and ∫ Y ij ε v(y) dy = l 1 + l   iε∫ ε(i−1) v(y1, jlε− 2−1lε) dy1 + lεj∫ lε(j−1) v(iε− 2−1ε, y2) dy2  , respectively. We prove now the statement on the change of integrals over the small-periodic network Gε by integrals over Ω = [0, 1]× [0, l] which we will need later. Proposition 1. For v ∈ H2(Ω) there exists a constant C independent of ε such that the following inequality holds: ∣∣∣∣∣∣ ∫ Ω v(x) dx− ε ∫ Gε v(x′) dx′ ∣∣∣∣∣∣ ≤ Cε2. P r o o f. Consider the linear functional l(v) = ∫ Ω v(x)dx− ε ∫ Gε v(x′)dx′. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 345 A.S. Krylova and G.V. Sandrakov By the definition, l(v) = N∑ i=1 N∑ j=1   ∫ Qij ε v(x) dx− ε ∫ Y ij ε v(y) dy   . We estimate the functional in brackets using the following lemma from [15]. Lemma (Bramble–Hilbert). Suppose Ω is an open convex bounded do- main in Rn and the linear functional l(u) is bounded on Hm+1(Ω), where m ∈ Z+ is fixed, that is \|l(u)\| ≤ M‖u‖Hm+1(Ω). If l(u) is equal to zero for every polynomial in variables x1, x2, . . . , xn of degree m, then there exists a constant M dependent only on Ω such that the following inequality holds: \|l(u)\| ≤ MM \|u\|Hm+1(Ω), where \|u\|Hm+1(Ω) = ‖∇m+1u‖L2(Ω) is a standard seminorm for Hm+1(Ω). Before using the embedding estimate for H2(Qij ε ) ⊂ C(Qij ε ) with some con- stant independent of ε, we change the variable y = x/ε and introduce the notation ṽ(y) = v(εy). Thus we get lij(v) = lij(ṽ) = ε2   ∫ Qij ṽ(y) dy − ∫ Y ij ṽ(y) dy   . Taking into account the embedding estimate max y∈Qij \|ṽ(y)\| ≤ C‖ṽ(y)‖H2(Qij) for the domain Qij , we have \|lij(ṽ)\| ≤ Cε2‖ṽ(y)‖H2(Qij). It is verified directly that the functional l(ṽ) becomes zero on the first-degree polynomials. In this case, by the Bramble–Hilbert lemma, we have the relevant estimate. In this estimate, we return again to the variable x and obtain \|lij(v)\| ≤ Mε2\|ṽ(y)\|H2(Qij) = Mε3\|v(x)\| H2(Qij ε ) . Therefore, we have \|l(v)\| ≤ Mε3 N∑ i=1 N∑ j=1 \|v(x)\| H2(Qij ε ) = Mε3N   N∑ i=1 N∑ j=1 \|v(x)\|2 H2(Qij ε )   1/2 =Mε2\|v(x)\|H2(Ω). 346 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 Homogenization of Spectral Problem on Small-Periodic Networks Thus, the integral change leads to the error O(ε2), and that is just what was to be proved. This analogue of the Riemann–Lebesgue lemma will be used later on. Proposition 2. Let U ∈ L2(Y ) be periodically extended to the whole network functions and v ∈ H2 per (Ω) be such that ‖ ∂2 x′v‖L2(Gε) ≤ c with the constant c independent of ε. Then the following inequality holds: ∣∣∣∣∣∣ ε ∫ Gε U ( x′ ε ) v(x′) dx′ − l−1 ∫ Y U(y) dy ∫ Ω v(x) dx ∣∣∣∣∣∣ ≤ Cε2, where the constant C is independent of ε. P r o o f. Let M(y) ∈ H2 per∗(Y ) be a solution of the equation ∂2 yM(y) = U(y)− l−1 ∫ Y U(y)dy and M(y) be periodically extend to the whole network. Using the equality ε2∂2 x′M ( x′ ε ) = ∂2 yM(y) ∣∣ y=x′ ε = U ( x′ ε ) − l−1 ∫ Y U(y)dy, we multiply it by the function v(x) and integrate the result over the