On free vibrations of a thick periodic junction with concentrated masses on the fine rods
Доведенi теореми про збiжнiсть та асимптотичнi оцiнки (коли ε → 0) для власних значень та власних функцiй крайової задачi для оператора Лапласа в плоскому густому перiодичному з’єднаннi з концентрованою масою. Це з’єднання складається з деякої областi i великої кiлько- стi N = O(ε ⁻¹ ) тонких стержн...
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irk-123456789-1772702021-02-15T01:26:49Z On free vibrations of a thick periodic junction with concentrated masses on the fine rods Mel'nyk, T. A. Доведенi теореми про збiжнiсть та асимптотичнi оцiнки (коли ε → 0) для власних значень та власних функцiй крайової задачi для оператора Лапласа в плоскому густому перiодичному з’єднаннi з концентрованою масою. Це з’єднання складається з деякої областi i великої кiлько- стi N = O(ε ⁻¹ ) тонких стержнiв. Густина з’єднання є величиною порядку O(ε ^(−α)) на стержнях (концентрацiя маси при α > 0) та O(1) поза стержнями. Можливi три якiсно рiзнi випадки в асимптотичнiй поведiнцi власних значень та власних функцiй: 0 ≤ α < 2, α = 2, α > 2. Головна увага придiляється першому випадку. Convergence theorems and asymptotic estimates (as ε → 0 ) are proved for eigenvalues and eigenfunctions of a boundary value problem for the Laplace operator in a plane thick periodic junction with concentrated masses. This junction consists of the junction’s body and a large number N = O(ε ⁻¹ ) of the fine rods. The density of the junction is order O(ε ^(−α)), α ≥ 0, on the rods (the concentrated masses if α > 0), and O(1) outside of them. There are three qualitatively different cases in the asymptotic behavior of the eigenvalues and eigenfunctions: 0 ≤ α < 2, α = 2, α > 2. The main attention is payed to the case 0 ≤ α < 2. 1999 Article On free vibrations of a thick periodic junction with concentrated masses on the fine rods / T.A. Mel’nyk // Нелінійні коливання. — 1999. — Т. 2, № 4. — С. 511-522. — Бібліогр.: 16 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177270 517.956 en Нелінійні коливання Інститут математики НАН України |
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Доведенi теореми про збiжнiсть та асимптотичнi оцiнки (коли ε → 0) для власних значень та власних функцiй крайової задачi для оператора Лапласа в плоскому густому перiодичному з’єднаннi з концентрованою масою. Це з’єднання складається з деякої областi i великої кiлько- стi N = O(ε ⁻¹ ) тонких стержнiв. Густина з’єднання є величиною порядку O(ε ^(−α)) на стержнях (концентрацiя маси при α > 0) та O(1) поза стержнями. Можливi три якiсно рiзнi випадки в асимптотичнiй поведiнцi власних значень та власних функцiй: 0 ≤ α < 2, α = 2, α > 2. Головна увага придiляється першому випадку. |
| format |
Article |
| author |
Mel'nyk, T. A. |
| spellingShingle |
Mel'nyk, T. A. On free vibrations of a thick periodic junction with concentrated masses on the fine rods Нелінійні коливання |
| author_facet |
Mel'nyk, T. A. |
| author_sort |
Mel'nyk, T. A. |
| title |
On free vibrations of a thick periodic junction with concentrated masses on the fine rods |
| title_short |
On free vibrations of a thick periodic junction with concentrated masses on the fine rods |
| title_full |
On free vibrations of a thick periodic junction with concentrated masses on the fine rods |
| title_fullStr |
On free vibrations of a thick periodic junction with concentrated masses on the fine rods |
| title_full_unstemmed |
On free vibrations of a thick periodic junction with concentrated masses on the fine rods |
| title_sort |
on free vibrations of a thick periodic junction with concentrated masses on the fine rods |
| publisher |
Інститут математики НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/177270 |
| citation_txt |
On free vibrations of a thick periodic junction with concentrated masses on the fine rods / T.A. Mel’nyk // Нелінійні коливання. — 1999. — Т. 2, № 4. — С. 511-522. — Бібліогр.: 16 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT melnykta onfreevibrationsofathickperiodicjunctionwithconcentratedmassesonthefinerods |
| first_indexed |
2025-07-15T15:18:39Z |
| last_indexed |
2025-07-15T15:18:39Z |
| _version_ |
1837726663917436928 |
| fulltext |
т. 2 •№ 4 • 1999
UDC 517 . 956
ON FREE VIBRATIONS OF A THICK PERIODIC JUNCTION
WITH CONCENTRATED MASSES ON THE FINE RODS
ВЛАСНI КОЛИВАННЯ ГУСТОГО ПЕРIОДИЧНОГО З’ЄДНАННЯ
З КОНЦЕНТРОВАНОЮ МАСОЮ НА ТОНКИХ СТЕРЖНЯХ
T.A. Mel’nyk
Inst. Math. A, Univ. Stuttgart,
Plaffenwaldring 57, Postfach 801140, D-70511 Stuttgart, Germany
e-mail: melnyk@mathematik.uni-stuttgart.de
Convergence theorems and asymptotic estimates (as ε → 0 ) are proved for eigenvalues and eigenfunc-
tions of a boundary value problem for the Laplace operator in a plane thick periodic junction with con-
centrated masses. This junction consists of the junction’s body and a large number N = O(ε−1) of the
fine rods. The density of the junction is order O(ε−α), α ≥ 0, on the rods (the concentrated masses if
α > 0), and O(1) outside of them. There are three qualitatively different cases in the asymptotic behavior
of the eigenvalues and eigenfunctions: 0 ≤ α < 2, α = 2, α > 2. The main attention is payed to the case
0 ≤ α < 2.
