In the paper I solve the problem of a multiple spectrum without any artificial conditions imposed and obtain whole asymptotic expansions for eigenvalues and eigen-functions of the general smooth boundary-value problem for an elliptic operator with non-uniformly oscillating coe±cients (whose homogenization is made in [7])

Teplinsky, A.Yu.

English

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

ASYMPTOTIC EXPANSIONS FOR EIGENVALUES ANDEIGENFUNCTIONS OF ELLIPTIC BOUNDARY-VALUEPROBLEMS WITH RAPIDLY OSCILLATING COEFFICIENTSc© A.Yu.Teplinsky1The problem of the construction of asymptotic expansions for solutions of such spec-tral problems stands in the homogenization theory [1–4] for about 10 years—since suchexpansions were constructed at first for the Sturm-Liouville problem [5]. But the methodproposed worked only for the case of a simple spectrum of the corresponding homoge-nized problem. At the time when the main trouble for boundary-value problems withrapidly oscillating coefficients was a construction of a boundary layer near the boundarythe main trouble of the corresponding spectral problems was multiplicity of the spec-trum. The only progress in this direction was an approach of [6] in which the authorovercame the both obstacles by imposing certain symmetry relations on the domainand on the coefficients of the equation. It is important to notice that all the troublesare arising namely in the process of formal construction of expansions and not in therejustification. The methods of the justification are worked out well (see [4], [6]).In the paper I solve the problem of a multiple spectrum without any artificial con-ditions imposed and obtain whole asymptotic expansions for eigenvalues and eigen-functions of the general smooth boundary-value problem for an elliptic operator withnon-uniformly oscillating coefficients (whose homogenization is made in [7]). I assumethat the boundary layers (the functions N2p,α in sequel) are constructed.In what follows we denote Di = ∂/∂ξi; 〈f〉 =∫(0;1)df(ξ)dξ; Dα = ∂|α|/∂x|α1|1 , . . . ,x|αd|d , |α| = |α1|+ · · ·+ |αd| for α = (α1, . . . , αd), and by repeated indices i, j we assumethe summation from 1 to d; d ≥ 1 is the dimension of our space.In a bounded domain Ω ⊂ Rd, d ≥ 1, consider the problem− ∂∂xi(Aij(x,xε) ∂∂xjuε(x))+A(x,xε)uε(x)= λερ(x,xε)uε(x), x ∈ Ω, (1)Bεuε = 0, (2)where A, ρ, Aij , 1 ≤ i ≤ d, 1 ≤ j ≤ d, are 1-periodic with respect to ξ = x/ε and forall x, ξ, η ∈ Rdρ(x, ξ) > 0, A(x, ξ) > 0, Aij(x, ξ)ηiηj ≥ κ|η|2,Aij(x, ξ) = Aji(x, ξ) for all 1 ≤ i ≤ d, 1 ≤ j ≤ d,where κ > 0 does not depend on x, ξ, η; Bε is a boundary condition operator (Dirichlet,Neumann and 1-periodic boundary conditions mixed); 0 < ε ¿ 1; all the elements ofthe problem are infinitely smooth (in the case of 1-periodic boundary condition withrespect to xl+1, . . . , xd, 0 ≤ l < d, we suppose that Ω ⊂ Rl × (0; 1)d−l; {x ∈ Ω : xi =0}+ei = {x ∈ Ω : xi = 1}, l < i ≤ d; not Ω but Ω+{0}l×Zd−l has a smooth boundary;other boundary conditions are adapted).Let us suppose that for each sequence Λ = (λ0, λ1, . . . ) we