small-periodic network Gε. Thus, we obtain ε3 ∫ Gε ∂2 x′M ( x′ ε ) v(x′) dx′ = ε ∫ Gε U ( x′ ε ) v(x′) dx′ − εl−1 ∫ Y U(y)dy ∫ Gε v(x′) dx′. Consider the left-hand side of the equality separately. Integrating by parts twice, we use the Cauchy–Bunyakovsky inequality. Taking into account the func- tion periodicity over the network, we get ε3 ∫ Gε ∂2 x′M ( x′ ε ) v(x′) dx′ = ε3 ∫ Gε M ( x′ ε ) ∂2 x′v(x′) dx′ ≤ ε2 √√√√ε ∫ Gε M2 ( x′ ε ) dx′ √√√√ε ∫ Gε ( ∂2 x′v(x′) )2 dx′ ≤ c ε2‖M‖L2(Y ), since ‖M ( x′ ε ) ‖2 L2(Gε) = N∑ i=1 N∑ j=1 ε ∫ Y ij ε M2 ( x′ ε ) dx′ = N∑ i=1 N∑ j=1 ε2 ‖M‖2 L2(Y ) = ‖M‖2 L2(Y ). Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 347 A.S. Krylova and G.V. Sandrakov Using Proposition 1, we get the estimate ∣∣∣∣∣∣ ε ∫ Gε U ( x′ ε ) v(x′) dx′ − l−1 ∫ Y U(y) dy ∫ Ω v(x) dx ∣∣∣∣∣∣ ≤ ε2C. Thus, we obtained the error O(ε2) that is defined by the values ‖M‖L2(Y ), ‖∂2 x′v‖L2(Gε) and ‖∂2 xv‖L2(Ω) (by Proposition 1). This concludes the proof. Here we use H2 per∗(Y ), H2 per(Ω), . . . that are defined in the standard way according to Sec. 1, [3, 15]. In what follows, the constants independent of ε and s are denoted by C, although the constants may be different in different formulas. P r o o f of Theorem1. The justification of the asymptotic expansions is real- ized by the minimax principle, the Rayleigh–Ritz method used from [14] and the known Vishik–Lyusternik theorem from [13, 16]. We recall the exact formulations of the statements. Suppose H1 and H0 are Hilbert spaces. The space H1 is embedded compactly in H0, and H−1 is a dual space for H1 with respect to the inner product of H0. The operator L : H1 → H−1 is continuous and such that 〈Lv, v〉H0 ≥ α‖v‖2 H0 ∀ v ∈ H1, where α is a positive constant, and L∗ = L. Then the eigenvalues λk of the operator L are real and define a nondecreasing sequence α ≤ λ1 ≤ λ2 ≤ ... (with the multiplicity being taken into account), and the eigenfunctions u1, u2, . . . are orthonormalized in H0. For the operator L, the following statements hold. Theorem 2 (Minimax principle). Let λk and uk be (ordered) eigenvalues and eigenfunctions of the operator L. Then the following equalities hold: λ1 = min v∈H0 v 6=0 〈Lv, v〉H0 ‖v‖2 H0 , λk = min v∈H0 k v 6=0 〈Lv, v〉H0 ‖v‖2 H0 , where H0 k = { v ∈ H0 : 〈v, u1〉H0 = 0, . . . , 〈v, uk−1〉H0 = 0 } for k = 2, 3, . . . . Theorem 3 (Rayleigh–Ritz method). Let Hd be a d-dimensional subspace of H0 and Pd be an orthogonal projector onto Hd with some natural number d. Then for the ordered eigenvalues µ1, . . . , µd of the operator PdLPd on Hd and the eigenvalues λ1, . . . , λd of the operator L the following inequalities hold: λ1 ≤ µ1, . . . , λd ≤ µd. Theorem 4 (Vishik–Lyusternik). Let λk and uk be eigenvalues and eigen- functions of the operator L. Assume that there exists a real µ ∈ R and u ∈ H1 such that ‖u‖H0 = 1 and ‖Lu− µu‖H0 ≤ β. 348 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 Homogenization of Spectral Problem on Small-Periodic