Доведенi теореми про збiжнiсть та асимптотичнi оцiнки (коли ε → 0) для власних значень
та власних функцiй крайової задачi для оператора Лапласа в плоскому густому перiодичному
з’єднаннi з концентрованою масою. Це з’єднання складається з деякої областi i великої кiлько-
стi N = O(ε−1) тонких стержнiв. Густина з’єднання є величиною порядку O(ε−α) на стержнях
(концентрацiя маси при α > 0) та O(1) поза стержнями. Можливi три якiсно рiзнi випадки в
асимптотичнiй поведiнцi власних значень та власних функцiй: 0 ≤ α < 2, α = 2, α > 2. Головна
увага придiляється першому випадку.
1. Introduction and statement of the problem. Vibration systems with a concentration of mass
on a small set of diameter O(ε) have been studied for a long time. It is experimentally establi-
shed that such concentration leads to the big reduction of the main frequency and to the big
localization of vibrations. The new impulse in these research was given by E. Sanchez-Palencia
in the paper [1] in which the effect of local vibrations was mathematically described. Then many
articles appeared (see [2 – 9] and other) that deal with the asymptotic behavior of vibrations of
a body containing a small region (many small regions) where the density is very much higher
than elsewhere.
In this paper we investigate free vibrations of a plane thick periodic junction Ωε with
concentrated masses on the fine rods. The asymptotic method developed in [10 – 13] for peri-
odic thick junctions is used. Some results have already been announced in [14].
The junction Ωε consists of the junction’s body
Ω0 = {x ∈ R2 : 0 < x1 < a, 0 < x2 < γ(x1)},
and a large number N of the fine rods Gε =
⋃N−1
j=0 Gjε,
Gjε = {x ∈ R2 : |x1 − ε (j + 1/2)| < εh/2, x2 ∈ (−1, 0]}, j = 0, 1, . . . , N − 1,
c© T. A. Mel’nyk, 1999 511
i.e., Ωε = Ω0 ∪ Gε. Here γ ∈ C∞([0, a]), 0 < γ0 = minx1∈[0,a] γ(x1); h is a fix number from
the interval (0, 1); N is a large positive integer, therefore ε = a/N is a small discrete parameter
which characterizes the distance between the rods and their thickness.
We consider the spectral boundary value problem
−∆x u(ε, x) = λ(ε) ρ1(ε, x)u(ε, x), x ∈ Ωε,
∂ν u(ε, x) = 0, x ∈ ∂Ωε ∩ {x : x2 ≥ 0},
u(ε, x) = 0, x ∈ Γε = ∂Ωε ∩ {x : x2 < 0},
(1)
where ∂ν = ∂/∂ν is the outward normal derivative; and ρ1(ε, x) = 1 if x ∈ Ω0, and ρ1(ε, x) =
= ε−α if x ∈ Gε; α is a nonnegative parameter.
For each 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}, that are orthonormali-
zed by the following way
(un, um)Ω0 + ε−α(un, um)Gε = 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 ei-
genfunctions {un(ε, ·) : n ∈ N} as ε → 0 (N → +∞). If α > 0, then the passage to the limit is
accompanied by the concentrated masses on the joined thin domains G0
ε, . . . , G
N−1
ε .
As we see in section 2, there are three qualitatively different cases in the asymptotic behavi-
or of the eigenvalues and eigenfunctions: 0 ≤ α < 2, α = 2, α > 2. Here, we consider the case
0 ≤ α < 2. Some remarks for the other cases are given in Remark 2 and in the conclusion.
2. Auxiliary inequalities. The case 0 ≤ α ≤ 2. Consider the space H1(Ωε,Γε) formed by
functions of the Sobolev spaceH1(Ωε) whose traces vanish on Γε. In this subspace we introduce
along with the norm ‖u‖1 = (
∫
Ωε
(|∇u|2 + ρ1u
2) dx)1/2 a new norm ‖ · ‖ε that is generated by the
scalar product
〈u, v〉ε =
∫
Ωε
∇u · ∇v dx.
