Asymptotic analysis of a vibrating system containing stiff-heavy and flexible-light parts
A model of a strongly inhomogeneous medium with simultaneous perturbation of the rigidity and mass density is studied. The medium has strongly contrasting physical characteristics in two parts with the ratio of rigidities being proportional to a small parameter ". Additionally, the ratio of mas...
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irk-123456789-1242622017-09-24T03:03:13Z Asymptotic analysis of a vibrating system containing stiff-heavy and flexible-light parts Babych, N. Golovaty, Yu. A model of a strongly inhomogeneous medium with simultaneous perturbation of the rigidity and mass density is studied. The medium has strongly contrasting physical characteristics in two parts with the ratio of rigidities being proportional to a small parameter ". Additionally, the ratio of mass densities is of order " ε⁻¹. We investigate the asymptotic behaviour of the spectrum and eigensubspaces as ε → 0. Complete asymptotic expansions of eigenvalues and eigenfunctions are constructed and justified. We show that the limit operator is nonself-adjoint in general and possesses two-dimensional Jordan cells in spite of the singular perturbed problem is associated with a self-adjoint operator in appropriated Hilbert space Lε. This may happen if the metric in which the problem is self-adjoint depends on small parameter " in a singular way. In particular, it leads to a loss of completeness for the eigenfunction collection. We describe how root spaces of the limit operator approximate eigenspaces of the perturbed operator. 2008 Article Asymptotic analysis of a vibrating system containing stiff-heavy and flexible-light parts / N. Babych, Yu. Golovaty // Нелинейные граничные задачи. — 2008. — Т. 18. — С. 194-217. — Бібліогр.: 22 назв. — англ. 0236-0497 MSC (2000): 35P20; 74H45; 35J25 http://dspace.nbuv.gov.ua/handle/123456789/124262 en Інститут прикладної математики і механіки НАН України |
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A model of a strongly inhomogeneous medium with simultaneous perturbation of the rigidity and mass density is studied. The medium has strongly contrasting physical characteristics in two parts with the ratio of rigidities being proportional to a small parameter ". Additionally, the ratio of mass densities is of order " ε⁻¹. We investigate the asymptotic behaviour of the spectrum and eigensubspaces as ε → 0. Complete asymptotic expansions of eigenvalues and eigenfunctions are constructed and justified. We show that the limit operator is nonself-adjoint in general and possesses two-dimensional Jordan cells in spite of the singular perturbed problem is associated with a self-adjoint operator in appropriated Hilbert space Lε. This may happen if the metric in which the problem is self-adjoint depends on small parameter " in a singular way. In particular, it leads to a loss of completeness for the eigenfunction collection. We describe how root spaces of the limit operator approximate eigenspaces of the perturbed operator. |
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Babych, N. Golovaty, Yu. |
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Babych, N. Golovaty, Yu. Asymptotic analysis of a vibrating system containing stiff-heavy and flexible-light parts |
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Babych, N. Golovaty, Yu. |
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Babych, N. |
| title |
Asymptotic analysis of a vibrating system containing stiff-heavy and flexible-light parts |
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Asymptotic analysis of a vibrating system containing stiff-heavy and flexible-light parts |
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Asymptotic analysis of a vibrating system containing stiff-heavy and flexible-light parts |
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Asymptotic analysis of a vibrating system containing stiff-heavy and flexible-light parts |
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Asymptotic analysis of a vibrating system containing stiff-heavy and flexible-light parts |
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asymptotic analysis of a vibrating system containing stiff-heavy and flexible-light parts |
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Інститут прикладної математики і механіки НАН України |
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2008 |
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Asymptotic analysis of a vibrating system containing stiff-heavy and flexible-light parts / N. Babych, Yu. Golovaty // Нелинейные граничные задачи. — 2008. — Т. 18. — С. 194-217. — Бібліогр.: 22 назв. — англ. |
| work_keys_str_mv |
AT babychn asymptoticanalysisofavibratingsystemcontainingstiffheavyandflexiblelightparts AT golovatyyu asymptoticanalysisofavibratingsystemcontainingstiffheavyandflexiblelightparts |
| first_indexed |
2025-07-09T01:09:01Z |
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2025-07-09T01:09:01Z |
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1837129627910275072 |
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194 Нелинейные граничные задачи 18, 194-217 (2008)
c©2008. N. Babych, Yu. Golovaty
ASYMPTOTIC ANALYSIS OF A VIBRATING
SYSTEM CONTAINING STIFF-HEAVY
AND FLEXIBLE-LIGHT PARTS
A model of a strongly inhomogeneous medium with simultaneous pertur-
bation of the rigidity and mass density is studied. The medium has strongly
contrasting physical characteristics in two parts with the ratio of rigidities being
proportional to a small parameter ε. Additionally, the ratio of mass densities
is of order ε
−1. We investigate the asymptotic behaviour of the spectrum and
eigensubspaces as ε → 0. Complete asymptotic expansions of eigenvalues and
eigenfunctions are constructed and justified.