have constructed suchfunctions Np,α(x, ξ), 0 ≤ |α| ≤ p, on Ω× ε−1Ω that the formal substitutionsλε =+∞∑p=0εpλp, vε(x) =+∞∑p=0εpvp(x),uε(x) =+∞∑p=0εp∑|α|≤pNp,α(x,xε)Dαvε(x), x ∈ Ω,transform the input problem (1), (2) into the sequence of problemsp∑n=0Lp−nvn(x) = 0, x ∈ Ω, (3)p∑n=0Bp−nvn = 0; p ≥ 0, (4)where differential operators Lp and boundary condition operators Bp, p ≥ 0, do notdepend on ε. The structure of the functions Np,α is the following: Np,α = N1p,α +N2p,α,where N1p,α are the 1-periodic components, N2p,α are the boundary layer functions.The functions N1p,α(x, ξ) are defined on Ω×Rd and are 1-periodic with respect to ξ.They are determined by the equation (1) only and determine the operators Lp by theformulaeN10,0 = 1;−Di(Aij(x, ξ)DjN1p,α)(x, ξ) = hˆp,α(x)− hp,α(x, ξ), ξ ∈ Rd,〈N1p,α(x, ·)〉= 0,hp,α = −(Di(AijN1p−1,α−δj ) +AijDiN1p−1,α−δj++AijN1p−2,α−δi−δj ) +AN1p−2,α −p−2∑t=|α|λp−2−tBN1t,α ,hˆp,α(x) = 〈hp,α(x, ·)〉 , p > 0, |α| ≤ p, x ∈ Ω;Lp =∑|α|≤p+2hˆp+2,α(x)Dα, p ≥ 0.Note that Lp = L˙p−λp 〈B〉 where operator L˙p does not depend on λk, k ≥ p; p ≥ 0.Proposition 1. All the operators Lp, p ≥ 0, are self-adjoint.This fact was not proved for p ≥ 1; the self-adjointness of L0 is proved in [7].Remark. Proposition 1 is also true for boundary-value (not spectrum) problems irre-spective of Ω and Bε. So the volume of calculations when computing the operators Lp,p ≥ 1, decreases twice.The functions N2p,α(x, ξ) tend to zero exponentially when moving from the boundary,they satisfy the equations−Di(AijDjN2p,α) = −(Di(AijN2p−1,α−δj ) +AijDiN2p−1,α−δj+AijN2p−2,α−δi−δj ) +AN2p−2,α −p−2∑t=|α|λp−2−tBN2t,α.These functions determine the operators Bp and are constructed only for a few domainsand boundary conditions. Always N20,0 = 0.The homogenized problem is the problem for p = 0 from the system (3), (4). Namelythe system (3), (4) was not solved in the case of a multiple spectrum of the homogenizedproblem. The next result shows some symmetry in this system and opens a way to itssolving.Lemma. Let the functions v0, . . . , vk and v˜0, . . . , v˜k satisfy equations (3) for p < k andboundary conditions (4) for p ≤ k with some fixed λ0, . . . , λk−1 ∈ R and k ≥ 1. Then∫Ω(k∑n=1Lk−nvn + L˙kv0)v˜0dx =∫Ω(k∑n=1Lk−nv˜n + L˙kv˜0)v0dx. (5)Denotation. Let Hλ0,...,λk be the set of such functions v0 that the part of the system(3), (4) for p ≤ k has a solution with these λ0, . . . , λk ∈ R, k ≥ 0.All these sets are finite-dimensional subspaces of C∞(Ω). In particular, Hλ0\{0} isthe set of all the eigenfunctions corresponding to the eigenvalue λ0 of the homogenizedproblem. AlwaysHλ0,...,λk ⊂ Hλ0,...,λk−1 ⊂ · · · ⊂ Hλ0 ⊂ C∞(Ω).In accordance with Lemma (for k + 1 instead of k) the expression(Pλ0,...,λkv0, v˜0) = λ0∫Ω(k+1∑n=1Lk+1−nvn + L˙k+1v0)v˜0dxdetermines a self-adjoint operator Pλ0,...,λk in Hλ0,...,λk , here the scalar product(u, v) =∫Ω(L˙0u)vdx (6)is equivalent to the standard one in Sobolev space W 12 (Ω).Proposition 2. Let Hλ0,...,λk 6= {0} for some λ0, . . . , λk ∈ R,k ≥ 0. Then the general solution of the part of the system (3), (4) for p ≤ k