Denote the space H1(Ωε,Γε) with this scalar product by Hε.
Lemma 1. For ε small enough, the norms ‖ · ‖1 and ‖ · ‖ε are equivalent, i.e., there exist
positive constants c1, ε0, such that for all ε ∈ (0, ε0) the following inequalities hold :
‖u‖ε ≤ ‖u‖1 ≤ c1‖u‖ε , u ∈ Hε. (4)
Proof. In (4), it is not obvious that the second inequality holds. Suppose the contrary. Then
there exist sequences {εm : m ∈ N}, {vm : m ∈ N} ∈ Hεm , such that limm→0 εm = 0 ,
‖vm‖1 = 1 , (5)
‖vm‖εm =
∫
Ωεm
|∇vm|2 dx < m−1. (6)
512 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 4
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 (6) it follows that the sequence {vm} is
a Cauchy sequence also in H1(Ω0) :
‖vm − vn;H1(Ω0)‖2 ≤ ‖vm − vn;L2(Ω0)‖2 +m−1 + n−1.
Hence, {vm} converges in this space to some element v0 ∈ H1(Ω0).
By virtue of the Friedrich inequality we have
ε−α
∫
Gεm
v2
m dx ≤ ε−α+2
∫
Ωεm
(∂1vm)2 dx , (7)
where ∂i = ∂/∂xi, i = 1, 2. Granting this estimate, we obtain from (5) and (6) that
1 = ‖vm‖1 −→
∫
Ω0
v2
0 dx as m→∞ ;
∫
Ω0
|∇v0|2 dx = 0 .
This means that v0 = const = |Ω0|−1/2 in Ω0 , where |Υ| is the measure of a domain Υ in R2.
On the one hand, from the trace theorem for functions in Sobolev spaces and the Corollary
1.7 [5], it follows that∫
Qεm
v2
m dx −→ h |Ω0| a as m→∞ ,
where Qε = Gε ∩ {x2 = 0}. On the other hand, we have∫
Qεm
v2
m dx ≤
∫
Gεm
(∂2vm)2 dx < m−1 → 0 as m→∞ .
The lemma is proved.
Remark 1. It should be noted that here and further all constants {ci} in asymptotic inequali-
ties are independent of the parameter ε.
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 v ∈ Hε the following integral identity holds:
〈u(ε, ·), v〉ε = λ(ε) (ρ1(ε, ·)u(ε, ·), v)Ωε . (8)
In this case the function u(ε, ·) is called an eigenfunction that corresponds to the eigenvalue λ(ε).
Define an operator A(1)
ε : Hε 7−→ Hε by
〈A(1)
ε u, v〉ε =
∫
Ωε
ρ1uv dx , u, v ∈ Hε . (9)
It is easy to verify that this operator is self-adjoint, positive, compact, and
‖A(1)
ε uε‖ε ≤ c1‖u‖ε , u ∈ Hε. (10)
ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 4 513
Now we can rewrite the integral identity (8) as the spectral problem for the operator A(1)
ε :
A
(1)
ε u(ε, ·) = λ−1(ε)u(ε, ·).
Thus, the eigenvalues of problem (1) form the sequence (2), with the classical convention of
repeated eigenvalues. Let us prove some inequalities for these eigenvalues. Let L0(ṽ1, . . . , ṽn)
be the n-dimensional subspace of Hε that is spanned on n linearly independent functions
ṽk =
{
v+
k , x ∈ Ω0;
0, x ∈ Gε ,
k = 1, . . . , n , (11)
where v+
1 , . . . , v
+
n are orthonormal in L2(Ω0) eigenfunctions of the problem
−∆x v
+
k (x) = µk v
+
k (x), x ∈ Ω0,
∂ν v
+
k (x) = 0, x ∈ ∂Ω0 ∩ {x : x2 > 0},
v+
k (x) = 0, x ∈ ∂Ω0 ∩ {x : x2 = 0}.
(12)
By virtue of the minimax principle for eigenvalues, we have
λn(ε) = min
E∈En
max
0 6=v∈E
∫
Ωε
|∇v|2 dx∫
Ωε
ρ1v2 dx
≤ max
0 6=v∈L0
∫
Ω0
|∇v|2 dx∫
Ω0
v2 dx
= µn . (13)
Here En is a set of all subspaces of Hε with the dimension n.
Taking into account conditions (3) and the second inequality (4), we obtain from the integral
identity (8) the lower estimates for the eigenvalues
λn(ε) = ‖un(ε, ·)‖2ε ≥ c0‖un(ε, ·)‖21 ≥ c0
∫
Ωε
ρ1(ε, x)u2
n(ε, x) dx = c0 > 0 , (14)
where c0 depend neither on ε nor on n.