We show that the limit operator is nonself-adjoint in general and possesses
two-dimensional Jordan cells in spite of the singular perturbed problem is asso-
ciated with a self-adjoint operator in appropriated Hilbert space Lε. This may
happen if the metric in which the problem is self-adjoint depends on small
parameter ε in a singular way. In particular, it leads to a loss of completeness for
the eigenfunction collection. We describe how root spaces of the limit operator
approximate eigenspaces of the perturbed operator.
Keywords and phrases: spectral analysis, asymptotic analysis, stiff problem,
eigenvalue
MSC (2000): 35P20; 74H45; 35J25
Introduction.
We consider a model of strongly inhomogeneous medium con-
sisting of two nearly homogeneous components. Assuming a strong
contrast of the corres ponding stiffness coefficients k1 � k2, we get
that their ratio k1/k2 has a small order, which we denote by ε. In
general, the mass densities r1 and r2 in two parts could be quite
different as well or could be the same. We model this assuming that
the density ratio r1/r2 is proportional to ε−m. We investigate how the
resonance vibrations of the medium change if the parameter ε tends
to 0. In the one-dimensional case we consider the spectral problem
d
dx
(
kε(x)
duε
dx
)
+ λε rε(x)uε = 0 in (a, b),
α1u
′
ε(a) + α0uε(a) = 0, β1u
′
ε(b) + β0uε(b) = 0,
where (a, b) is an interval in R containing the origin and
kε(x)=
{
k(x) for x ∈ (a, 0)
εκ(x) for x ∈ (0, b),
rε(x)=
{
ε−mr(x) for x ∈ (a, 0)
ρ(x) for x ∈ (0, b).
(1)
Asymptotic analysis of vibrating system 195
Here k, r and κ, ρ are smooth positive functions in intervals [a, 0] and
[0, b] respectively. At point x = 0 of discontinuity of the coefficients we
assume that transmission conditions uε(−0) = uε(+0), (ku′ε)(−0) =
ε(κu′ε)(+0) hold.
Of course, the limit properties of spectrum depend on the power
m characterizing the density ratio. Intuitively, we expect that for
large values of m the mass density perturbation has to be dominating
whereas for small m the rigidity perturbation has to be leading.
Then it has to be at least one critical point m separating the cases.
It appears to be truth exactly for m = 1, when the mass density
perturbation is strictly inverse to the stiffness one.
This paper is devoted to the critical case m = 1. We consider
the Dirichlet problem
(k(x) u′ε)
′
+ ε−1λε r(x) uε = 0, x ∈ (a, 0), (2)
ε (κ(x) u′ε)
′
+ λε ρ(x) uε = 0, x ∈ (0, b), (3)
uε(−0) = uε(+0), (ku′ε)(−0) = ε (κu′ε)(+0), (4)
uε(a) = 0, uε(b) = 0 (5)
and investigate the asymptotic behavior of eigenvalues λε and eigen-
functions uε as ε→ 0.
