Asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients
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 homogeni...
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| Дата: | 1999 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
1999
|
| Назва видання: | Нелинейные граничные задачи |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/169278 |
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| Цитувати: | Asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients / A.Yu. Teplinsky // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 98-102. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1692782020-06-10T01:26:28Z Asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients Teplinsky, A.Yu. 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]). 1999 Article Asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients / A.Yu. Teplinsky // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 98-102. — Бібліогр.: 7 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169278 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
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]). |
| format |
Article |
| author |
Teplinsky, A.Yu. |
| spellingShingle |
Teplinsky, A.Yu. Asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients Нелинейные граничные задачи |
| author_facet |
Teplinsky, A.Yu. |
| author_sort |
Teplinsky, A.Yu. |
| title |
Asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients |
| title_short |
Asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients |
| title_full |
Asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients |
| title_fullStr |
Asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients |
| title_full_unstemmed |
Asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients |
| title_sort |
asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/169278 |
| citation_txt |
Asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients / A.Yu. Teplinsky // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 98-102. — Бібліогр.: 7 назв. — англ. |
| series |
Нелинейные граничные задачи |
| work_keys_str_mv |
AT teplinskyayu asymptoticexpansionsforeigenvaluesandeigenfunctionsofellipticboundaryvalueproblemswithrapidlyoscillatingcoefficients |
| first_indexed |
2025-07-15T04:02:09Z |
| last_indexed |
2025-07-15T04:02:09Z |
| _version_ |
1837684102494420992 |
| fulltext |
ASYMPTOTIC EXPANSIONS FOR EIGENVALUES AND
EIGENFUNCTIONS OF ELLIPTIC BOUNDARY-VALUE
PROBLEMS WITH RAPIDLY OSCILLATING COEFFICIENTS
c© A.Yu.Teplinsky
1
The 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 such
expansions were constructed at first for the Sturm-Liouville problem [5]. But the method
proposed 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 with
rapidly oscillating coefficients was a construction of a boundary layer near the boundary
the 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 author
overcame the both obstacles by imposing certain symmetry relations on the domain
and on the coefficients of the equation. It is important to notice that all the troubles
are arising namely in the process of formal construction of expansions and not in there
justification. 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 with
non-uniformly oscillating coefficients (whose homogenization is made in [7]). I assume
that the boundary layers (the functions N2
p,α in sequel) are constructed.
In what follows we denote Di = ∂/∂ξi; 〈f〉 =
∫
(0;1)d f(ξ)dξ; Dα = ∂|α|/∂x
|α1|
1 , . . . ,
x
|αd|
d , |α| = |α1|+ · · ·+ |αd| for α = (α1, . . . , αd), and by repeated indices i, j we assume
the 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,
x
ε
) ∂
∂xj
uε(x)
)
+ A
(
x,
x
ε
)
uε(x)
= λερ
(
x,
x
ε
)
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 for
all x, ξ, η ∈ Rd
ρ(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 of
the problem are infinitely smooth (in the case of 1-periodic boundary condition with
respect 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 such
functions Np,α(x, ξ), 0 ≤ |α| ≤ p, on Ω× ε−1Ω that the formal substitutions
λε =
+∞∑
p=0
εpλp, vε(x) =
+∞∑
p=0
εpvp(x),
uε(x) =
+∞∑
p=0
εp
∑
|α|≤p
Np,α
(
x,
x
ε
)
Dαvε(x), x ∈ Ω,
transform the input problem (1), (2) into the sequence of problems
p∑
n=0
Lp−nvn(x) = 0, x ∈ Ω, (3)
p∑
n=0
Bp−nvn = 0; p ≥ 0, (4)
where differential operators Lp and boundary condition operators Bp, p ≥ 0, do not
depend on ε. The structure of the functions Np,α is the following: Np,α = N1
p,α + N2
p,α,
where N1
p,α are the 1-periodic components, N2
p,α are the boundary layer functions.
The functions N1
p,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 the
formulae
N1
0,0 = 1;
−Di(Aij(x, ξ)DjN
1
p,α)(x, ξ) = ĥp,α(x)− hp,α(x, ξ), ξ ∈ Rd,
〈
N1
p,α(x, ·)〉 = 0,
hp,α = −(Di(AijN
1
p−1,α−δj
) + AijDiN
1
p−1,α−δj
+
+ AijN
1
p−2,α−δi−δj
) + AN1
p−2,α −
p−2∑
t=|α|
λp−2−tBN1
t,α ,
ĥp,α(x) = 〈hp,α(x, ·)〉 , p > 0, |α| ≤ p, x ∈ Ω;
Lp =
∑
|α|≤p+2
ĥp+2,α(x)Dα, p ≥ 0.