Using inequality (13) and conditions (3), we deduce from (8) the following estimates for the
eigenfunctions∫
Ωε
|∇un(ε, x)|2 dx ≤ c(n) . (15)
The case α > 2. Let us consider the following n-dimensional subspace Lε(φ1, . . . , φn) of
Hε that is spanned on the linearly independent functions
φk =
0, x ∈ Ω0;
sin
π(2x1 − ε(1 + 2j − h))
2εh
sinπkx2, x ∈ Gjε,
k = 1, . . . , n .
Then we get
λn(ε) = min
E∈En
max
0 6=v∈E
∫
Ωε
|∇v|2 dx∫
Ωε
ρ1v2 dx
≤ max
0 6=v∈Lε
∫
Gε
|∇v|2 dx
ε−α
∫
Gε
v2 dx
≤ c(n)εα−2 . (16)
514 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 4
Taking into account inequality (16), we make the following change of spectral parameter
λ(ε) = εα−2λ̃(ε) (17)
in problem (1). As a result, we have the problem
−∆x u(ε, x) = λ̃(ε) ρ2(ε, x)u(ε, x), x ∈ Ωε,
∂ν u(ε, x) = 0, x ∈ ∂Ωε ∩ {x : x2 ≥ 0},
u(ε, x) = 0, x ∈ Γε,
(18)
where ρ2(ε, x) = εα−2 if x ∈ Ω0, and ρ2(ε, x) = ε−2 if x ∈ Gε.
By analogy with Lemma 1 we prove the following lemmas.
Lemma 2. For ε small enough, the norms ‖u‖2 = (
∫
Ωε
(|∇u|2 + ρ2u
2) dx)1/2 and ‖ · ‖ε are
equivalent.
Lemma 3. For ε small enough, the following inequality holds (u, u)Ω0 ≤ ‖u‖ε, u ∈ Hε.
Changing ρ2(ε, ·) instead of ρ1(ε, ·), we can repeat definition 1 for problem (18), define an
operator A(2)
ε : Hε 7−→ Hε by formula (9) and obtain for one estimate (10). By repeating the
previous argument and using Lemma 2, we deduce the following estimates
0 < c0 ≤ λ̃n(ε) ≤ c(n), ‖un‖2ε ≤ c(n) (19)
for eigenvalues {0 < λ̃1(ε) ≤ . . . ≤ λ̃n(ε) ≤ . . . } of problem (18) and corresponding ei-
genfunctions, but in this case these eigenfunctions are orthonormalized by the following way
εα−2(un, um)Ω0 + ε−2(un, um)Gε = δn,m, {n, m} ∈ N. (20)
Remark 2. According to Lemma 3 and estimates (19) for the eigenfunctions, the first term
in (20) tends 0 as ε → 0. Taking into account (7), (15), we can state the same for the second
term in (3), if 0 ≤ α < 2. Thus, there are three qualitatively different cases in the asymptotic
behavior of the eigenvalues and the eigenfunctions: 0 ≤ α < 2, α = 2, α > 2. As we see below,
in the first case the energy of the free vibrations is concentrated in the junction’s body. It should
be noted that in the other cases the energy is concentrated both in the junction’s body and in
the fine rods.
3. Junction-layer problems. Let us introduce the „rapid” coordinates η = ε−1x in problem
(1). Passing to ε = 0 , we see that the plane cylinder G0
ε is transformed into the semi-infinite
strip Π− = Ih × (−∞, 0], where Ih =
(1− h
2
,
1 + h
2
)
; and the set Ω0 is transformed into the
first octant {η : ηi > 0 , i = 1, 2}. Taking into account the periodicity of the cylinders {Gjε : j =
= 0, . . . , N−1},we can regard that the union Π of the semi-strips Π− and Π+ = (0, 1)×(0,+∞)
is the base domain in which the junction-layer problems have to be considered. Obviously,
solutions of these junction-layer problems must be 1-periodic in η1, i. e.,
∂kη1
Z(η)|η1=0 = ∂kη1
Z(η)|η1=1 , η ∈ ∂Π+, η2 > 0, k = 0, 1. (21)
Let us investigate some properties of solutions to the following junction-layer problem
−∆ηη Z(η) = F (η) , η ∈ Π,
Z(η) = 0 , η ∈ ∂Π− \ Ih,
∂η2Z(η1, 0) = 0 , (η1, 0) ∈ ∂Π+ \ Ih,
∂kη1
Z(η)|η1=0 = ∂kη1
Z(η)|η1=1 , η ∈ ∂Π+, η2 > 0, k = 0, 1.
