Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction
A spectral boundary-value problem is considered in a plane thick two-level junction $\Omega_{\varepsilon}$, which is the union of a domain $\Omega_{0}$ and a large number $2N$ of thin rods with thickness of order $\varepsilon = \mathcal{O} (N^{-1})$. The thin rods are divided into two levels depen...
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2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509539911073792 |
|---|---|
| author | Mel'nik, T. A. Мельник, Т. А. |
| author_facet | Mel'nik, T. A. Мельник, Т. А. |
| author_sort | Mel'nik, T. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:54:47Z |
| description | A spectral boundary-value problem is considered in a plane thick two-level junction $\Omega_{\varepsilon}$, which is the union of a
domain $\Omega_{0}$ and a large number $2N$ of thin rods with thickness of order $\varepsilon = \mathcal{O} (N^{-1})$. The thin rods are divided into
two levels depending on their length. In addition, the thin rods from each level are $\varepsilon$-periodically alternated.
The Fourier conditions are given on the lateral boundaries of the thin rods. The asymptotic behavior of the eigenvalues
and eigenfunctions is investigated as $\varepsilon \rightarrow 0$, i.e., when the number of the thin rods infinitely increases and their thickness tends to zero.
The Hausdorff convergence of the spectrum is proved as $\varepsilon \rightarrow 0$, the leading terms of asymptotics are constructed and the
corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions. |
| first_indexed | 2026-03-24T02:42:43Z |
| format | Article |
| fulltext |
UDC 517.956 + 517.43
T. A. Mel’nyk (Kyiv Nat. Taras Shevchenko Univ.)
ASYMPTOTIC BEHAVIOR OF EIGENVALUES
AND EIGENFUNCTIONS OF THE FOURIER PROBLEM
IN A THICK MULTILEVEL JUNCTION
АСИМПТОТИЧНА ПОВЕДIНКА ВЛАСНИХ ЗНАЧЕНЬ
ТА ВЛАСНИХ ФУНКЦIЙ ЗАДАЧI ФУР’Є
В ГУСТОМУ БАГАТОРIВНЕВОМУ З’ЄДНАННI
A spectral boundary-value problem is considered in a plane thick two-level junction Ωε, which is the
union of a domain Ω0 and a large number 2N 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 Fourier conditions are given on the lateral boundaries of the thin rods. The
asymptotic behavior of the eigenvalues and eigenfunctions is investigated as ε→ 0, i.e., when the number
of the thin rods infinitely increases and their thickness tends to zero. The Hausdorff convergence of the
spectrum is proved as ε → 0, the leading terms of asymptotics are constructed and the corresponding
asymptotic estimates are justified for the eigenvalues and eigenfunctions.
Розглядається спектральна крайова задача у плоскому дворiвневому з’єднаннi Ωε, яке є об’єднанням
областi Ω0 та великого числа 2N тонких стержнiв товщиною порядку ε = O(N−1). Тонкi стержнi
роздiлено на два рiвнi в залежностi вiд їх довжини. Крiм того, тонкi стержнi з кожного рiвня
ε-перiодично чергуються. На вертикальних сторонах тонких стержнiв задано крайовi умови Фур’є.
Вивчено асимптотичну поведiнку власних значень та власних функцiй при ε → 0, тобто коли
число тонких стержнiв необмежено зростає, а їх товщина прямує до нуля. Доведено хаусдорфову
збiжнiсть спектра при ε → 0, побудовано першi члени асимптотики та обґрунтовано вiдповiднi
асимптотичнi оцiнки для власних значень та власних функцiй.
1. Introduction and statement of the problem. As has been stated in [1], multiscale
modeling and computation is a rapidly evolving area of research that will have a
fundamental impact on computational science and applied mathematics. This is connected
with the prospect of development of more efficient methods that should be symbiosis
of a new class of numerical and analytical modeling techniques. There is a long
history in mathematics for the study of multiscale problems. One class of multiscale
problems is boundary-value problems in perturbed domains. There are many kinds of the
domain perturbations and we need different asymptotic methods to study boundary-value
problems in perturbed domains (see, e.g., [2 – 11] and references there).
Perturbed spectral boundary-value problems deserve special attention, since the
asymptotic behaviour of the spectrum is highly sensitive to the perturbation and it
is unexpected (see, e.g., [12]). If the perturbation is smooth and in some sense small,
then with the help of a family of diffeomorphisms we can reduce a perturbed spectral
problem to investigation of behaviour of the spectrum of operators defined in some fixed
domain. But there are many problems with singular perturbed domains and it is not
possible to use above-mentioned approach. The extensive review of such problems was
presented in [13].
In this paper a new kind of perturbed domains, namely, thick multilevel junctions
is considered. Boundary-value problems in thick one-level junctions (thick junctions)
are very intensively investigated in the last time. As was shown in the papers [10, 14],
c© T. A. MEL’NYK, 2006
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 2 195
196 T. A. MEL’NYK
Fig. 1. The thick two-level junction Ωε.
such problems lose the coercitivity and compactness as ε → 0. This creates special
difficulties in the asymptotic investigation. In [13, 15 – 22], classification of thick one-
level junctions was given and basic results were obtained both for boundary-value and
spectral problems in thick junctions of different types. It was shown that qualitative
properties of solutions 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 [13, 15 – 22]. Here we mention only the pioneer papers
[7, 23, 24], where the asymptotic behaviour of Green’s function of the Neumann problem
for the Helmholtz equation in unbounded thick junctions was studied.
1.1. Statement of the problem. Let a, d1, d2, b1, b2, h1, h2 be positive real
numbers and let d1 ≥ d2, 0 < b1 < b2 < 1, 0 < b1 − h1/2, b1 + h1/2 < b2 − h2/2,
b2 + h2/2 < 1. The last restrictions mean that the intervals Ih1(b1) :=
(
b1 − h1/2, b1 +
+ h1/2
)
and Ih2(b2) := (b2 − h2/2, b2 + h2/2) belong to (0, 1) and don’t intersect. Let
us divide the segment I0 := [0, a] on N equal segments [εj, ε(j+1)], j = 0, . . . , N −1.
Here N is a large integer, therefore, the value ε = a/N is a small discrete parameter.
A model plane thick two-level junction Ωε (Fig. 1) consists of the junction’s body
Ω0 =
{
x ∈ R2 : 0 < x1 < a, 0 < x2 < γ(x1)
}
,
where γ ∈ C1([0, a]), γ(0) = γ(a), min[0,a] γ > 0, and a large number of the thin rods
G
(1)
j (ε) =
{
x ∈ R2 : |x1 − ε(j + b1)| <
εh1
2
, x2 ∈ (−d1, 0]
}
,
G
(2)
j (ε) =
{
x ∈ R2 : |x1 − ε(j + b2)| <
εh2
2
, x2 ∈ (−d2, 0]
}
,
j = 0, 1, . . . , N − 1,
i.e., Ωε = Ω0 ∪ G(1)
ε ∪ G(2)
ε , where G(1)
ε = ∪N−1
j=0 G
(1)
j (ε), G(2)
ε = ∪N−1
j=0 G
(2)
j (ε).
We see that the number of the thin rods is equal to 2N and they are divided into two
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ASYMPTOTIC BEHAVIOR OF EIGENVALUES AND EIGENFUNCTIONS... 197
levels G(1)
ε and G
(2)
ε depending on their length. The parameter ε characterizes the
distance between the neighboring thin rods and their thickness. The thickness of the
rods from the first level is equal to εh1 and it is equal to εh2 for the rods from the second
one. These thin rods from each level are ε-periodically alternated along the segment
I0 = {x : x1 ∈ [0, a], x2 = 0} (the joint zone of this thick two-level junction).
Denote by Υ(i,±)
j (ε) the lateral sides of the thin rod G(i)
j (ε), the signs “+" or “−"
indicate the right or left side respectively; the base of G(i)
j (ε) will be denoted by Θ(i)
j (ε).
Also we introduce the following notations:
Υ(i,±)
ε := ∪N−1
j=0 Υ(i,±)
j (ε), Θ(i)
ε := ∪N−1
j=0 Θ(i)
j (ε), i = 1, 2.
In Ωε we consider the following spectral problem:
−∆x u(ε, x) = λ(ε)u(ε, x), x ∈ Ωε,
∂νu(ε, x) = −εk1u(ε, x), x ∈ Υ(1,±)
ε ,
∂νu(ε, x) = −εk2u(ε, x), x ∈ Υ(2,±)
ε ,
∂p
x1
u(ε, 0, x2) = ∂p
x1
u(ε, a, x2), x2 ∈ [0, γ(0)], p = 0, 1,
∂νu(ε, x) = 0, x ∈ Γε.