Networks Then there exists λs such that \|µ − λs\| ≤ β, and for every σ > β there exists ũ ∈ H1 such that ‖ũ‖H0 = 1 and ‖u− ũ‖H0 ≤ 2βσ−1, where ũ is a linear combination of the eigenvectors of the operator L, which correspond to eigenvalues from the interval (µ− σ, µ + σ). Applying the operator L = −ε2∂2 x′ to the approximate function us a = us 0+ε2us 2 and using the differentiation rule (7), we obtain Lus a ( x′, x′/ε ) = −ε2∇2us 0 − ε2∂2 yus 2 − 2ε3∇∂yu s 2 − ε4∇2us 2 = ε2λsus 0 + ε4λsus 2 + ε3Ks = ε2λsus a ( x′, x′/ε ) + ε3Ks(x′, x′/ε), (12) where for ws 3/2 = ∂3 xvs 0 + λs∂xvs 0 and ws 2 = ∂4 xvs 0 + 2λs∂2 xvs 0 + λsλsvs 0 we denote Ks(x, y) = −2∇∂yu s 2 − ε∇2us 2 − ελsus 2 = −2(∂yN2)ws 3/2 − ε(N2)ws 2. For large λs, directly from (11) we have ∥∥∂xvs 0 ∥∥2 L2(Ω) = O(λs) and ∥∥∂2 xvs 0 ∥∥2 L2(Ω) = O((λs)2), ∥∥∂2 xvs 0 + λsvs 0 ∥∥2 L2(Ω) = O((λs)2), ∥∥ws 3/2 ∥∥2 L2(Ω) = O((λs)3), ∥∥ws 2 ∥∥2 L2(Ω) = O((λs)4), ∥∥∂2 x((ws 3/2) 2) ∥∥ L2(Ω) = O((λs)4), ∥∥∂2 x((ws 2) 2) ∥∥ L2(Ω) = O((λs)5). The similar estimates ∥∥∂x′v s 0 ∥∥2 L2(Gε) = O(λs), . . . are valid for the correspond- ing norms on Gε, since a differentiation of vs 0 by ∂x′1 and ∂x′2 gives a multiplier equivalent to (λs)1/2, \|vs 0\| ≤ 2l−1/2 by the definition, and, for example, we have ε ∫ Gε 1 dx′ = l. Using Proposition 2 (with the constant dependence on v being taken into account), the regularity of functions N2(y), and the smoothness of functions vs 0(x), we get ε ∫ Gε ( Ks(x′, x′/ε) )2 dx′ ≤ 4ε ∫ Gε ( ∂yN2w s 3/2 )2 dx′ + 2ε3 ∫ Gε ( N2w s 2 )2 dx′ ≤ C ∫ Ω ( ws 3/2 )2 dx + ε2C ∫ Ω ( ws 2 )2 dx + ε2C(λs)4 + ε4C(λs)5, (13) where the constant C is independent of ε and s that is essential for large λs. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 349 A.S. Krylova and G.V. Sandrakov Besides, according to Proposition 2, we obtain the following relations: ∣∣∣ ‖us a(x ′, x′/ε)‖2 L2(Gε) − 1 ∣∣∣ ≤ Cε2λs + ∫ Ω vs 0(x)2dx− 1 + Cε4(λs)2 + 2ε2 ∫ Y N0(y)N2(y)dy ∫ Ω vs 0(x) ( ∂2 xvs 0(x) + λsvs 0(x) ) dx + Cε6(λs)3 + ε4 ∫ Y N2(y)2dy ∫ Ω ( ∂2 xvs 0(x) + λsvs 0(x) )2 dx ≤ Cε2λs + Cε4(λs)2 + Cε6(λs)3 ≤ Cε2λs (14) for λs ¿ ε−2 (that is for λs ≤ C ε−2+σ with 0 < σ ≤ 2), where the constant C is independent of ε and s, since ∫ Ω vs 0(x)2dx = 1 and ∫ Y N0(y)N2(y)dy = 0 by the definition. Similarly, we can verify that ∣∣∣∣∣∣ ε ∫ Gε us a(x ′, x′/ε) uj a(x ′, x′/ε) dx′ ∣∣∣∣∣∣ ≤ Cε2λs + Cε2λj for s 6= j and λs, λj ¿ ε−2. This means that the functions u1 a, u2 a,. . . , u s a are almost orthonormalized in L2(Gε) (in the sense of last two inequalities) and are linearly independent for λs ¿ ε−2. Here, it is important that the system of the eigenfunctions {vs 0}∞s=1 is orthonormalized in L2(Ω). According to (14), we get ‖us a‖L2(Gε) 6= 0 for λs ¿ ε−2. Therefore, defining ûs a = ‖us a‖−1 L2(Gε) us a, we obtain ‖ûs a‖L2(Gε) = 1. By shifting the term with the eigenvalue λs to the left-hand side of equal- ity (12), raising the result to the second power, multiplying by ε