(22)
ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 4 515
At first we study the solvability of this problem. In this connection we use the scheme gi-
ven in [12]. Let Ĉ∞0 (Π) be a space of infinitely differentiable functions in Π that satisfy the
periodical condition (21), the Dirichlet condition on ∂Π− \ Ih, and are finite in η2, i. e., ∀v ∈
∈ Ĉ∞0 (Π) ∃R > 0 ∀η ∈ Π |η2| ≥ R : v(η) = 0. Let H be the completion of the space
Ĉ∞0 (Π) by norm ‖u‖H = (‖∇ηu‖2L2(Π) + ‖ρ0u‖2L2(Π))
1/2, where ρ0(η2) = (1 + η2)−1 if η2 ≥ 0,
and ρ0(η2) = 1 if η2 < 0.
We will call a function Z a generalized solution of problem (22) if for all functions v ∈ H
the following integral identity holds∫
Π
∇ηZ · ∇ηv dη =
∫
Π
Fv dη . (23)
Lemma 4. Let ρ−1
0 F ∈ L2(Π). Then there exists a unique solution Z ∈ H of problem (22).
Proof. We rewrite identity (23) in the form
〈Z, v〉 −
∫
Π0,2
Zv dη =
∫
Π
Fv dη , (24)
where Πα,β = {η ∈ Π : α < η2 < β}, and
〈u, v〉 =
∫
Π
∇ηu · ∇ηv dη +
∫
Π0,2
uv dη . (25)
We show that the new scalar product (25) generates an equivalent norm in H. It is obvious
that 〈u, u〉 ≤ c ‖u‖2H , u ∈ H. The inverse inequality with another constant follows from
Friedrich’s inequality; from Hardy’s inequality
+∞∫
0
(1 + η2)−2φ2(η2) dη2 ≤ 4
+∞∫
0
|∂η2φ|2 dη2 ∀φ ∈ C1([0,+∞)), φ(0) = 0;
and the following inequality∫
Π
ρ2
0(η2)u2(η) dη ≤
∫
Π−
u2 dη +
∫
Π0,2
ρ2
0 u
2 dη +
∫
Π
ρ2
0((1− χ(η2))u)2 dη ≤
≤ c1
∫
Π−
(∂η1u)2 dη +
∫
Π0,2
ρ2
0u
2 dη +
∫
Π+
(∂η2u)2 dη
≤ c2〈u, u〉. (26)
Here χ ∈ C∞(R), 0 ≤ χ ≤ 1, and
χ(η2) =
{
1, |η2| ≤ 1;
0, |η2| ≥ 2 .
(27)
516 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 4
Due to the conditions of Lemma 4 and to inequality (26), the right-hand side of identity
(23) defines a linear continuous functional in H. As the embedding H ⊂ L2(Π0,2) is compact,
there exists a self-adjoint positive compact operator A : H 7→ H such that
〈Au, v〉 =
∫
Π0,2
uv dη , {u, v} ∈ H .
Thus, we can rewrite identity (24) as the operator equation Z −AZ = f, and apply to it the
Fredholm’s theorems. It is obvious that every solution of the homogeneous problem (22) in the
spaceH is trivial. Therefore, the lemma is proved.
Remark 3. Let exp(δ0|η2|)F ∈ L2(Π) , δ0 > 0. Taking into account the properties of soluti-
ons to elliptic problems in semi-cylinders, it is easily seen that the solution Z to problem (22)
has the following asymptotics
Z(η) =
{
C +O(exp(−δ1η2)), η2 → +∞;
O(exp(δ1η2)), η2 → −∞ ,
(28)
where δ1 is some positive number.
Remark 4. If the function F from Lemma 4 is even or odd in η1 with respect to 1/2, then
the solution Z has the same symmetry. In fact, let for example F be even in η1 with respect to
1/2, i. e., F (η1, η2) = F (1 − η1, η2). Then, due to the symmetry of the domain Π and with the
substitution η1 = 1− η′1 in problem (22), we obtain that the difference Z(η1, η2) – Z(1− η1, η2)
is a solution of the homogeneous problem (22). By virtue of the uniqueness of such solution in
the spaceH, it follows that this difference vanishes.
Corollary 1. The homogeneous problem (22) has a solution Ξ0 /∈ H with the asymptotics
Ξ0(η) =
{
C0 + η2 +O(exp(−δ2η2)), η2 → +∞,
O(exp(δ2η2)), η2 → −∞,
(29)
and this solution is even in η1 with respect to 1/2.
Proof. The solution Ξ0 is sought in the form of a sum
Ξ0(η) = χ+(η2)η2 + Z0(η),
where Z0 ∈ H, and Z0 is the solution to the problem (22) with the right-hand sides F (η) =
= 2χ′+(η2) + χ′′+(η2)η2 =: F0(η2); χ+ is a smooth cut-off function that equals 1 if η2 ≥ 2, and
vanishes if η2 ≤ 1.