(1)
Here ∂ν =
∂
∂ν
is the outward normal derivative; ∂x1 =
∂
∂x1
; the constants k1 and k2
are positive; Γε = Θ(1)
ε ∪Θ(2)
ε ∪(I0∩∂Ωε)∪Γγ , where Γγ = {x : x2 = γ(x1), x1 ∈ I0}.
It is well known that for each fixed ε > 0 there is a sequence of eigenvalues of
problem (1)
0 < λ1(ε) ≤ λ2(ε) ≤ . . . ≤ λn(ε) ≤ . . .→ +∞ as n→∞, (2)
and a sequence of the corresponding eigenfunctions {un(ε, ·) : n ∈ N} can be
orthonormalized by the following way:
(un, um)Ωε = δn,m, {n,m} ∈ N, (3)
where (·, ·)Υ is the scalar product in L2(Υ), and δn,m is the Kronecker delta.
Our aim is to describe the asymptotic behavior of eigenvalues {λn(ε) : n ∈ N} and
eigenfunctions {un(ε, ·) : n ∈ N} as ε → 0 (N → +∞), to find other limiting points
of the spectrum of problem (1), and to describe corresponding eigenfunctions.
1.2. Features of the investigation. As was showed in [13, 15 – 22], the corresponding
limit problem for a boundary-value problem in a thick one-level junction is derived from
the limit problems for each domain forming the thick junction with the help of the
solutions to junction-layer problems around the joint zone. However, the junction-layer
solutions behave as powers (or logarithm) at infinity and do not decrease exponentially.
Therefore, they influence directly the leading terms of the asymptotics. The model
problems describing the junction-layer phenomenon are posed in unbounded domains
having outlets to infinity. The principal terms of the inner expansion is nontrivial
solutions to the corresponding homogeneous junction-layer problem. In the case of a
thick one-level junction such a solution is identically defined. But for a thick p-level
junction, dimension of the kernel of the corresponding homogeneous junction-layer
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 2
198 T. A. MEL’NYK
problem is equal to p + 1 and the problem is how to define the principal terms of
the inner expansion. This fact very complicates the construction of the asymptotic
approximation for the solutions. We should modify the view of the inner expansion
and consider outer expansions in each thin domains from each level. Matching these
asymptotic expansions, we deduce the nonstandard limiting spectral boundary-value
problem (41) in an anisotropic Sobolev vector-space.
In this paper we consider the Fourier conditions ∂νuε = −εkiuε on the lateral
boundaries Υ(i,±)
ε , i = 1, 2, of the thin rods. At first sight it seems that there is no
difference between these Fourier condition and the homogeneous Neumann conditions
since the terms kiuε, i = 1, 2, are multiplied by the factor ε. But this is quite false. As
was mentioned above the boundary conditions on the boundaries of the attached thin
domains of thick junctions have essentially influence on the asymptotic behaviour of
the solutions. For problem (1) this leads to the appearance of special coefficients in the
differential operator of the limit problem.
The Fourier conditions or the nonhomogeneous Neumann conditions make the
process of homogenization and approximation more complicated. For this the method
of the integral identities was proposed in [20, 22].
For the first time a boundary-value problem in a plane thick multilevel junction
was considered in [25], where some results for problem (1) were announced. Then
the development of rigorous asymptotic methods for boundary-value problems in thick
multilevel junctions of different types have been continued in [26 – 29].
2. Auxiliary inequalities. In the subspace Hε := {u ∈ H1(Ωε) : u(0, x2) =
u(a, x2), x2 ∈ [0, γ(0)]} we introduce a new norm ‖ · ‖ε,k1,k2 that is generated by the
following scalar product:
〈u, v〉ε,k1,k2 =
∫
Ωε
∇u · ∇vdx+ εk1
∫
Υ
(1,±)
ε
uvdx2 + εk2
∫
Υ
(2,±)
ε
uvdx2.
Lemma 1. For ε small enough, the usual norm ‖ · ‖H1(Ωε) in the Sobolev space
H1(Ωε) and the norm ‖v‖ε,k1,k2 are uniformly equivalent, i.e., there exist constants
C1 > 0, C2 > 0 and ε0 such that for all values ε ∈ (0, ε0) and any function v ∈ Hε the
following inequalities hold:
C1‖v‖H1(Ωε) ≤ ‖v‖ε,k1,k2 ≤ C2‖v‖H1(Ωε). (4)
Remark 1. Here and further all constants {ci, Ci} in asymptotic inequalities are
independent of the parameter ε.
Proof. It follows from the assumptions made for the numbers b1, b2, h1, h2 that
there exists a such number δ0 that b1 + h1/2 < δ0 < b2 − h2/2. Defined the following
function:
Y (t) =
−t+ b1, t ∈ [0, δ0),
−t+ b2, t ∈ [δ0, 1),
(5)
and then periodically extend it into R. Integrating by parts in the integral
ε
∫
G
(i)
ε
Y (x1/ε)∂x1v dx, i = 1, 2,
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 2
ASYMPTOTIC BEHAVIOR OF EIGENVALUES AND EIGENFUNCTIONS... 199
we get the identity
ε2−1hi
∫
Υ
(i,±)
ε
vdx2 =
∫
G
(i)
ε
vdx− ε
∫
G
(i)
ε
Y
(x1
ε
)
∂x1vdx ∀v ∈ Hε, i = 1, 2. (6)
Since maxR |Y | ≤ 1, it follows from (6) that
‖
√
εv‖
L2
(
Υ
(1,±)
ε ∪Υ
(2,±)
ε
) ≤ C2‖v‖H1
(
G
(1)
ε ∪G
(2)
ε
)
for any v ∈ Hε. Therefore, the right inequality in (4) holds.
Using (6), we obtain
‖v‖2H1(Ωε) =
∫
Ωε
|∇v|2dx+
∫
Ω0
v2dx+
+ε2−1
2∑
i=1
hi
∫
Υ
(i,±)
ε
v2dx2 + ε
∫
G
(1)
ε ∪G
(2)
ε
Y
(x1
ε
)
2v∂x1vdx ≤
≤ c3‖v‖2ε,k1,k2
+
∫
Ω0
v2dx+ ε
∫
G
(1)
ε ∪G
(2)
ε
v2dx,
whence
‖v‖2H1(Ωε) ≤ c4
‖v‖2ε,k1,k2
+
∫
Ω0
v2dx
. (7)
Now let us show that there exists a positive constant c5 such that for ε small enough∫
Ω0
v2dx ≤ c5‖v‖2ε,k1,k2
∀ v ∈ Hε. (8)
We argue by contradiction. Then there exist sequences {εm : m ∈ N} and {vm} ⊂ Hεm
such that limm→0 εm = 0, ∫
Ω0
v2
mdx = 1, (9)
∫
Ωεm
|∇vm|2dx+ εm
2∑
i=1
ki
∫
Υ
(i)
εm
v2
mdx2 <
1
m
. (10)
Since the sequence {vm} is bounded in H1(Ω0), we may assume without loss of
generality that it is a Cauchy sequence in L2(Ω0). From inequality (10) it follows that
{vm} is a Cauchy sequence also in H1(Ω0) : ‖vm − vn‖2H1(Ω0)
≤ ‖vm − vn‖2L2(Ω0)
+
+
1
m
+
1
n
.Hence, {vm} converges to some element v0 ∈ H1(Ω0).Obviously, v0 ≡ const
in H1(Ω0). Due to (9), v0 = |Ω0|−1/2, where |Ω0| denotes the measure of the domain
Ω0. Then, the sequence of the traces of {vm} converges to v0 in L2(∂Ω0) as well and
it is easy to verify that
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 2
200 T. A. MEL’NYK
∫
I0(εm)
v2
m(x1, 0)dx1 =
2∑
i=1
∫
I0
χi(x1/εm)v2
m(x1, 0)dx1 →
→
2∑
i=1
hi
∫
I0
v2
0(x1, 0)dx1 = (h1 + h2) |Ω0|−1a 6= 0 as m→∞, (11)
where I0(ε) := I0 ∩ Ωε and χi(·) is 1-periodic function such that
χi(t) =
1, t ∈ [bi − hi/2, bi + hi/2] ,
0, t ∈ [0, 1] \ [bi − hi/2, bi + hi/2] ,
i = 1, 2. (12)
Obviously, that χi(x1/ε) →
∫ 1
0
χi(t)dt = hi weakly in L2(0, a) as ε→ 0.