and integrating the resulting relation over the network Gε, for λs ¿ ε−2 we obtain ε ∫ Gε ( Lus a − ε2λsus a )2 dx′ = ∥∥Lus a − ε2λsus a ∥∥2 L2(Gε) = ε7 ∫ Gε ( Ks(x′, x′/ε) )2 dx′ ≤ ε6C(λs)3 + ε8C(λs)4 + ε10C(λs)5 ≤ ε6C(λs)3 in accordance with (13) and (14). Thus, we have the following inequality: ∥∥Lûs a − ε2λsûs a ∥∥ L2(Gε) ≤ ε3C(λs)3/2, where the constant C is independent of ε and s for λs ¿ ε−2. Therefore, by Theorem 4, there exists an eigenvalue λ k(s) ε of problem (5)–(6) such that ∣∣∣λk(s) ε − ε2λs 2 ∣∣∣ ≤ ε3C(λs)3/2. (15) 350 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 Homogenization of Spectral Problem on Small-Periodic Networks Thus, the estimate of Theorem 1 for eigenvalues of problem (5)–(6) will follow from inequality (15) if we verify that k(s) = s for every s = 1, 2 . . . . By Theorem 2, for the first eigenvalue of problem (5)–(6) we have the relation λ1 ε = min 0 6=u∈H1 per∗(Gε) (Lu, u)L2(Gε) ‖u‖2 L2(Gε) . (16) By the definition, u1 ε satisfies the equality ( u1 ε, 1 ) L2(Gε) = 0. The constructed asymptotic expansion u1 a for u1 ε may not comply with the orthogonality condition for a constant. Therefore, we subtract the constant A1 ε = εl−1 ∫ Gε u1 adx′ from u1 a to have u1 a −A1 ε ∈ H1 per∗(Gε). Then, according to Proposition 2, we get εl−1 ∫ Gε u1 adx′ = ∫ Ω v1 0dx + ε2 ∫ Ω ( ∂2 xv1 0 + λ1v1 0 ) dx ∫ Y N0N2dy + O(ε2) = O(ε2), since ∫ Ω v1 0(x)dx = 0 and ∫ Y N0N2(y)dy = 0 by the definition. Thus, we can write that A1 ε = ε2Ã1 ε, where \|Ã1 ε\| ≤ C with C independent of ε. Furthermore, denoting ũ1 a = u1 a − ε2Ã1 ε, we obtain ũ1 a ∈ H1 per∗(Gε) by the definition of the constant Ã1 ε. Thus, we can substitute the obtained function ũ1 a(x ′, x′/ε) into (16). Applying the operator L to ũ1 a and using the differentiation rule (7), we obtain the result (which is similar to (12) for s = 1) in the following form: Lũ1 a ( x′, x′/ε ) = ε2λ1 ( u1 0 + ε2u1 2 − ε2Ã1 ε ) + ε3K̃1(x′, x′/ε), where we denote K̃1(x, y) = K1(x, y) + ε2Ã1 ελ 1 and use the relations from Sec- tion 3. Multiplying the obtained result by εũ1 a(x ′, x′/ε) and integrating over the network, we get ( Lũ1 a, ũ 1 a ) L2(Gε) = ε ∫ Gε Lũ1 aũ 1 adx′ = ε2λ1 ‖ũ1 a‖2 L2(Gε) + ε4 ∫ Gε K̃1ũ1 adx′. (17) Consider the last term of relation (17). Using the estimate (13) and the Cauchy– Bunyakovsky inequality, we obtain ε ∫ Gε K̃1ũ1 adx′ ≤  ε ∫ Gε (K̃1)2dx′   1/2  ε ∫ Gε (ũ1 a) 2dx′   1/2 ≤ C‖ũ1 a‖L2(Gε). The subtraction of the constant ε2Ã1 ε from the function u1 a does not influence on estimate (14) essentially, hence we can write ‖ũ1 a‖L2(Gε) = 1+O(ε2). Substituting Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 351 A.S. Krylova and G.V. Sandrakov the result in (16), multiplying the identity (17) by ‖ũ1 a‖−2 L2(Gε) , and using the estimate for the last term of the identity, we get the estimate from above for the first eigenvalue of problem (5)–(6) in the following form: λ1 ε ≤ ε2λ1 + Cε3. We fix a natural number d > 1 and denote U1 