By virtue of Remarks 3 and 4, this solution Z0 is even in η1 with respect to 1/2, and has the
asymptotics
Z0(η) =
{
C0 +O(exp(−δ2η2)), η2 → +∞;
O(exp(δ2η2)), η2 → −∞ .
(30)
In order to find the constant C0 in (29) and (30), it is necessary to substitute the function
Ξ0 and Z0 into Green’s formula in Π−R,R, and to pass to the limit as R → ∞. As a result, we
obtain
C0 =
∫
Π
Ξ0(η)F0(η2) dη .
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Remark 5. The solutions Z0 and Ξ0 have singularities in the points η1 = (1 ± h)/2 (see
[15]). Nevertheless, taking into account the order of these singularities, we can apply to Z0 and
Ξ0 Green’s formula.
Remark 6. By analogy, we can show that the constant C in (28) equals
C =
∫
Π
Ξ0(η)F (η) dη.
4. Asymptotic estimates in the case 0≤ α < 2. Asymptotic approximations. Let µn and v+
n
be an eigenvalue and eigenfunction of problem (12). Define the function ṽn by formula (11),
and construct the approximation
Un(ε, x) = ṽn(x) + εχ0(x2)∂2v
+
n (x1, 0)Ξ̃0(x/ε) , x ∈ Ωε , (31)
where χ0(x2) = χ(2x2/r0) , r0 = min(γ0, 1), the function χ is defined by (27);
Ξ̃0(η) =
{
Ξ0(η)− η2 , η ∈ Π+;
Ξ0(η) , η ∈ Π−,
and Ξ0 is the solution to the homogeneous problem (22) with asymptotic (29).
It is easily seen that Un(ε, ·) ∈ Hε, and due to characteristics of Ξ0, the function Un(ε, ·)
satisfy the boundary conditions of problem (1). Substituting {Un(ε, ·), µn} into problem (1) in
place of {u(ε, ·), λ(ε)}, we find that for any ψ ∈ Hε∫
Ωε
(∇Un · ∇ψ − µnρ1Unψ) dx = Φε(ψ) , (32)
where
Φε(ψ) = −εµn
∫
Ωε
ρ1(ε, x)χ0(x2)∂2v
+
n (x1, 0)Ξ̃0(x/ε)ψ(x) dx+
+ε
∫
Ωε
χ′0(x2)∂2v
+
n (x1, 0)
(
Ξ̃0(x/ε)∂2ψ(x)− ε−1∂η2(Ξ̃0)(x/ε)ψ(x)
)
dx+
+ε
∫
Ωε
χ0(x2)∂2
12v
+
n (x1, 0)
(
Ξ̃0(x/ε)∂1ψ(x)− ε−1∂η1(Ξ̃0)(x/ε)ψ(x)
)
dx. (33)
In order to estimate the terms in (33), we use the following lemma.
Lemma 5. Assume that Z is a function, 1-periodic in η1, belonging to the space L2(Π) and
exponentially decreasing at infinity, i.e., there exist positive constantsC0, R0, β0 such that |Z(η)| ≤
C0 exp(−β0|η2|) if |η2| ≥ R0. Then for any δ > 0 there exist positive constants C1, ε0 such that
for all ε ∈ (0, ε0) the following inequality is valid:∣∣∣∫
Ωε
Z(x/ε)ψ(x) dx
∣∣∣ ≤ C1ε
1−δ‖ψ‖ε, ψ ∈ Hε.
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Proof. Set Bε,δ = Ωε ∩ (0, a)× (−ε1−2δ, ε1−2δ), δ > 0. Then∣∣∣∫
Ωε
Z(x/ε)ψ(x) dx
∣∣∣ ≤ ∣∣∣ ∫
Bε,δ
Z(x/ε)ψ(x) dx
∣∣∣+
∣∣∣ ∫
Ωε\Bε,δ
Z(x/ε)ψ(x) dx
∣∣∣.
The properties of the function Z lead us to the conclusion that the second summand in this
inequality decreases exponentially as ε → 0. With the help of Lemma 1.5 [5], we estimate the
first summand:∣∣∣ ∫
Bε,δ
Z(x/ε)ψ(x) dx
∣∣∣ ≤ ( ∫
Bε,δ
Z2(x/ε) dx
)1/2
‖ψ‖L2(Bε,δ) ≤ cε
1−δ‖Z‖L2(Π)‖ψ‖ε.
The lemma is proved.
Using (29) and Lemma 5, we deduce that
|Φε(ψ)| ≤ c(δ) ε1−δ‖ψ‖ε for any δ > 0 . (34)
Remark 7. The constant c(δ) in inequality (34) depends on the quantities
maxx1∈[0,a] |∂i+1
i2 v+
n (x1, 0)|, i = 0, 1. Applying the even extension, with respect to the line
x1 = 0 and x1 = a, to problem (12), we establish that the function v+
n and its derivatives have no
singularities at the points (0, 0) and (0, a). Then, by virtue of classical results on the smoothness
of solutions to boundary value problems, the quantities mentioned above are bounded.