On the other hand, from (6) and (10) it follows that ‖vm‖2
H1
(
G
(1)
εm∪G
(2)
εm
) ≤ c6
m
and,
therefore,
∫
I0(εm)
v2
m(x1, 0)dx1 ≤
c7
m
, where the constants c6, c7 are independent of m.
This means that ∫
I0(εm)
v2
m(x1, 0)dx1 → 0 as m→∞. (13)
However (13) is at variance with (11). This contradiction establishes estimate (8).
Thus, by virtue of (8) and (7), we obtain the left inequality in (4).
The lemma is proved.
Definition 1. A number λ(ε) is called an eigenvalue of problem (1) if there exists
a function u(ε, ·) ∈ Hε \ {0} such that for all functions ϕ ∈ Hε the following integral
identity:
〈u, ϕ〉ε,k1,k2 = λ(ε)(u, ϕ)Ωε
(14)
holds. The function u(ε, ·) is called the eigenfunction that corresponds to λ(ε).
Define the operator Aε : Hε 7−→ Hε by the following equality
〈Aεu, v〉ε,k1,k2 = (u, v)Ωε ∀u, v ∈ Hε. (15)
It is easy to verify that Aε is self-adjoint, positive, compact, and the spectral problem (1)
is equivalent to the spectral problem Aεu = λ−1(ε)u in Hε. Due to Lemma 1, there
exist positive constants C1 and ε0 such that for all ε ∈ (0, ε0) ‖Aε‖ ≤ C1. Therefore,
C−1
1 ≤ λn(ε) ∀ n ∈ N. (16)
Denote by Di the rectangle {x : x1 ∈ (0, a), x2 ∈ (−di, 0)} which is filled up by
the thin rods G(i)
j (ε), j = 0, 1, . . . , N − 1, in the limit passage as ε → 0 (N → +∞);
i = 1, 2. Let Ln(φ̃1, . . . , φ̃n) be the n-dimensional subspace of Hε that is spanned on
n linearly independent functions φ̃k, k = 1, . . . , n, such that φ̃k = 0 in Ω0 ∪ G(1)
ε and
φ̃k = φk in G
(2)
ε , where φ1, . . . , φn are orthonormal in L2(D2) eigenfunctions of a
mixed boundary-value problem for the Laplace operator in the rectangle D2 with the
Neumann conditions on the vertical sides and the Dirichlet conditions on the horizontal
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 2
ASYMPTOTIC BEHAVIOR OF EIGENVALUES AND EIGENFUNCTIONS... 201
ones. Denote by {µn} the corresponding eigenvalues of this problem. By virtue of the
minimax principle for eigenvalues and Lemma 1, we have
λn(ε) = min
E∈En
max
v∈E, v 6=0
‖v‖2ε,k1,k2
‖v‖2Ωε
≤
≤ C2
2 min
E∈En
max
v∈E, v 6=0
∫
Ωε
|∇v|2 dx∫
Ωε
v2dx2
+ 1
≤
≤ C3 max
06=v∈Ln
∫
Ωε
|∇v|2 dx∫
Ωε
v2dx2
+ 1
= C3
µn max
06=v∈Ln
∫
D2
v2dx∫
G
(2)
ε
v2dx2
+ 1
.
Here En is a set of all subspaces of Hε with dimension n. By the same arguments as
we have proved (8), we can show that for ε small enough
max
0 6=v∈Ln
∫
D2
v2dx∫
G(2)(ε)
v2dx2
≤ C4.
Thus, for any fixed n ∈ N there exists a constant C1(n) such that for ε small enough,
we have
λn(ε) ≤ C1(n). (17)
From (3), (14), Lemma 1 and (17) it follows that
‖un(ε, ·)‖H1(Ωε) ≤ C2(n). (18)
3. Formal asymptotics of the solution on the thin rods. 3.1. Outer expansions.
Because of (16) – (18), we seek the leading terms for λn(ε) in the form
λ(ε) ≈ µ0 + εµ1 + . . . , (19)
and for the corresponding eigenfunction un(ε, ·), restricted to Ω0, in the form
u(ε, x) ≈ v+
0 (x) +
∞∑
k=1
εkv+
k (x, ε), (20)
and, restricted to each thin rod G(i)
j (ε), in the form
u(ε, x) ≈ vi,−
0 (x) +
∞∑
k=1
εkvi,−
k (x, ξ1 − j),
ξ1 = ε−1x1, j = 0, . . . , N − 1, i = 1, 2.
(21)
Hereafter the index n is omitted. The expansions (20) and (21) are usually called outer
expansions. Substituting series (20) and (19) in the equation of problem (1), in the
boundary conditions on ∂Ω0 and collecting coefficients of the same powers of ε, we get
the following relations for function v+
0 and number µ0 :
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 2
202 T. A. MEL’NYK
−∆x v
+
0 (x) = µ0v
+
0 (x), x ∈ Ω0,
∂p
x1
v+
0 (0, x2) = ∂p
x1
v+
0 (a, x2), x2 ∈ [0, γ(0)], p = 0, 1,
∂νv
+
0 (x) = 0, x ∈ Γγ .
(22)
Now we find limiting relations in the rectangle Di, i = 1, 2. Assuming for the
moment that the functions vi,−
k in (21) are smooth, we write their Taylor series with
respect to the x1 at the point x1 = ε(j+ bi) and pass to the "fast" variable ξ1 = ε−1x1.
Then (20) takes the form
u(ε, x) ≈ vi,−
0 (ε(j + bi), x2) +
+∞∑
k=1
εkV i,j
k (ξ1, x2), x ∈ G(i)
j (ε), (23)
where
V i,j
k (ξ1, x2) = vi,−
k (ε(j + bi), x2, ξ1 − j) +
+
k∑
m=1
(ξ1 − j − bi)m
m!
∂mvi,−
k−m
∂xm
1
(ε(j + bi), x2, ξ1 − j) . (24)
Let us substitute µ0 and (23) into (1) instead of λ(ε) and u(ε, ·) respectively. Since
the Laplace operator takes the form ∆x = ε−2 ∂
2
∂ξ21
+
∂2
∂x2
2
, the collection of coefficients
of the same power of ε gives us one dimensional boundary value problems with respect
to ξ1.
The first problem is the following:
∂2
ξ1ξ1
V i,j
1 (ξ1, x2) = 0, ξ2 ∈ Ihi
(bi), ∂ξ1V
i,j
1 (bi ± hi/2, x2) = 0, (25)
where ∂ξ1 =
∂
∂ξ1
, ∂2
ξ1ξ1
=
∂2
∂ξ21
. From (25) it follows that function V i,j
1 doesn’t depend
on ξ1. We restrict ourselves to the leading term of the asymptotics and set V i,j
1 ≡ 0.
Then, due to (24), we have
vi,−
1
(
ε(j + bi), x2, ξ1 − j
)
= −∂x1v
i,−
0
(
ε(j + bi), x2
)(
ξ1 − j − bi
)
.
The problem for the function V i,j
2 is as follows:
−∂2
ξ1ξ1
V i,j
2 (ξ1, x2) =
= ∂2
x2x2
vi,−
0 (ε(j + bi), x2) + µ0v
i,−
0 (ε(j + bi), x2), ξ1 ∈ Ihi
(bi),
∂ξ1V
i,j
2 (bi ± hi/2, x2) = ±kiv
i,−
0 (ε(j + bi), x2) .
(26)
The solvability condition for problem (26) is given by the differential equation
−hi∂
2
x2x2
vi,−
0 (ε(j + bi), x2) + 2kiv
i,−
0 (ε(j + bi), x2) = hiµ0v
i,−
0 (ε(j + bi), x2) .
(27)
Due to the Neumann conditions for the eigenfunction u(ε, ·) on the bases Θ(i)(ε), we
must require from vi,−
0 to satisfy the following condition:
∂x2v
i,−
0 (ε(j + bi),−di) = 0. (28)
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ASYMPTOTIC BEHAVIOR OF EIGENVALUES AND EIGENFUNCTIONS... 203
To find conditions in points of the joint zone I0, we use the method of matched
asymptotic expansions for the outer expansions (20), (21) and an inner expansion that is
constructed in the following subsection.