a = ũ1 a‖ũ1 a‖−1 L2(Gε) , then we have ‖U1 a‖L2(Gε) = 1 and U1 a ∈ H1 per∗(Gε). In what follows, we orthonormalize the functions U1 a (x′, x′/ε), u2 a(x ′, x′/ε), . . . , ud a(x ′, x′/ε) in the space L2∗(Gε). Define the constant Ai ε = εl−1 ∫ Gε ui adx′ for i = 2, . . . , d. Then, we can write Ai ε = ε2Ãi ε as in the case i = 1, where \|Ãi ε\| ≤ C with C independent of ε. Thus, we have ũi a = ui a − ε2Ãi ε ∈ H1 per∗(Gε) for i = 2, . . . , d. Define also the constant A21 ε = ε ∫ Gε ũ2 aU 1 adx′. Then, according to Proposition 2, we obtain A21 ε = ∫ Ω v2 0v 1 0 dx + O(ε2) = O(ε2), since ∫ Ω v2 0(x)v1 0(x) dx = 0 by the definition. Thus, A21 ε = ε2Ã21 ε , where \|Ã21 ε \| ≤ C with C independent of ε. Denote ŭ2 a = ũ2 a − ε2Ã21 ε U1 a . Then ŭ2 a is orthogonal to U1 a and satisfies the relations similar to (12)–(14) and (17). Thus, U2 a = ŭ2 a ‖ŭ2 a‖−1 L2(Gε) is well de- fined and orthogonal to U1 a , ‖U2 a‖L2(Gε) = 1 and U2 a ∈ H1 per∗(Gε). Furthermore, by induction we can find the orthonormalized U1 a , U2 a ,. . . , Ud−1 a and define the function ŭd a = ũd a − ε2Ãd,d−1 ε Ud−1 a − · · · − ε2Ãd1 ε U1 a . The function is orthogonal to U i a when ε2Ãdi ε = ε ∫ Gε ũd aU i adx′ with \|Ãdi ε \| ≤ C for i = 1, . . . , d − 1 (it is helpful here that the system of the eigenfunctions v1 0, . . . , vd 0 is orthonormalized) and it satisfies the relations similar to (12)–(14) and (17). Thus, Ud a = ŭd a ‖ŭd a‖−1 L2(Gε) is defined and orthogonal to the functions U1 a , U2 a , . . . , Ud−1 a . Moreover, we get ‖Ud a‖L2(Gε) = 1 and Ud a ∈ H1 per∗(Gε). Define the d-dimensional subspace Hd ⊂ H1 per∗(Gε) as a linear span of the functions U1 a (x′, x′/ε), U2 a (x′, x′/ε), . . . , Ud a (x′, x′/ε) and the orthogonal projector Pd onto Hd. By the definition, for U ∈ H1 per∗(Gε) we have PdU = d∑ i=1 U i a (U i a, U)L2∗(Gε). Therefore, PdU i a = U i a for i = 1, . . . , d. Thus, there exist d-element sets of the eigenvalues µ1 ε, µ 2 ε, . . ., µ d ε and of the orthonormalized eigenfunctions w1 ε , w 2 ε , . . ., w d ε of the operator Ld = PdLPd such that the inequality µ1 ε ≤ µ2 ε ≤ . . . ≤ µd ε is valid 352 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 Homogenization of Spectral Problem on Small-Periodic Networks (with the multiplicity being taken into account). Besides, according to Theo- rem 3, we obtain λ1 ε ≤ µ1 ε, . . . , λ d ε ≤ µd ε . Using the relations similar to (12)–(14) and (17) for the orthonormalized functions U1 a (x′, x′/ε), U1 a (x′, x′/ε), . . . , Ud a (x′, x′/ε), we get LdU i a(x ′, x′/ε) = PdLU i a = ε2λiU i a + O(ε3) and ∥∥LdU i a − ε2λiU i a ∥∥ L2(Gε) ≤ ε3C, where i = 1, . . . , d, and the constant C is independent of ε. Therefore, according to Theorem 4, there exists an eigenvalue µ j(i) ε of the operator Ld such that ∣∣∣µj(i) ε − ε2λi ∣∣∣ ≤ ε3C, (18) where i = 1, . . . , d, and the constant C is independent of ε. Here the dependence of C on i (which is clear from the relations similar to (12)–(14)) is not