After a proper change of spectral parameter problem (2)-(5) can
be represented as a problem with perturbation of the transmission
conditions only (cf. the example with constant coefficients below). At
first blush, the problem looks very simple. But the point is that the
problem shows a complicated picture of the eigenspace bifurcation. In
Section we prove that the limit behavior of the spectrum is described
in terms of a nonself-adjoint operator that has in general multiple
eigenvalues and two-dimensional root spaces. At the same time, (2)-
(5) is associated with a self-adjoint operator in the weighted space
Lε with the following scalar product and norm
(φ, ψ)ε = ε−1(rφ, ψ)L2(a,0) + (ρφ, ψ)L2(0,b), ‖φ‖ε =
√
(φ, φ)ε . (6)
It is obvious that for each fixed ε > 0 the spectrum of (2)-(5) is
real, discrete and simple, 0 < λε
1 < λε
2 < · · · < λε
j < · · · → ∞ as
j → ∞ and the corresponding real-valued eigenfunctions {uε,j}∞j=1
form an orthogonal basis in Lε. How may it happen? The metric
in Lε for which the perturbed problem is self-adjoint, depends on
small parameter ε in a singular way. In Sections , we construct and
196 N. Babych, Yu. Golovaty
justify the complete asymptotic expansions of eigenvalues and eigen-
functions. Therefore there exist pairs of closely adjacent eigenvalues
λε
j and λε
j+1 being the bifurcation of double limit eigenvalues. Al-
though the corresponding eigenfunctions uε,j and uε,j+1 remain ortho-
gonal in Lε for all ε > 0, they make an infinitely small angle between
them in L2(a, b) with the standard metric and stick together at the
limit. In particular, it leads to the loss of completeness in L2(a, b) for
the limit eigenfunction collection. Nevertheless both uε,j and uε,j+1
converge to the same limit, a plane π(ε) being the linear span of
these eigenfunctions has regular asymptotic behaviour as ε → 0. In
fact, a root space π corresponding the double eigenvalue is the limit
position of plane π(ε) as ε→ 0, as is shown in Theorem 5. We actually
prove that the completeness property of the perturbed eigenfunction
collection passes into the completeness of eigenfunctions and adjoined
functions of the limit nonself-adjoint operator.
This work was motivated by [1, Ch.8], where the similar problem
for the Laplace operator has been considered. The authors have
handled the limit operator as the direct sum of two self-adjoint ope-
rators that nevertheless does not entirely explain the bifurcation
picture in perturbation theory of operators. The aim of this paper is
to present more rigorous and detailed study of the case in operator
framework.
Finally, let us remark that the vibrating systems with singularly
perturbed stiffness and mass density have been considered in many
papers. In the case of purely stiff models (with homogeneous mass
density), the asymptotic behavior of spectra have been studied in [6]
- [12]. Referring to problems with purely density perturbation often
involving domain perturbations, we mention [13]- [18] with the latter
including a broad literature overview in the area. Spectral properties
of vibrating systems with mass entirely neglected in a subdomain
were also studied in [19], [20]. The asymptotic results for the problems
with simultaneous perturbations of mass density and stiffness were
obtained in [21], [22].
1. Preliminaries.
We demonstrate an example where eigenvalue bifurcation is cal-
culated explicitly. If all coefficients in (2), (3) are constant we get the
Asymptotic analysis of vibrating system 197
eigenvalue problem
y′′ε + ω2
ε yε = 0, x ∈ (a, 0) ∪ (0, b), (7)
yε(a) = 0, yε(b) = 0, yε(−0) = yε(+0), y′ε(−0) = εy′ε(+0),(8)
where ω2
ε = ε−1λε. Then each non-zero solution can be represented
by
yε =
{
Aε sinωε(x− a) for x ∈ (a, 0),
Bε sinωε(x− b) for x ∈ (0, b),
with ωε > 0 and Aε, Bε ∈ R. By virtue of (8) we have
Aε sinωεa− Bε sinωεb = 0 and Aε cosωεa− εBε cosωεb = 0.
Looking for a non-zero solution of the algebraic system, we get the
characteristic equation
cosωεa sinωεb = ε sinωεa cosωεb. (9)
The latter easily gives existence of the limit ωε → ω as ε → 0
such that
cosωa sinωb = 0. (10)
Moreover, the root ω has to be positive. Obviously, if we suppose,
contrary to our claim, that ωε goes to 0 as ε → 0, then (9) can be
written in the equivalent form
cosωεa sinωεb
cosωεb sinωεa
= ε
for sufficiently small ε. A passage to the limit as ε → 0 and ωε → 0
leads to a contradiction, because the left-hand side converges towards
the negative number b/a.
If a and b are incommensurable number, then all roots of (10)
are simple. In fact, multiple roots exist iff 2n|a| = (2l−1)b for certain
natural l and n. Let us consider the case a = −1 and b = 2. Then
the lowest positive root ω = π/2 of (10) has multiplicity 2. On the
other hand, equation (9) admits the factorization
(
cosωε −
√
ε
2 + 2ε
) (
cosωε +
√
ε
2 + 2ε
)
sinωε = 0.