3.2. Inner expansion. In a neighborhood of the joint zone I0 we introduce the
"rapid" coordinates ξ = (ξ1, ξ2), where ξ1 = ε−1x1 and ξ2 = ε−1x2. The Laplace
operator takes the following form ε−2∆ξ in the coordinates ξ. We seek the leading
terms of the inner expansion in a neighborhood of the joint zone I0 in the form
uε(x) ≈ v+
0 (x1, 0) + ε
(
Z1 (x/ε) ∂x1v
+
0 (x1, 0) + Z2 (x/ε) ∂x2v
+
0 (x1, 0)
)
+ . . . , (29)
where functions Z1(ξ) and Z2(ξ), ξ ∈ Π, are 1-periodic with respect to ξ1. Here Π is
the union of semiinfinite strips Π+ = (0, 1)× (0,+∞), Π−h1
= Ih1(b1)× (−∞, 0] and
Π−h2
= Ih2(b2) × (−∞, 0]. Substituting (29) in the differential equation of problem (1)
and in the corresponding boundary conditions, collecting the coefficients of the same
power of ε, we arrive junction-layer problems for the functions Z1 and Z2 :
−∆ξ Zi(ξ) = 0, ξ ∈ Π,
∂ξ2Zi(ξ1, 0) = 0, ξ1 ∈ (0, 1) \ (Ih1(b1) ∪ Ih2(b2)) ,
∂ξ1Zi(ξ) = −δ1i, ξ ∈
(
∂Π−h1
\ Ih1(b1)
)
∪
(
∂Π−h2
\ Ih2(b2)
)
,
∂p
ξ1
Zi(0, ξ2) = ∂p
ξ1
Zi(1, ξ2), ξ2 > 0, p = 0, 1.
(30)
The main asymptotic relations for the functions {Zi} can be obtained from general
results about the asymptotic behaviour of solutions to elliptic problems in domains with
different exits to infinity [6, 30, 31]. The proofs simplify substantially if the polynomial
property of the corresponding sesquilinear forms is employed [32]. However, for the
domain Π,we can define more exactly the asymptotic relations and detect other properties
of the junction-layer solutions Z1, Z2 similarly as in the papers [16, 17].
Statement 1. There exist two solutions Ξ1, Ξ2 ∈ H1
],loc(Π) to the homogeneous
problem (30) (i = 2), which have the following differentiable asymptotics:
Ξ1(ξ) =
ξ2 +O(exp(−2πξ2)), ξ2 → +∞, ξ ∈ Π+,
h−1
1 ξ2 + α
(1)
1 +O(exp(πh−1
1 ξ2)), ξ2 → −∞, ξ ∈ Π−h1
,
α
(2)
1 +O(exp(πh−1
2 ξ2)), ξ2 → −∞, ξ ∈ Π−h2
,
(31)
Ξ2(ξ) =
ξ2 +O(exp(−2πξ2)), ξ2 → +∞, ξ ∈ Π+,
α
(1)
2 +O(exp(πh−1
1 ξ2)), ξ2 → −∞, ξ ∈ Π−h1
,
h−1
2 ξ2 + α
(2)
2 +O(exp(πh−1
2 ξ2)), ξ2 → −∞, ξ ∈ Π−h2
.
(32)
Any other solution to the homogeneous problem (30), which has polynomial grow at
infinity, can be presented as a linear combination β0 + β1Ξ1 + β2Ξ2.
The solution Z1 to problem (30) at i = 1 has the following asymptotics:
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204 T. A. MEL’NYK
Z1(ξ) =
O(exp(−2πξ2)), ξ2 → +∞, ξ ∈ Π+,
−ξ1 + b1 + α
(1)
3 +O(exp(πh−1
1 ξ2)), ξ2 → −∞, ξ ∈ Π−h1
,
−ξ1 + b2 + α
(2)
3 +O(exp(πh−1
2 ξ2)), ξ2 → −∞, ξ ∈ Π−h2
.
(33)
Here H1
],loc(Π) = {u : Π → R | u(0, ξ2) = u(1, ξ2) for any ξ2 > 0, u ∈ H1(ΠR) for
any R > 0}, where ΠR = Π ∩ {ξ : −R < ξ2 < R}; α(i)
1 , α
(i)
2 , α
(i)
3 , i = 1, 2, are some
fixed constants.
Now we verify the matching conditions for the outer expansions (20), (21) and
the inner expansion (29), namely, the leading terms of the asymptotics of the outer
expansions as x2 → ±0 must coincide with the leading terms of the inner expansion
as ξ2 → ±∞ respectively. Near the point (ε(j + bi), 0) ∈ I0 the function v+
0 has the
following asymptotics:
v+
0 (ε(j + bi), 0) + ε ξ2∂x2v
+
0 (ε(j + bi), 0) +O(ε2ξ22), x2 → 0 + 0.
We see that the matching condition is satisfied for the expansions (20) and (29) if Z2 =
= β1Ξ1 + (1− β1)Ξ2.
The asymptotics of (21) is equal to
vi,−
0 (ε(j + bi), 0)+
+ε
(
(−ξ1 + bi + j) ∂x1v
i,−
0 (ε(j + bi), 0) + ξ2∂x2v
i,−
0 (ε(j + bi), 0)
)
+ . . . (34)
as x2 → 0− 0, x ∈ G(i)
j (ε), i = 1, 2.
The asymptotics of (29) is equal to
v+
0 (ε(j + b1), 0) + ε
((
−ξ1 + j + b1 + α
(1)
3
)
∂x1v
+
0 (ε(j + b1), 0)+
+
{
β1
(
h−1
1 ξ2 + α
(1)
1
)
+ (1− β1))α
(1)
2
}
∂x2v
+
0 (ε(j + b1), 0)
)
+ . . . (35)
as ξ2 → −∞, ξ ∈ Π−h1
,
and it is equal to
v+
0 (ε(j + b2), 0) + ε
((
−ξ1 + j + b2 + α
(2)
3
)
∂x1v
+
0 (ε(j + b2), 0)+
+
{
(1− β1)
(
h−1
1 ξ2 + α
(2)
2
)
+ β1α
(2)
1
}
∂x2v
+
0 (ε(j + b2), 0)
)
+ . . . (36)
as ξ2 → −∞, ξ ∈ Π−h2
.
Comparing the first terms of (34), (35), and (36), we get
v+
0 (ε(j + bi), 0) = vi,−
0 (ε(j + bi), 0), j = 0, 1, . . . , N − 1, i = 1, 2. (37)
Comparing the second terms of (34) and (35), and (34) and (36), we find
β1∂x2v
+
0 (ε(j + b1), 0) = h1∂x2v
1,−
0 (ε(j + b1), 0), (38)
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ASYMPTOTIC BEHAVIOR OF EIGENVALUES AND EIGENFUNCTIONS... 205
(1− β1) ∂x2v
+
0 (ε(j + b2), 0) = h2∂x2v
2,−
0 (ε(j + b2), 0), j = 0, 1, . . . , N − 1. (39)
Since the segments {x : x1 = ε(j + bi), x2 ∈ [−di, 0]}, j = 0, 1, . . . , N − 1, fill
out the rectangle Di in the limit passage as ε → 0 (N → +∞) both for i = 1 and for
i = 2, we can spread the equation (27) into rectangle D1 = I0 × (−d1, 0) for i = 1 and
into rectangle D2 for i = 2. On the basis of the same arguments, we spread the relations
(28), (37), (38), and (39) into all interval I0. From the limiting relations (38) and (39) it
follows that
∂x2v
+
0 (x1, 0) = h1∂x2v
1,−
0 (x1, 0) + h2∂x2v
2,−
0 (x1, 0), x1 ∈ I0.
Now define the following vector function:
v0(x) =
v+
0 (x), x ∈ Ω0,
v1,−
0 (x), x ∈ D1,
v2,−
0 (x), x ∈ D2.
(40)
As follows from the foregoing the components of this function must satisfy the relations
−∆x v
+
0 (x) = µ0v
+
0 (x), x ∈ Ω0,
∂p
x1
v+
0 (0, x2) = ∂p
x1
v+
0 (a, x2), p = 0, 1, x2 ∈ [0, γ(0)],
∂νv
+
0 (x) = 0, x ∈ Γγ ,
−h1∂
2
x2x2
v1,−
0 (x) + 2k1v
1,−
0 (x) = h1µ0v
1,−
0 (x), x ∈ D1,
∂x2v
1,−
0 (x1,−d1) = 0, x1 ∈ I0,
−h2∂
2
x2x2
v2,−
0 (x) + 2k2v
2,−
0 (x) = h2µ0v
2,−
0 (x), x ∈ D2,
∂x2v
2,−
0 (x1,−d2) = 0, x1 ∈ I0,
v+
0 (x1, 0) = vi,−
0 (x1, 0), i = 1, 2, x1 ∈ I0,
h1∂x2v
1,−
0 (x1, 0) + h2∂x2v
2,−
0 (x1, 0) = ∂x2v
+
0 (x1, 0), x1 ∈ I0.