Thus, the right-hand side of integral equality (32) is a linear bounded functional on the
space Hε, and its norm is bounded by c(δ) ε1−δ, δ > 0. On the basis of the definition of the
operator A(1)
ε (see (9)) and the Riesz theorem, we get from (32) the inequality
‖Un(ε, ·)− µnA(1)
ε Un(ε, ·)‖ε ≤ c(δ) ε1−δ , δ > 0 , (35)
which, by virtue of the first part of Lemma 12 [16], partially justifies the constructed asymptotics
for the solutions of problem (1) :
min
k∈N
|µ−1
n − λ−1
k (ε)| ≤ ‖Un‖−1
ε ‖A(1)
ε Un − µ−1
n Un‖ε = O(ε1−δ) . (36)
Convergence theorem and asymptotic estimates. To prove the convergence theorem, first
we observe that there exists an extension operator
Pε : Hε 7→ H1(Ω,Γ−1) such that
∫
Ω
|∇Pεu|2 dx ≤ c‖u‖ε, u ∈ Hε. (37)
Here Ω is the interior of the union Ω0 ∪ D; D = (0, a) × (−1, 0); Γ−1 = {x : 0 < x1 < a,
x2 = −1}; and functions that belong to the subspace H1(Ω,Γ−1) of H1(Ω) vanish on Γ−1.
We construct this operator in the following way. At first a function u ∈ Hε is prolonged by
zero on the set Ωε ∪Dε, where Dβ = [0, a]× [−1,−β]. Further extension of u to Ω is performed
similarly as for perforated domains [5].
Theorem 1. Let {λn(ε) : n ∈ N} and {0 < µ1 < µ2 ≤ . . . ≤ µn . . . } be the ordered
sequences sequences of eigenvalues of problems (1) and (12) respectively; let {un(ε, ·) : n ∈ N}
be the corresponding sequence of eigenfunctions satisfying condition (3). Then for any n ∈ N
λn(ε) → µn as ε → 0 ;
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there is a subsequence of {ε} (still denoted by {ε}) such that
Pεun(ε, ·)→ ϕ̃n weakly in H1(Ω,Γ−1) as ε→ 0 ,
where ϕ̃n(x) = ϕ+
n (x) if x ∈ Ω0, and ϕ̃n(x) = 0 if x ∈ D; {ϕ+
n } are eigenfunctions of problem
(12) that are orthonormal in the space L2(Ω0).
Proof. Bearing in mind the boundedness of λn(ε) in ε for fixed n (see (13), (14)), and
the inequalities (15) and (37), with the help of the diagonal process, one can choose a
subsequence of {ε} ( still denoted by {ε}) such that λn(ε) → µ∗n , and Pεun(ε, ·) →
→ ϕ̃n weakly in H1(Ω,Γ−1) as ε→ 0. From inequality (14), it follows that 0 < µ∗1 ≤ µ∗2 ≤
≤ . . . ≤ µ∗n ≤ . . . .
According to the Remark 2, we have that
δn,m = (un, um)Ω0 + ε−α(un, um)Gε → (ϕ+
n , ϕ
+
m)Ω0 as ε → 0 ,
whence ϕ+
n 6= 0.
Write the integral identity (8) for the eigenfunction un(ε, ·) with a test function v ∈ H1(Ω0)
that is equal to 0 on the interval [0, a] and on the set Gε, and pass to the limit as ε → 0. We get∫
Ω0
∇ϕ+
n (x) · ∇v(x) dx = µ∗n
∫
Ω0
ϕ+
n (x)v(x) dx.
This means that µ∗n is an eigenvalue of problem (12), and ϕ+
n is the corresponding eigenfunction.
Now we write (8) with the following test function
v(x) =
{
0 , x ∈ Ω0;
un(ε, x)ψ(x) , x ∈ Gε ,
where ψ ∈ C∞(D); ψ(x) > 0 if x ∈ [0, a]× [−1, 0) and ψ(x1, 0) = 0, x1 ∈ [0, a]. We obtain that∫
Gε
ψ|∇un|2 dx = −
∫
Gε
un∇un · ∇ψ dx+ λn(ε)ε−α
∫
Gε
ψ u2
n dx. (38)
Fixing some β > 0 and taking into account (7), (13), (15), we deduce from (38)
0 < cβ
∫
Dβ
|∇Pεun|2 dx = cβ
∫
Gε∩{x: x2<−β}
|∇un|2 dx ≤ Cβ(ε+ ε2−α)‖un‖2ε,
if ε is small enough. Since Pεun → 0 in H1(Dβ ,Γ−1) as ε → 0 and β is arbitrary positive
number, ϕ̃n = 0 in D.