Hence the lowest eigenvalues ωε,1 = π
2
− arcsin
√
ε
2+2ε
, ωε,2 = π
2
+
arcsin
√
ε
2+2ε
are closely adjacent and converge to the same limit
198 N. Babych, Yu. Golovaty
π/2. The corresponding eigenfunctions yε,1 and yε,2 are defined up to
a constant factor as
yε,j(x) =
{
(−1)j
√
2ε/(1 + ε) sinωε,j(x+ 1) for x ∈ (−1, 0),
sinωε,j(x− 2) for x ∈ (0, 2).
(11)
We see at once that the angle in L2(−1, 2) between the eigenfunctions
yε,1 and yε,2 is infinitely small as ε→ 0, because both eigenfunctions
converge towards the same function
y∗(x) =
{
0 for x ∈ (−1, 0),
sin
π
2
(x− 2) for x ∈ (0, 2).
The point of the example is that the collection of eigenfunctions
{uε,j}∞j=1 loses the completeness property at the limit on account of
the double eigenvalues. We now turn to perturbed problem (2)-(5) in
the general case. To shorten formulas below, we introduce notation
Ia = (a, 0), Ib = (0, b) and
K(x) =
{
k(x) for x ∈ Ia
κ(x) for x ∈ Ib,
R(x) =
{
r(x) for x ∈ Ia
ρ(x) for x ∈ Ib.
Proposition 1. For each number j ∈ N eigenvalue λε
j of (2)-(5) is a
continuous function of ε ∈ (0, 1) and c ε < λε
j ≤ Cj ε with constants
c, Cj being independent of ε.
Proof. The continuity of eigenvalues with respect to the small para-
meter follows immediately from the mini-max principle
λε
j = min
Ej
max
v∈Ej
v 6=0
∫ 0
a
kv′2 dx+ ε
∫ b
0
κv′2 dx
ε−1
∫ 0
a
rv2 dx+
∫ b
0
ρv2 dx
, (12)
where the minimum is taken over all the subspaces Ej ⊂ H1
0 (a, b) with
dimEj = j. We consider the eigenfunctions v1, . . . , vj corresponding
to the lowest eigenvalues µ1, . . . , µj of the problem
(κ(x)v′)′ + µρ(x)v = 0, x ∈ Ib, v(0) = v(b) = 0. (13)
Extending each vk by zero to (a, 0) we get that the span M of
v1, . . . , vj is an j-dimensional subspace of H1
0 (a, b). Then
λε
j ≤ max
v∈M
∫ 0
a
kv′2 dx+ ε
∫ b
0
κv′2 dx
ε−1
∫ 0
a
rv2 dx +
∫ b
0
ρv2 dx
= max
v∈M
ε
∫ b
0
κv′2 dx
∫ b
0
ρv2 dx
= εµj, (14)
Asymptotic analysis of vibrating system 199
which establishes the upper estimate. Next, by the same mini-max
principle
λε
j > λε
1 = min
H1
0
(a,b)
∫ 0
a
kv′2 dx + ε
∫ b
0
κv′2 dx
ε−1
∫ 0
a
rv2 dx+
∫ b
0
ρv2 dx
≥
≥ k∗
r∗
min
H1
0
(a,b)
∫ 0
a
v′2 dx+ ε
∫ b
0
v′2 dx
ε−1
∫ 0
a
v2 dx+
∫ b
0
v2 dx
=
εk∗ ω
2
ε,1
r∗
≥ cε,
where k∗ = minx∈(a,b) K(x), r∗ = maxx∈(a,b)R(x) and ω2
ε,1 is the first
eigenvalue of problem (7)-(8) with constant coefficients. It remains
to note that ωε,1 → π/2. 2
2. Convergence Results and Properties of Limit
Problem .