(41)
These relations form the spectral limiting problem for problem (1); here µ0 is the spectral
parameter. Let us investigate its spectrum.
4. The resulting limit problem and its spectrum. Denote by V0 the vector-space
L2(Ω0)× L2(D1)× L2(D2) with the following scalar product:
(u,v)V0
=
∫
Ω0
u0v0dx+
2∑
i=1
hi
∫
Di
uividx,
where u = (u0, u1, u2) and v = (v0, v1, v2) belong to V0. Also we define the Hilbert
space H0 = {u ∈ V0 : u0 ∈ H1(Ω0), u0(0, x2) = u0(a, x2) for x2 ∈ (0, γ(0));
∃ ∂x2u1 ∈ L2(D1); ∃ ∂x2u2 ∈ L2(D2); u0(x1, 0) = u1(x1, 0) = u2(x1, 0) for x1 ∈ I0}
with the following scalar product:
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206 T. A. MEL’NYK
(u,v)H0
=
∫
Ω0
∇u0 · ∇v0dx+
2∑
i=1
∫
Di
(
hi∂x2ui∂x2vi + 2kiuivi
)
dx.
Obviously, H0 continuously embeds in V0. If we define the operator A0 : H0 7−→ H0
by the following equality:
(A0u,v)H0
= (u,v)V0
∀ u,v ∈ H0, (42)
then problem (41) is equivalent to the spectral problem A0v0 = µ−1
0 v0 in H0. It is
easy to verify that A0 is self-adjoint, positive, continuous, noncompact and 0 /∈ σ(A0).
Thus σ(A0) ⊂ (c0,+∞), where c0 is some positive constant.
Next we assume that c0 ≥ max
(
2k1
h1
,
2k2
h2
)
; the other cases we will be discussed in
Remark 2. Solving the ordinary differential equations of problem (41) in the rectangles
D1 and D2 with regard of the first conjugation condition in the joint zone I0 and the
Neumann conditions on the opposite sides of these rectangles, we get
vi,−
0 (x) =
v+
0 (x1, 0)
cos
(
di
√
µ0 − 2kih
−1
i
) cos
(√
µ0 − 2kih
−1
i (x2 + di)
)
, i = 1, 2.
(43)
Substituting these relations into the second conjugation condition, we obtain the following
spectral problem:
−∆x v
+
0 (x) = µ0v
+
0 (x), x ∈ Ω0,
∂p
x1
v+
0 (0, x2) = ∂p
x1
v+
0 (a, x2), x2 ∈ [0, γ(0)], p = 0, 1,
∂νv
+
0 (x) = 0, x ∈ Γγ ,
∂x2v
+
0 (x1, 0) =
= −v+
0 (x1, 0)
2∑
i=1
hi
√
µ0 − 2kih
−1
i tan
(
di
√
µ0 − 2kih
−1
i
)
, x1 ∈ I0,
(44)
with the spectral parameter µ0 occurring both in the differential equation and in the
boundary condition on I0, where it enters in a nonlinear way. Problem (44) is called the
resulting problem for problem (1).
Multiplying the differential equation of problem (44) with an arbitrary function
ψ ∈ H1
],x1
(Ω0) = {u ∈ H1(Ω0) : u is 1-periodic with respect to x1} and integrating
by parts in Ω0, we reduce the nonlinear spectral problem (44) to the spectral problem
L(µ)v+
0 = 0 in H1
],x1
(Ω0), µ ∈ [c0,+∞),
for the following operator-function:
L(µ) := (µ+ 1)A1 +
2∑
i=1
hi
√
µ− 2ki
hi
tan
(
di
√
µ− 2ki
hi
)
A2 − I, (45)
where I is the identity operator in H1
],x1
(Ω0); A1, A2 are self-adjoint, compact operators
in H1
],x1
(Ω0) such that for all ϕ,ψ ∈ H1
],x1
(Ω0)
(A1ϕ,ψ)H1(Ω0) =
∫
Ω0
ϕ(x)ψ(x)dx,
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ASYMPTOTIC BEHAVIOR OF EIGENVALUES AND EIGENFUNCTIONS... 207
(A2ϕ,ψ)H1(Ω0) =
a∫
0
ϕ(x1, 0) ψ(x1, 0)dx1.
Theorems on existence and concentration of the spectrum for such self-adjoint
discontinuous operator-functions and minimax principles for the eigenvalues were proved
in [33, 34]. From these results it follows the following theorem.
Theorem 1. The spectrum of L consists of normal eigenvalues and points {Pm :
m ∈ N} of the essential spectrum, which are poles of the functions
tan
(
di
√
µ− 2kih
−1
i
)
, i = 1, 2, µ ∈ (c0,+∞).
These points divide the eigenvalues into the sequences
c0 < µ
(1)
1 ≤ . . . ≤ µ(1)
n ≤ . . .→ P1,
Pm−1 < µ
(m)
1 ≤ . . . ≤ µ(m)
n ≤ . . .→ Pm as n→∞.
We recall that an eigenvalue is called normal eigenvalue if it has finite multiplicity
and the corresponding eigenvectors have no Jordan chain.
Remark 2. Consider for example the case
2k1
h1
≤ c0 <
2k2
h2
. Then v1,−
0 is represented
by (43) and
v2,−
0 (x) =
v+
0 (x1, 0)
cosh
(
di
√
2k2h
−1
2 − µ0
) cosh
(√
2k2h
−1
2 − µ0(x2 + d2))
)
.
Using these representations, we similarly as before reduce problem (41) to the nonlinear
spectral problem for the following operator-function:
L(µ) := (µ+ 1)A1 +
(
h1
√
µ− 2k1
h1
tan
(
d1
√
µ− 2k1
h1
)
+
+h2
√
2k2
h2
− µ tanh
(
d2
√
2k2
h2
− µ
))
A2 − I, µ ∈
(
c0,
2k2
h2
)
.
It follows from [33, 34] that the spectrum of L on
(
c0,
2k2
h2
)
consists of normal
eigenvalues and points of the essential spectrum, which are poles of function
tan
(
d1
√
µ− 2k1h
−1
1
)
on
(
c0,
2k2
h2
)
. In addition, the points of the essential spectrum
are left accumulation points of the normal eigenvalues. Thus, in fact, Theorem 1
describes structure of the spectrum of problem (41) in all cases.
5. Asymptotic approximations. 5.1. The case of the discrete spectrum. Let
µ0 be an eigenvalue of the limiting problem (41) and v0 = (v+
0 , v
1,−
0 , v2,−
0 ) is the
corresponding eigenfunction, i.e., v+
0 is the eigenfunction of problem (44) and vi,−
0 ,
i = 1, 2, are defined by (43). With the help of v0 and the junction-layer solutions
Z1,Ξ1,Ξ2 (see Section 3), we define the leading terms in (20), (21), and (29). Then
matching these expansions, we construct an asymptotic approximation Rε belonging to
Hε. It is equal to
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208 T. A. MEL’NYK
R+
ε (x) := v+
0 (x) + εχ0(x2)N+
(x
ε
, x1
)
, x ∈ Ω0, (46)
Ri,−
ε := vi,−
0 (x) + ε
(
Y1
(x1
ε
)
∂x1v
i,−
0 (x) + χ0(x2)N−
(x
ε
, x1
))
, (47)
x ∈ G(i)
ε , i = 1, 2.
Here
N+(ξ, x1) = Z1(ξ)∂x1v
+
0 (x1, 0) +
(
β1Ξ1(ξ) + (1− β1)
(
Ξ2(ξ)− ξ2
))
∂x2v
+
0 (x1, 0),
N−(ξ, x1) = (Z1(ξ)− Y1(ξ1)) ∂x1v
+
0 (x1, 0)+
+
(
β1Ξ1(ξ) + (1− β1)
(
Ξ2(ξ)− Y2(ξ2)
))
∂x2v
+
0 (x1, 0),
where ξ = x/ε, Y1 and Y2 are 1-periodic functions with respect to ξ1 and on the
corresponding cells of periodicity they are equal to
Y1(ξ1) =
−ξ1 + b1 + α
(1)
3 , ξ1 ∈ [0, δ0),
−ξ1 + b2 + α
(2)
3 , ξ1 ∈ [δ0, 1),
Y2(ξ2) =
β1(h−1
1 ξ2 + α
(1)
1 ) + (1− β1)α
(1)
2 , ξ ∈ Π−h1
,
β1α
(2)
1 + (1− β1)(h−1
2 ξ2 + α
(2)
2 ), ξ ∈ Π−h2
,
the number β1 is defined from relation (38), (39) and it is equal to
β1 =
h1
√
µ0 −
2k1
h1
tan
(
d1
√
µ0 −
2k1
h1
)
∑2
i=1 hi
√
µ0 −
2ki
hi
tan
(
di
√
µ0 −
2ki
hi
) ,
the function χ0 is a smooth cut-off function such that χ0(x2) = 1 for |x2| ≤ α0/2 and
χ0(x2) = 0 for |x2| ≥ α0, where 0 < α0 < 2−1 min{d1, d2,min[0,a] γ(x)}.