essen- tial, since in order to complete the proof of the estimate of Theorem 1 for the eigenvalues of problem (5)–(6), we have to verify that k(s) = s in (15) for every s = 1, 2, . . . , d. Following [13] and [16], we verify that j(i) = i in (18) for every i = 1, . . . , d. Indeed, if the eigenvalues λ1, . . . , λd are simple, then we have j(i) = i for every i = 1, . . . , d, since d ordered values µ1 ε, . . . , µ d ε are in the ε3-neighborhoods of d strictly ordered values ε2λ1, . . . , ε2λd which is possible only if j(i) = i. However, the multiplicity of the eigenvalue λ1 is equal either to two or four, depending on l. Consider the first case, for example, then λ3 is separated from λ1 by some positive constant δ (for example, δ = 1 for l = 1/2). Choosing d = 2 in (18), we conclude that j(1) = 1 and j(2) = 2 (what is to be proved, since λ1 = λ2 ) or j(1) = 2 and j(2) = 2. In the latest case, there exists a constant σ > 0 such that µ1 ε < µ2 ε− ε2σ < µ2 ε + ε2σ < µ3 ε, and on the interval (ε2λ1− ε2σ, ε2λ1 + ε2σ) there exists only one eigenvalue µ2 ε of the operator L2. Thus, by Theorem 4, we have ‖U1 a − w2 ε‖L2(Gε) ≤ εC, ‖U2 a − w2 ε‖L2(Gε) ≤ εC. But, it is impossible [13] since the normalized function w2 ε approximates two orthonormalized functions U1 a and U2 a simultaneously. Thus, the equality j(i) = i for i = 1, 2 is proved. In the same way, we can prove that k(s) = s in (15) for s = 1, 2 (when the multiplicity of λ1 is 2). Indeed, there exist only two eigenvalues λ1 ε, λ 2 ε of problem (5)–(6) on the segment [αε2, µ2 ε], since αε2 ≤ λ1 ε ≤ λ2 ε ≤ µ2 ε due to Theorem 3. Moreover, inequality (15) is valid. Therefore, we have k(1) = 1 and k(2) = 2 or k(1) = 2 and k(2) = 2. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 353 A.S. Krylova and G.V. Sandrakov In the last case, there exists a constant σ > 0 such that only one eigenvalue λ2 ε of the operator L is on the interval (ε2λ1 − ε2σ, ε2λ1 + ε2σ). Hence, according to Theorem 4, almost orthonormalized functions û1 a and û2 a (in the sense of (14)) are approximated by one normalized function u2 ε, which is impossible. Next, we consider, for example, the case when the multiplicities of λ1 and λ3 are equal to 2 and r, respectively. Choosing d = 3 + r − 1 in (18), we have that the eigenvalues µ1 ε, µ 2 ε are in a ε3-neighborhood of ε2λ1, and µ3+r−1 ε is in a ε3-neighborhood of the value ε2λ3 at least. If µ3 ε is not in the ε3-neighborhood of the value ε2λ3, then r orthonormalized functions U3 a , . . . , U3+r−1 a can be approxi- mated by (r− 1) orthonormalized functions w4 ε , . . . , w3+r−1 ε , which is impossible. Thus, the equality j(i) = i for i = 1, . . . , 3+ r−1 is proved. Similarly, we can prove that k(s) = s in (15) for s = 1, . . . , 3 + r − 1. Due to inequality (15) and Theorem 3, this proof can be continued by the induction over d for λd ¿ ε−2. This proves the estimate of Theorem 1 for the eigenvalues of problem (5)–(6). It is useful here