Note that Lp = L̇p−λp 〈B〉 where operator L̇p 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 N2
p,α(x, ξ) tend to zero exponentially when moving from the boundary,
they satisfy the equations
−Di(AijDjN
2
p,α) = −(Di(AijN
2
p−1,α−δj
) + AijDiN
2
p−1,α−δj
+ AijN
2
p−2,α−δi−δj
) + AN2
p−2,α −
p−2∑
t=|α|
λp−2−tBN2
t,α.
These functions determine the operators Bp and are constructed only for a few domains
and boundary conditions. Always N2
0,0 = 0.
The homogenized problem is the problem for p = 0 from the system (3), (4). Namely
the system (3), (4) was not solved in the case of a multiple spectrum of the homogenized
problem. The next result shows some symmetry in this system and opens a way to its
solving.
Lemma. Let the functions v0, . . . , vk and ṽ0, . . . , ṽk satisfy equations (3) for p < k and
boundary conditions (4) for p ≤ k with some fixed λ0, . . . , λk−1 ∈ R and k ≥ 1. Then
∫
Ω
(
k∑
n=1
Lk−nvn + L̇kv0
)
ṽ0dx =
∫
Ω
(
k∑
n=1
Lk−nṽn + L̇kṽ0
)
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} is
the set of all the eigenfunctions corresponding to the eigenvalue λ0 of the homogenized
problem. Always
Hλ0,...,λk
⊂ Hλ0,...,λk−1 ⊂ · · · ⊂ Hλ0 ⊂ C∞(Ω).
In accordance with Lemma (for k + 1 instead of k) the expression
(Pλ0,...,λk
v0, ṽ0) = λ0
∫
Ω
(
k+1∑
n=1
Lk+1−nvn + L̇k+1v0
)
ṽ0dx
determines a self-adjoint operator Pλ0,...,λk
in Hλ0,...,λk
, here the scalar product
(u, v) =
∫
Ω
(L̇0u)vdx (6)
is equivalent to the standard one in Sobolev space W 1
2 (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 get
in the form: vm ∈ v∗m + Hλ0,...,λk−m
, 0 ≤ m ≤ k, where v∗0 = 0 and v∗m is determining
algorithmically 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 decomposition
Hλ0,...,λk
=
⊕
λk+1∈Sλ0,...,λk
Hλ0,...,λk+1 , where Sλ0,...,λk
is a spectrum, Hλ0,...,λk+1 is a
subspace 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 correspondence
with the sequence λ0
(1) ≤ λ0
(2) ≤ . . . of eigenvalues of the homogenized problem (they
are taken with an account of a multiplicity) and get the corresponding decomposition
{u ∈ W 1
2 (Ω) : B0u = 0} =
⊕+∞
n=1 Hsn , where s1 = 1, Λ(s) = Λ(sn) and Hs = Hsn for
sn ≤ s < sn+1, sn+1 = sn + dim Hsn
, n ≥ 1, and Hs =
⋂+∞
k=0 Hλ0
(s),...,λk
(s) , s ≥ 1.
Let the eigenvalues of the input problem (1), (2) be enumerated too (with an account
of a multiplicity): λ
(1)
ε ≤ λ
(2)
ε ≤ . . . The justification procedure (see it in [6] for different
problems) gives the next result.
Theorem. For every s ≥ 1 we have
λ(s)
ε ≈
+∞∑
p=0
εpλ(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=0
εp
p∑
n=0
∑
|α|≤p−n
Np−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 is
corresponding to the introduced scalar product.
References
1. A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,
Northen Holland, Amsterdam, 1978.
2. L. Tartar, Homogénéisation: Cours Peccot, Collège 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 Strongly
Non-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 and
eigenfunctions of Sturm-Liouville problem with rapidly oscillating coefficients, Vestnik Moscovskogo
Universiteta. Ser. math.-mech (1985), no. 6, 22–35. (in Russian)
6. T. A. Melnik, Asymptotic expansions of eigenvalues and eigenfunctions for elliptic boundary-value
problems 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 eigenfunctions
of the boundary value problems in perforated domains for elliptic equations with non-uniformly
oscillating coefficients, in Current Topics in Partial Differential Equations, Kinokuniya Co., Tokyo,
1986, pp. 187–216.
Ukraine, 281900
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Kosmonavtiv str., 2, 48.
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