5.1.1. Discrepancies in the domain Ω0. Taking into account the properties of the
functions Z1, Ξ1, Ξ2 and v+
0 , we conclude that R+
ε is a-periodic with respect to x1,
∂νR
+
ε = 0 on Γγ , and ∂x2R
+
ε (x1, 0) = 0 for any x1 ∈ I0 \ I0(ε). Thus R+
ε satisfies all
boundary conditions for problem (1) on ∂Ω0 ∩ ∂Ωε. Putting R+
ε and µ0 in the equation
of problem (1), we get
−∆xR
+
ε − µ0R
+
ε =
=
(
−χ′0∂ξ2N+(ξ, x1)− χ0∂
2
x1ξ1
N+(ξ, x1)− ε∂x2
(
χ′0N+(x/ε, x1)
)
−
−εχ0∂x1
((
∂x1N+(ξ, x1)
)
|ξ=x/ε
)
− εµ0χ0N+(ξ, x1)
)∣∣∣
ξ=x/ε
, x ∈ Ω0. (48)
Further, the arguments of functions involved in calculations are indicated only if their
absence may cause confusion. We multiply the identity (48) by a test function ψ ∈ Hε
and integrate by parts in Ω0 :
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ASYMPTOTIC BEHAVIOR OF EIGENVALUES AND EIGENFUNCTIONS... 209
−
∫
I0(ε)
∂x2R
+
ε (x1, 0)ψdx1 +
∫
Ω0
∇xR
+
ε ·∇xψdx−µ0
∫
Ω0
R+
ε ψdx =
5∑
i=1
I+
i (ε, ψ), (49)
where
I+
1 (ε, ψ) = −
∫
Ω0
χ′0
(
∂ξ2N+(ξ, x1)
) ∣∣∣
ξ= x
ε
ψdx,
I+
2 (ε, ψ) = −
∫
Ω0
χ0
(
∂2
x1ξ1
N+(ξ, x1)
) ∣∣∣
ξ= x
ε
ψdx,
I+
3 (ε, ψ) = ε
∫
Ω0
χ′0N+
(x
ε
, x1
)
∂x2ψdx,
I+
4 (ε, ψ) = ε
∫
Ω0
χ0
(
∂x1N+(ξ, x1)
) ∣∣∣
ξ= x
ε
∂x1ψdx,
I+
5 (ε, ψ) = −εµ0
∫
Ω0
χ0(x2)N+(ξ, x1)
∣∣∣
ξ= x
ε
ψdx.
5.1.2. Discrepancies in the thin rods. It is easy to calculate that ∂x2R
i,−
ε (x1,
−di) = 0,
∂x2R
i,−
ε (x1, 0) = εY1
(x1
ε
)
∂2
x2x1
vi,−
0 (x1, 0) + ∂x2R
+
ε (x1, 0), x1 ∈ I0 ∩G(i)
ε ,
(50)
∂νR
i,−
ε (x) = ±ε
(
Y1
(x1
ε
)
∂2
x1x1
vi,−
0 (x) + χ0(x2)
(
∂x1N−(ξ, x1)
) ∣∣∣
ξ= x
ε
)
, (51)
x ∈ Υ(i,±)
ε , i = 1, 2.
Putting Ri,−
ε and µ0 in the differential equation of problem (1), we obtain
−∆xR
i,−
ε (x)− µ0R
i,−
ε (x) =
= −χ′0(x2)
(
∂ξ2N−(ξ, x1)
)
|ξ= x
ε
− χ0(x2)
(
∂2
x1ξ1
N−(ξ, x1)
)
|ξ= x
ε
−
−ε∂x2
(
χ′0(x2)N−
(x
ε
, x1
))
− εχ0∂x1
((
∂x1N−(ξ, x1)
)
|ξ= x
ε
)
−
−div
(
Y1
(x1
ε
)
∇x
(
∂x1v
i,−
0
))
− εµ0
(
Y1
(x1
ε
)
∂x1v
i,−
0 (x) + χ0N−
(x
ε
, x1
))
−
−2kih
−1
i vi,−
0 (x), x ∈ G(i)
ε , i = 1, 2. (52)
Using (6) and taking into account the boundary values of ∂νR
i,−
ε (see (60), (51)),
we multiply (52) by a test function ψ ∈ Hε and integrate by parts in G
(i)
ε , i = 1, 2.
This yields
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 2
210 T. A. MEL’NYK∫
I0(ε)
∂x2R
+
ε (x1, 0)ψdx1 +
∫
G
(i)
ε
∇xR
i,−
ε · ∇xψdx+
+εki
∫
Υ
(i)
ε
Ri,−
ε ψdx2 − µ0
∫
G
(i)
ε
Ri,−
ε (x)ψdx =
= Ii,−
1 (ε, ψ) + . . .+ Ii,−
7 (ε, ψ), (53)
where
Ii,−
1 = −
∫
G
(i)
ε
χ′0
(
∂ξ2N−(ξ, x1)
)
|ξ= x
ε
ψdx,
Ii,−
2 = −
∫
G
(i)
ε
χ0
(
∂2
x1ξ1
N−(ξ, x1)
)
|ξ= x
ε
ψdx,
Ii,−
3 = ε
∫
G
(i)
ε
χ′0N−
(x
ε
, x1
)
∂x2ψdx,
Ii,−
4 = ε
∫
G
(i)
ε
χ0
(
∂x1N−(ξ, x1)
)∣∣∣
ξ= x
ε
∂x1ψdx,
Ii,−
5 (ε, ψ) = −εµ0
∫
G
(i)
ε
(
Y1
(x1
ε
)
∂x1v
i,−
0 (x) + χ0N−
(x
ε
, x1
))
ψdx,
Ii,−
6 (ε, ψ) = ε
∫
G
(i)
ε
Y1
(x1
ε
)
∇x
(
∂x1v
i,−
0
)
· ∇xψdx,
Ii,−
7 (ε, ψ) = kiε
∫
Υ
(i)
ε
Ri,−
ε ψdx2 − kiε
∫
Υ
(i,±)
ε
vi,−
0 ψdx2−
−2kih
−1
i ε
∫
G
(i)
ε
Y
(x1
ε
)
∂x1(v
i,−
0 ψ)dx.
Summing (49) and (53), we see that the function Rε constructed by formulas (46)
and (47) satisfies the following integral identity:∫
Ωε
∇xRε ·∇xψdx+ε
2∑
i=1
ki
∫
Υ(i)(ε)
Rεψdx2−µ0
∫
Ωε
Rεψdx = Fε(ψ) ∀ψ ∈ Hε, (54)
where
Fε(ψ) = I±1 (ε, ψ) + . . .+ I±5 (ε, ψ) + I−6 (ε, ψ) + I−7 (ε, ψ),
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 2
ASYMPTOTIC BEHAVIOR OF EIGENVALUES AND EIGENFUNCTIONS... 211
I±j (ε, ψ) = I+
j (ε, ψ) + I−j (ε, ψ),
I−j (ε, ψ) = I1,−
j (ε, ψ) + I2,−
j (ε, ψ), j = 1, . . . , 7.
Using (6), Lemma 1 and doing similar calculations as in the paper [16], we can show that
for any positive fixed number δ and for any ψ ∈ Hε the following inequality |Fε(ψ)| ≤
≤ c(δ)ε1−δ‖ψ‖Hε
holds. Then with the help of the definition of operator Aε and the
Riesz theorem, we deduce from (54) that for any δ > 0
‖Rε − µ0AεRε‖Hε ≤ c(δ)ε1−δ. (55)
5.2. The case of the essential spectrum. Let µ0 ∈ σess(A0), i.e., µ0 coincides
with one of the numbers {Pm : m ∈ N}
(
they are poles of the functions
tan
(
di
√
µ− 2kih
−1
i
)
, µ ∈ (c0,+∞), i = 1, 2; see Theorem 1
)
. For definiteness
we assume that i = 1. Then we choose the following approximation function:
Wε(x) =
=
√
2
ε(h1 + k1)d1(µ0 − 2k1h
−1
1 )
cos
√
µ0 − 2k1h
−1
1 (x2 + d1), x ∈ G(1)
j0
(ε),
0, x ∈ Ωε\G(1)
j0
(ε),
(56)
whereG(1)
j0
(ε) is an arbitrary rod from the first level. It is easy to verify that ‖Wε‖Hε
= 1.