that for every d and ε the relations αε2 ≤ λ1 ε ≤ λ2 ε ≤ · · · ≤ λd ε ≤ µd ε hold, which provides a control over the number of eigenvalues for problem (5)–(6) on the concrete segment [αε2, µd ε ] ⊂ [αε2, ε2λd +ε2C(λd)3/2]. It is useful, because the function k(s) in (15) can depend on ε. To be definite, Theorem 4 guaranties only that the number k(s) in (15) is defined for fixed s and ε. Next, we consider some eigenvalue λs of problem (11) with the multiplicity r (which can be 2, 4 or 8). By the definition, we have the relations λs−1 < λs = λs+1 = · · · = λs+r−1 < λs+r. Denote by σs the smallest number of (λs−1 + λs)/2 and (λs + λs+r)/2. It follows from inequality (15) that only eigenvalues λs ε, . . . , λs+r−1 ε of problem (5)–(6) are in the interval (ε2λs − ε2σs, ε 2λs + ε2σs). Thus, according to Theorem 4, there exist constants αj i (possibly, dependent on ε) for i, j = s, s + r − 1 such that ∥∥ûs a − αs su s ε − · · · − αs s+r−1u s+r−1 ε ∥∥ L2(Gε) ≤ C ε (λs)3/2σ−1 s , . . . . . . . . . ,∥∥ûs+r−1 a − αs+r−1 s us ε − · · · − αs+r−1 s+r−1u s+r−1 ε ∥∥ L2(Gε) ≤ C ε (λs)3/2σ−1 s . (19) Moreover, by the conditions of Theorem 4, we have ∥∥αs su s ε + · · ·+ αs s+r−1u s+r−1 ε ∥∥ L2(Gε) = 1, . . . , ∥∥αs+r−1 s us ε + · · ·+ αs+r−1 s+r−1u s+r−1 ε ∥∥ L2(Gε) = 1. (20) 354 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 Homogenization of Spectral Problem on Small-Periodic Networks The functions us ε, . . . , us+r−1 ε are orthonormalized. Hence, we get ( αs s )2 + · · ·+ ( αs s+r−1 )2 = 1, . . . , ( αs+r−1 s )2 + · · ·+ ( αs+r−1 s+r−1 )2 = 1 in accordance with (20). This means that the matrix { αi j } i,j=s,s+r−1 is orthogo- nal. Define the functions ǔs a, . . . , ǔs+r−1 a as the orthogonal transform of the func- tions ûs a, . . . , ûs+r−1 a by the matrix { αi j } i,j=s,s+r−1 . Then, it follows from (19) that ‖ǔs a − us ε‖L2(Gε) ≤ Cε (λs)3/2σ−1 s , . . . , ∥∥ǔs+r−1 a − us+r−1 ε ∥∥ L2(Gε) ≤ Cε (λs)3/2σ−1 s , which concludes the proof of Theorem 1 (since it can be assumed that λs = σs for large s). Here, as the eigenfunction vs of problem (11) from the estimate of Theorem 1, we can take the relevant linear combination of the eigenfunctions 2l−1/2 cos 2πnx1 cos 2πmx2l −1, 2l−1/2 sin 2πnx1 cos 2πmx2l −1, 2l−1/2 cos 2πnx1 sin 2πmx2l −1, 2l−1/2 sin 2πnx1 sin 2πmx2l −1 (21) with the coefficients located in a line of the matrix { αi j } i,j=s,s+r−1 and the rel- evant n and m. We emphasize that for the eigenvalue λs of problem (11) with multiplicity r there exists some arbitrariness in choosing the eigenfunctions vs 0, . . . , vs+r−1 0 , which is determined by some orthogonal matrix. Thus, it is necessary to use the lines of the corresponding orthogonal matrix { αi j } i,j=s,s+r−1 . In conclusion, the asymptotics of the eigenvalues and eigenfunctions for prob- lem (5)–(6) is constructed and Theorem 1 for the sth eigenvalue λs ε and sth eigenfunction us ε of this problem is proved, where λs is the eigenvalue of the homogenized problem (11) with the eigenfunction vs, which is suitably orthonor- malized, i.e., vs is the relevant linear combination of the eigenfunctions from (21). References [1] V.G. Maz’ya and A.S. Slutskii, Homogenization of a Differential Operator on a Fine Periodic Curvilinear Mesh. — Math. Nachr. 