we getin the form: vm ∈ v∗m +Hλ0,...,λk−m , 0 ≤ m ≤ k, where v∗0 = 0 and v∗m is determiningalgorithmically by already chosen v0, . . . , vm−1 for 1 ≤ m ≤ k.Proceeding to the next problem from (3), (4) (for p = k + 1) we get a decompositionHλ0,...,λk =⊕λk+1∈Sλ0,...,λk Hλ0,...,λk+1 , where Sλ0,...,λk is a spectrum, Hλ0,...,λk+1 is asubspace of the eigenfunctions of Pλ0,...,λk corresponding to the eigenvalue λk+1.As a result we obtain a sequence of sequences Λ(1) ≤ Λ(2) ≤ . . . in correspondencewith the sequence λ0(1) ≤ λ0(2) ≤ . . . of eigenvalues of the homogenized problem (theyare taken with an account of a multiplicity) and get the corresponding decomposition{u ∈ W 12 (Ω) : B0u = 0} =⊕+∞n=1Hsn , where s1 = 1, Λ(s) = Λ(sn) and Hs = Hsn forsn ≤ s < sn+1, sn+1 = sn + dimHsn , n ≥ 1, and Hs =⋂+∞k=0Hλ0(s),...,λk(s) , s ≥ 1.Let the eigenvalues of the input problem (1), (2) be enumerated too (with an accountof a multiplicity): λ(1)ε ≤ λ(2)ε ≤ . . . The justification procedure (see it in [6] for differentproblems) gives the next result.Theorem. For every s ≥ 1 we haveλ(s)ε ≈+∞∑p=0εpλ(s)p , ε→ 0.For each r ≥ 0 and for the functions v0, . . . , vr satisfying the part of the system (3),(4) for p ≤ r with λ0(s), . . . , λ(s)r we have‖uε −r∑p=0εpp∑n=0∑|α|≤p−nNp−n,α(·, ·ε)Dαvn‖ = O(εr), ε→ 0,here uε is a linear combination of the eigenfunctions of the problem (1), (2) correspond-ing to the eigenvalues λ(t)ε for sn ≤ t < sn+1 (where sn ≤ s < sn+1); the norm iscorresponding to the introduced scalar product.References1. A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,Northen Holland, Amsterdam, 1978.2. L. Tartar, Homoge´ne´isation: Cours Peccot, Colle`ge de France, Paris, 1977.3. N. S. Bakhvalov, and G. P. Panasenko, Homogenization of Processes in Periodic Media, Nauka,Moscow, 1981. (in Russian)4. O. A. Oleinik, G. A. Yosifian, and A. S. Shamaev, Mathematical Problems of Theory of StronglyNon-Homogenious Ellastic Media, Moscow Univ. Publ., Moscow, 1990. (in Russian)5. G. A. Yosifian, O. A. Oleinik, and A. S. Shamaev, Asymptotical expansions for eigenvalues andeigenfunctions of Sturm-Liouville problem with rapidly oscillating coefficients, Vestnik MoscovskogoUniversiteta. Ser. math.-mech (1985), no. 6, 22–35. (in Russian)6. T. A. Melnik, Asymptotic expansions of eigenvalues and eigenfunctions for elliptic boundary-valueproblems with rapidly oscillating coefficients in a perforated cube, Trudy Seminara imeni G. P. Petro-vskogo 17 (1994); English translation in Jour. of Math. Sciences 75 (1995), no. 3.7. O. A. Oleinik, A. S. Shamaev, and G. A. Yosifian, Homogenization of eigenvalues and eigenfunctionsof the boundary value problems in perforated domains for elliptic equations with non-uniformlyoscillating coefficients, in Current Topics in Partial Differential Equations, Kinokuniya Co., Tokyo,1986, pp. 187–216.Ukraine, 281900Kamyanets-Podilsky,Kosmonavtiv str., 2, 48.

Asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients

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Asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)