Substituting the function Wε and the number µ0 in problem (1) instead of u(ε, ·) and
λ(ε) respectively, we find residuals and deduce that there exist constants c > 0 and ε0
such that for any values ε ∈ (0, ε0) the following inequality is satisfied:
‖Wε − µ0AεWε‖Hε
≤ c ε
1
4 . (57)
6. Justification and asymptotic estimates. To justify the constructed asymptotic
approximations we use the scheme proposed in [13], where an abstract scheme of
investigation of the asymptotic behaviour of eigenvalues and eigenvectors of some
family of abstract operators {Aε : ε > 0} acting in different spaces was proposed.
This scheme generalizes the procedure of justification of the asymptotic behaviour of
eigenvalues and eigenvectors of boundary value problems in perturbed domains.
In our case this is the family of the operators {Aε : ε > 0} acting in the spaces
{Hε : ε > 0} and they are defined by (15). Recall that operator Aε corresponds to
problem (1) and operator A0 : H0 7−→ H0, which is defined by (42) corresponds to the
limiting problem (41).
Then we should define special coupling operators Pε and Sε. For better understanding,
we write the diagram
Hε ⊂⊂ Vε
Pε
y xSε
Z0⊂ H0 ⊂V0
in which the imbedding H ⊂ V means that the space H is densely and only continuously
embedded into V, but the imbedding H ⊂⊂ V is compact in addition. Here Z0 =
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 2
212 T. A. MEL’NYK
= {u = (u0, u1, u2) ∈ V0 : u0 ∈ H1(Ω0), u0(0, x2) = u0(a, x2) for x2 ∈ (0, γ(0));
u1 ∈ H1(D1); u2 ∈ H1(D2); u0(x1, 0) = u1(x1, 0) = u2(x1, 0) for x1 ∈ I0} is a
Hilbert space with the scalar product (u,v)Z0
= (u0, v0)H1(Ω0) + (u1, v1)H1(D1) +
+(u1, v1)H1(D1). Obviously, that Z0 ⊂⊂ V0.
The operator Sε : V0 7→ Vε assigns to any vector-function v = (v0, v1, v2) from V0
a function Sεv, which is equal to v0 in Ω0 and to vi|G(i)
ε
, i = 1, 2, where vi|G(i)
ε
is the
restriction of vi on G(i)
ε . It is easy to verify that operator Sε is uniformly bounded with
respect to ε. Thus the condition (C1) in the scheme [13] is satisfied.
The operator Pε from condition (C2) is associated with special extension operator
Pε =
(
P(1)
ε ,P(2)
ε
)
, where P(1)
ε : H1(Ω0 ∪ G(1)(ε)) 7→ H1(Ω1) and P(2)
ε : H1(Ω0 ∪
∪ G(2)(ε)) 7→ H1(Ω2), where Ωi is the interior of Ω0 ∪ Di, i = 1, 2. The operators
P(1)
ε and P(2)
ε can be constructed similarly as in [16] (see also [26]). Thus operator
Pε : Hε 7→ Z0 every u from Hε puts in the correspondence a vector-function u =
=
(
u|Ω0 ,P
(1)
ε u|D1 ,P
(2)
ε u|D2
)
from Z0. Despite the fact that the norm of this operator
takes an infinitely large value as ε → 0, the norm of its restriction to an arbitrary finite
combination of eigenfunctions of problem (1) is uniformly bounded with respect to ε, i.e.,
the following statement is true: ∀n ∈ N ∃c > 0 ∃ε0 > 0 ∀ε ∈ (0, ε0) : ‖Pεun(ε, ·)‖Z0 ≤
≤ c(n)‖un(ε, ·)‖Hε
. Furthermore, this operator is also uniformly bounded on sequences
from condition (C2) (the proof of this fact is analogous to the corresponding part of the
proof of Theorem 5.4 [18]).
Conditions (C5) and (C6), in fact, have been verified in the previous section. The
result of the action of the operator Rε from the condition (C5) is the construction of
the approximation function Rε (see (46) and (47)) on the basis of an eigenfunction of
the limit spectral problem (41). In addition, this approximation function satisfies the
estimate (55), which coincides with similar estimate from condition (C5). The estimate
(57) coincides with similar estimate from condition (C6). To verify conditions (C3) and
(C4) we prove the following theorem.
Theorem 2. Let {λ(ε) : ε > 0} be a sequence of eigenvalues of problem (1)
such that lim
ε→0
λ(ε) = Λ and
1
Λ
/∈ σess(A0); let {uε} be the corresponding sequence
of eigenfunctions such that ‖uε‖L2(Ωε) = 1 for any value ε and Pεu
ε → u∗ =
=
(
u+
0 , u
1,−
0 , u2,−
0
)
weakly in Z0 as ε→ 0.
Then Λ is the eigenvalue of the limiting problem (41) and u∗ is the corresponding
eigenfunction.
Proof. Using operator Pε and the functions χ1 and χ2 defined in (12), we can
rewrite the equality (uε, uε)Ωε
= 1 in the following form:
1 =
∫
Ω0
(uε)2 dx+
∫
D1
χ1(x1/ε)
(
P(1)
ε uε
)2
dx+
∫
D2
χ2(x1/ε)
(
P(2)
ε uε
)2
dx.
Passing to the limit in this relation as ε→ 0, we obtain 1 = ‖u∗‖2V0
, whence u∗ 6= 0.
With the help of the identity (6), the extension operators P(i)
ε and the functions χi,
i = 1, 2, we rewrite identity (14) in the following way:
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 2
ASYMPTOTIC BEHAVIOR OF EIGENVALUES AND EIGENFUNCTIONS... 213∫
Ω0
∇uε · ∇ϕ0dx+
+
2∑
i=1
∫
Di
χi(x1/ε)∇
(
P(i)
ε uε
)
· ∇ϕidx+
2ki
hi
∫
Di
χi(x1/ε)P(i)
ε uεϕidx
−
− 2ε
2∑
i=1
ki
hi
∫
G
(i)
ε
Y
(x1
ε
)
∂x1 (uεϕi) dx =
= λ(ε)
∫
Ω0
u∗(x)ϕ(x)dx+
2∑
i=1
∫
Di
χi(x1/ε)
(
P(1)
ε uε
)
(x)ϕi(x)dx
(58)
∀ (ϕ0, ϕ1, ϕ2) ∈ Z0.
Obviously, that the last summand in the left-hand side of (58) vanishes as ε → 0.
Now, passing to the limit in (58) and taking the theorem conditions into account, we
obtain∫
Ω0
∇u+
0 · ∇ϕ0dx+
2∑
i=1
∫
Di
2∑
j=1
σ
(i)
j (x)∂xj
ϕi(x)dx+ 2ki
∫
Di
ui,−
0 ϕidx
=
= Λ
∫
Ω0
u0(x)ϕ0(x)dx+
2∑
i=1
hi
∫
Di
ui,−
0 (x)ϕi(x)dx
(59)
∀ (ϕ0, ϕ1, ϕ2) ∈ Z0,
where σ(i)
j is the weak limit of the sequence χi
(x1
ε
)
∂xj
(
P(i)
ε uε
)
, j = 1, 2, i = 1, 2.
Next we should find these limits.
In order to determine σ(i)
1 , i = 1, 2, we consider the integral identity (14) with the
following test functions :
ψ1(x) = ε
0, x ∈ Ω0 ∪G(2)
ε ,
Y (x1/ε)φ1(x), x ∈ G(1)
ε ,
ψ2(x) = ε
0, x ∈ Ω0 ∪G(1)
ε ,
Y (x1/ε)φ2(x), x ∈ G(2)
ε ,
where φ1 and φ2 are arbitrary functions from C∞0 (D1) and C∞0 (D2) respectively. It is
obvious that ψ1 and ψ2 belong to Hε. As a result, we get∫
D1
χ1
(x1
ε
)
∂x1P
(1)
ε (uε)φ1dx = O(ε),
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 2
214 T. A. MEL’NYK∫
D2
χ(2)
ε (x)∂x1P
(2)
ε (uε)φ2dx = O(ε), ε→ 0,
whence σ(1)
1 ≡ 0 and σ(2)
1 ≡ 0.