In order to complete the proof, it remains to show that
µ∗n = µn , n ∈ N . (39)
Let µk = µk+1 = . . . = µk+q−1 be an eigenvalue of multiplicity q. Let us show that there
exist exactly q eigenvalues of problem (1) with regard to multiplicity which tend to µk as ε→ 0.
This will mean that relations (39) are true.
520 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 4
Assume that there exist r eigenvalues {λni(ε) : i = 1, . . . , r} of problem (1) which tend to
µk and r > q. By the preceding arguments, we have for the corresponding eigenfunctions that
uni(ε, ·) → ϕ+
ni weakly in H1(Ω0) as ε → 0, where {ϕ+
ni : i = 1, . . . , r} are orthonormal
in L2(Ω0) eigenfunctions of problem (12). Thus, the eigenvalue µk has multiplicity r, but it is a
contradiction.
Now, let r be less then q and let v+
k+i , i = 0, 1, . . . , q − 1 be eigenfunctions of problem
(12) that correspond to the eigenvalue µk. With the help of these eigenfunctions, we construct
the approximations Uk+i , i = 0, 1, . . . , q − 1 , by formula (31), and arrive at inequality (35).
Applying the second part of Lemma 12 [16] to this inequality, we conclude that there exists a
linear combination of the eigenfunctions un1 , . . . , unr of problem (1)
R(i)
ε =
r∑
j=1
dij(ε)unj (ε, ·) , 0 < c1 ≤
r∑
j=1
d2
ij(ε) ≤ c2 , r < q,
such that ‖v+
k+i−R
(i)
ε ;L2(Ω0)‖ ≤ ciε1−δ , i = 1, . . . , q. Passing to the limit in these inequalities
over a suitable subsequence of {ε}, we get
v+
k+i(x) =
r∑
j=1
d∗ijϕ
+
nj (x), x ∈ Ω, 0 < c1 ≤
r∑
j=1
(d∗ij)
2 ≤ c2, i = 0, 1, . . . , q − 1.
But this contradicts to the linear independence of the functions v+
k , . . . , v
+
k+q−1.
Since the above reasoning holds for any subsequence of {ε} chosen at the beginning of the
proof, we have λn(ε) → µn as ε → 0. The theorem is proved.
The above theorem allows us to obtain asymptotic estimates for the eigenvalues and ei-
genfunctions immediately from (36) and Lemma 12 [16].
Theorem 2. For any δ > 0, n ∈ N, and ε small enough, we have
|λn(ε)− µn| ≤ c1(n, δ) ε1−δ.
Theorem 3. Assume that µn = µn+1 = . . . = µn+q is an eigenvalue of problem (12) with
multiplicity q, and that v+
n , . . . , v
+
n+q−1 are the corresponding eigenfunctions. Then there exist
constants ε0 , c2(n, δ), and {dik} such that for ε ∈ (0, ε0) the following inequalities hold :
∥∥∥Un+i(ε, ·)−
q−1∑
k=0
dik un+k(ε, ·)
∥∥∥
ε
≤ c2(n, δ) ε1−δ, i = 0, 1, . . . , q − 1 ,
where {Un+i : i = 0, 1, . . . , q − 1} are defined by (31).
It follows from Theorem 3, Lemma 1, and Lemma 5 the following corollary.
Corollary 2. Let µn be a simple eigenvalue of problem (12). Then∥∥∥‖Un‖−1
ε Un(ε, ·)− λ−1/2
n (ε) un(ε, ·)
∥∥∥
H1(Ω0)
≤ c2(n, δ) ε1−δ,
∫
Gε
|∇un(ε, ·)|2 dx ≤ c3(n, δ) ε1−δ, δ > 0.
ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 4 521
Conclusion. It follows from above obtained results that the energy of the free vibrations is
concentrated in the junction’s body, and there exists no reduction of the frequencies in the case
0 ≤ α < 2. Similar situations were observed in [1 – 10], when the density on small sets is not so
big.
In the other case we observe the big reduction. If α = 2, then the eigenvalues {λn(ε)} tend
to π2h−2 (h is the width of the strip Π−) as ε→ 0, and their „splitting” occurs only in the second
term of the asymptotics, i.e., λn(ε) = π2h−2 + ε2τn +O(ε3), where τn is an eigenvalue of some
operator-function. If α > 2, then all eigenvalues {λn(ε)} tend to zero, and have the asymptotics
λn(ε) = εα−2π2h−2 + εαβn +O(εα+1).
Acknowledgement. The author is grateful to the Alexander von Humboldt Foundation and
Prof. W.L. Wendland for the possibility to carry out this research at the University of Stuttgart.
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Received 19.04.99
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