133 (1986), 107–133. [2] N.S. Bakhvalov and G.P. Panasenko, Homogenization: Averaging Processes in Periodic Media. Kluwer, Dordrecht–Boston–London, 1989. [3] A. Gavrilov, S. Nicaise, and O. Penkin, Poincares Inequality on the Stratified Sets and Applications. — Progress in Nonlinear Differential Equations and Their Applications 55 (2003), 195–213. [4] S. Nicaise and O. Penkin, Relationship Between the Lower Frequency Spectrum of Plates and Networks of Beams. — Math. Meth. Appl. Sci. 23 (2000), 1389–1399. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 355 A.S. Krylova and G.V. Sandrakov [5] Yu.V. Pokornyi, O.M. Penkin, and V.L. Pryadiev, Differential Equations on Geo- metric Graphs. Fizmatlit, Moscow, 2004. (Russian) [6] O.A. Oleinik, A.S. Shamaev, and G.A. Yosifian, Mathematical Problems in Elas- ticity and Homogenization. North-Holland, Amsterdam,1992. [7] T.A. Melnik, Asymptotic Expansions of Eigenvalues and Eigenfunctions for Ellip- tic Boundary-Value Problems with Rapidly Oscillating Coefficients in a Perforated Cube. — J. Math. Sci. 75 (1995), 1646–1671. [8] V.A. Marchenko and E.Ya. Khruslov, Homogenization of Partial Differential Equa- tions. Progr. Math. Phys., 46, Springer, Berlin, 2005. [9] G. Allaire and C. Conca, Bloch Wave Homogenization and Spectral Asymptotic Analysis. — J. Math. Pures et Appli. 77 (1998), 153–208. [10] Yu.D. Golovaty and S.S. Man’ko, Schrödinger Operator with δ′-potential. — Dopov. Nats. Akad. Nauk Ukr, Mat. Pryr. Tekh. Nauky 5 (2009), 16–21. (Ukrainian) [11] A.S. Krylova and G.V. Sandrakov, Investigation of Eigenvalues and Eigenfunctions for Arbitrary Fragments of Networks. — Journal of Numerical and Applied Mathe- matics 101 (2010), No. 2, 81–96. (Ukrainian) [12] G.V. Sandrakov, Averaging Principles for Equations with Rapidly Oscillating Co- efficients. — Math. USSR-Sb. 68 (1991) No. 2, 503–553. [13] L.A. Lyusternik, On Difference Approximations of the Laplace Operator. — Usp. Mat. Nauk 9 (1954), No. 2(60), 3–66. (Russian) [14] M. Reed and B. Simon, Methods of Modern Mathematical Physics: Vol. 4. Analysis of Operators. Academic Press, New York, 1978. [15] A.A. Samarskii, R.D. Lazarov, and V.L. Makarov, Difference Schemes for Diffe- rential Equations having Generalized Solutions. Vysshaya Shkola, Moskow, 1987. (Russian) [16] M.I. Vishik and L.A. Lyusternik, Regular Degeneration and Boundary Layer for Linear Differential Equations with Small Parameter. — Usp. Mat. Nauk 12 (1957), No. 5(77), 3–122. (Russian) 356 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4

Homogenization of Spectral Problem on Small-Periodic Networks

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