Next let us define σ(i)
2 , i = 1, 2. Take any function φ ∈ C∞0 (Di) and pass to the
limit in the following relation:∫
Di
χi(x1/ε)∂x2
(
P(i)
ε uε
)
φ(x)dx = −
∫
Di
χi(x1/ε)
(
P(i)
ε uε
)
∂x2φdx. (60)
As a result, we get that σ(i)
2 (x) = hi∂x2u
i,−
0 (x), x ∈ Di, i = 1, 2.
Thus, we obtain that u∗ satisfies the following identity (u∗,v)H0
= Λ (u∗,v)V0
for
any vector-function v = (ϕ0, ϕ1, ϕ2) ∈ Z0. This identity is the corresponding integral
identity for the spectral limiting problem (41) (see (42)). This means that Λ is the
eigenvalue of problem (41) and u∗ is the corresponding eigenfunction.
The theorem is proved.
Thus, all conditions (C1) – (C6) of the scheme from [13] are satisfied for problems
(1) and (41). Applying this scheme, we get the following theorems.
Theorem 3 (the Hausdorff convergence). Only the points of the spectrum of problem
(41) are accumulation points for the spectrum of problem (1) as ε→ 0.
The eigenvalues {λn(ε)} at fixed indices n, are usually called low eigenvalues (see
[21]); the corresponding eigenfunctions are called low frequency oscillations.
Definition 2 [21]. The value T := supn∈N limε→0λn(ε) is called the threshold of
the low eigenvalues of problem (1).
Theorem 4 (low-frequency convergence). Let {λn(ε) : n ∈ N0} be the ordered
sequence (2) of eigenvalues of problem (1), let {un(ε, ·) : n ∈ N} be the corresponding
sequence of eigenfunction orthonormalized by condition (3), and let c0 < µ
(1)
1 ≤ . . .
. . . ≤ µ
(1)
n ≤ . . . → P1 be the first series of eigenvalues of the limiting problem (41)
(see Theorem 1).
Then the threshold of the low eigenvalues of problem (1) is equal to P1, and for any
n ∈ N λn(ε) → µ
(1)
n as ε→ 0. There exists a subsequence of the sequence {ε} (again
denoted by {ε}) such that Pεun(ε, ·) → v(0)
n weakly in Z0 as ε→ 0, where {v(0)
n } are
the corresponding eigenfunctions of the limiting problem (41) that satisfy the condition(
v(0)
n ,v(0)
m
)
V0
= δn,m.
Theorem 5. Let µ(1)
n = µ
(1)
n+1 = . . . = µ
(1)
n+r−1 be an r-multiple eigenvalue of
problem (41) from the first series (see Theorem 1) and let v(1)
n , . . . ,v(1)
n+r−1 be the
corresponding eigenfunction orthonormalized in V0.
Then for any δ > 0 and i ∈ {0, 1, . . . , r − 1}, there exist ε0 > 0, Ci > 0, and
{αik(ε), k = 0, 1, . . . , r − 1} ⊂ R, such that for any ε ∈ (0, ε0) :∥∥∥R(n+i)
ε −
r−1∑
k=0
αik(ε)un+k(ε, ·)
∥∥∥
H1(Ωε)
≤ Ci(n, δ) ε1−δ,
0 < c1 <
r−1∑
k=0
(αik(ε))2 < c2,
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 2
ASYMPTOTIC BEHAVIOR OF EIGENVALUES AND EIGENFUNCTIONS... 215
where {R(n+i)
ε } is approximation function defined by (46) and (47) with the help of
v(1)
n+i.
For any δ > 0 and n ∈ N and sufficiently small ε, we have |λn(ε) − µ
(1)
n | ≤
≤ c0(n, δ)ε1−δ.
Theorem 6. Let µ(m)
n = µ
(m)
n+1 = . . . = µ
(m)
n+r−1 be an r-multiple eigenvalue
of problem (41) from the m-th series (see Theorem 1) and v(m)
n , . . . ,v(m)
n+r−1 be the
corresponding eigenfunction orthonormalized in V0.
Then, for any δ > 0, there exist εn,m > 0 and c > 0 such that for all value of
the parameter ε ∈ (0, εn,m) in the interval In,m(ε) =
(
µ
(m)
n − cε1−δ, µ
(m)
n + cε1−δ
)
contains exactly r eigenvalues of problem (41).
For the approximation function Rn+i,m
ε , i = 0, 1, . . . , r−1, constructed by (46) and
(47) on the basis of v(m)
n+i, the following asymptotic estimate is true:∥∥∥∥∥ Rn+i,m
ε
‖Rn+i,m
ε ‖Hε
− Ũi(ε, ·)
∥∥∥∥∥
Hε
≤ c(n,m, δ)ε1−δ, ‖Ũi(ε, ·)‖Hε = 1,
where Ũi(ε, ·) is a linear combination of the eigenfunctions of problem (1) that correspond
to the eigenvalues from the interval In,m(ε).
Theorem 7. Let µ0 coincides with one of the points of the essential spectrum
{Pm : m ∈ N} of the limiting problem (41).
Then there exist c0 > 0 and ε0 > 0 such that for all values of the parameter ε ∈
∈ (0, ε0), the interval
(
1
µ0
− c0ε
1
4 ,
1
µ0
+ c0ε
1
4
)
contains finitely many eigenvalues of
the operator Aε.
There exists a finite linear combination Ũε (‖Ũε‖ε = 1) of the eigenfunction uε
k(ε)+i,
i = 0, p(ε), that correspond, respectively, to the eigenvalues
(
λk(ε)+i(ε)
)−1
of the
operator Aε from the segment
[
1
µ0
− c0ε
1
8 ,
1
µ0
+ c0ε
1
8
]
, such that
∥∥∥Wε − Ũε
∥∥∥
Hε
≤
≤ 2ε
1
8 , where Wε is defined by (56).
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Received 01.11.2005
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 2
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| id | umjimathkievua-article-3447 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:42:43Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fe/83a00ddc01b9cced0c13bfd01d73f7fe.pdf |
| spelling | umjimathkievua-article-34472020-03-18T19:54:47Z Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction Асимптотична поведінка власних значень та власних функцій задачі Фур'є в густому багаторівневому з'єднанні Mel'nik, T. A. Мельник, Т. А. A spectral boundary-value problem is considered in a plane thick two-level junction $\Omega_{\varepsilon}$, which is the union of a domain $\Omega_{0}$ and a large number $2N$ of thin rods with thickness of order $\varepsilon = \mathcal{O} (N^{-1})$. The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are $\varepsilon$-periodically alternated. The Fourier conditions are given on the lateral boundaries of the thin rods. The asymptotic behavior of the eigenvalues and eigenfunctions is investigated as $\varepsilon \rightarrow 0$, i.e., when the number of the thin rods infinitely increases and their thickness tends to zero. The Hausdorff convergence of the spectrum is proved as $\varepsilon \rightarrow 0$, the leading terms of asymptotics are constructed and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions. Розглядається спектральна крайова задача у плоскому дворівнєвому з'єднанні $\Omega_{\varepsilon}$, яке є об'єднанням області $\Omega_{0}$ та великого числа $2N$ тонких стержнів товщиною порядку $\varepsilon = \mathcal{O} (N^{-1})$. Тонкі стержні розділено на два рівні в залежності від їх довжини. Крім того, тонкі стержні з кожного рівня $\varepsilon$-періодично чергуються. На вертикальних сторонах тонких стержнів задано крайові умови Фур'є. Вивчено асимптотичну поведінку власних значень та власних функцій при $\varepsilon \rightarrow 0$, тобто коли число тонких стержнів необмежено зростає, а їх товщина прямує до нуля. Доведено хаусдорфову збіжність спектра при $\varepsilon \rightarrow 0$, побудовано перші члени асимптотики та обґрунтовано відповідні асимптотичні оцінки для власних значень та власних функцій. Institute of Mathematics, NAS of Ukraine 2006-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3447 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 2 (2006); 195–216 Український математичний журнал; Том 58 № 2 (2006); 195–216 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3447/3632 https://umj.imath.kiev.ua/index.php/umj/article/view/3447/3633 Copyright (c) 2006 Mel'nik T. A. |
| spellingShingle | Mel'nik, T. A. Мельник, Т. А. Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction |
| title | Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction |
| title_alt | Асимптотична поведінка власних значень та власних функцій задачі Фур'є в густому багаторівневому з'єднанні
|
| title_full | Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction |
| title_fullStr | Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction |
| title_full_unstemmed | Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction |
| title_short | Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction |
| title_sort | asymptotic behavior of eigenvalues and eigenfunctions of the fourier problem in a thick multilevel junction |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3447 |
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