Let us consider the eigenvalue problem
{
(K(x)u′)′ + µR(x)u = 0, x ∈ Ia ∪ Ib,
u(a) = 0, u(b) = 0, u(−0) = u(+0), u′(−0) = 0,
(15)
that will be referred to as the limit spectral problem. The spectrum
of (15) is discrete and real (see Th. 1 below). We introduce the space
H = {f ∈ H1
0 (a, b) : fa ∈ H2(a, 0) and fb ∈ H2(0, b)}, where fa and
fb are the restrictions of f to intervals Ia and Ib resp. Problem (15)
admits the variational formulation: to find µ ∈ C and a nontrivial
u ∈ H such that
∫ b
a
K u′φ′ dx+ κ(0)u′(+0)φ(0) = µ
∫ b
a
Ruφ dx (16)
for all φ ∈ C∞
0 (a, b). We first prove a conditional results.
Proposition 2. Given eigenvalue λε and the corresponding eigen-
function uε of (2)-(5), if ε−1λε → µ∗ and uε → u∗ in H2 weakly
on each intervals Ia, Ib and u∗ is different from zero, then µ∗ is an
eigenvalue of (15) with the eigenfunction u∗.
Proof. We make a change of spectral parameter λε = εµε in (2)-(5),
whereat we can reduce equation (3) by the first order of ε. Then each
pair (µε, uε) satisfies the integral identity
∫ b
a
K u′εφ
′ dx+ (1 − ε)κ(0)u′ε(+0)φ(0) = µε
∫ b
a
Ruεφ dx (17)
200 N. Babych, Yu. Golovaty
for all φ ∈ C∞
0 (a, b). The weak convergence of uε in H2(0, b) gives
the convergence uε → u∗ in C1(0, b), in particular, u′ε(+0) → u′∗(+0)
as well as u′ε(−0) → 0. Moreover, the limit function u∗ belongs to
H, since each uε is a continuous function at x = 0. A passage to the
limit in (17) implies that pair (µ∗, u∗) satisfies identity (16). Recall
that u∗ is different from zero, which completes the proof. 2
Before improving the convergent results, we first compute the
spectrum of the limit problem. Let us introduce space L = L2(r, Ia)⊕
L2(ρ, Ib), where L2(g, I) is a weighted L2-space with the norm ‖v‖ =
(∫
I
g|v|2
)1/2
. We consider two operators
A1 = −1
r
d
dx
k d
dx
in L2(r, Ia),
D(A1) =
{
u ∈ H2(Ia) : u(a) = 0, u′(0) = 0
}
,
A2 = −1
ρ
d
dx
κ
d
dx
in L2(ρ, Ib),
D(A2) =
{
u ∈ H2(Ib) : u(b) = 0
}
.
For problem (15) we assign the matrix operator
A =
(
A1 0
0 A2
)
in L,
D(A) =
{
(u1, u2) ∈ D(A1) ⊕D(A2) : u1(0) = u2(0)
}
.
The operator A is nonself-adjoint. Actually, it is easy to check that
A∗ =
(
Â1 0
0 Â2
)
,
D(A∗) =
{
(v1, v2) ∈ D(Â1) ⊕D(Â2) : (kv′1)(0) = (κv′2)(0)
}
,
where Â1 is the extension of operator A1 to D(Â1) =
{
u ∈ H2(a, 0) :
u(a) = 0
}
and Â2 is the restriction of A2 to D(Â2) =
{
u ∈ D(A2) :
u(0) = 0
}
. Let σ(A) and %(A) denote the spectrum and the resolvent
set of an operator A respectively. Let Rµ(A) denote the resolvent
(A− µI)−1 of an operator A, where I is the identity operator in L.
Definition. Let u be an eigenvector of A with eigenvalue µ. A solu-
tion u∗ to (A−µI)u∗ = u is called an adjoined vector of A (correspon-
ding to the eigenvalue µ).
Theorem 1.
Asymptotic analysis of vibrating system 201
(i) σ(A) = σ(A1) ∪ σ(Â2).
(ii) If µ belongs to σ(A) \
(
σ(A1) ∩ σ(Â2)
)
, then µ is a simple
eigenvalue. If µ ∈ σ(A1)∩ σ(Â2), then µ has multiplicity 2 and
the corresponding root space is generated by an eigenvector and
an adjoined vector of A.
(iii) The set of eigenvectors and adjoined vectors of A forms a com-
plete system in L.
Proof.
(i) Let us consider the equation (A−µI)u = f for fixed f ∈ L. In
the coordinate representation we have A1 u1−µu1 = f1, A2 u2−µu2 =
f2. If µ 6∈ σ(A1), then u1 = Rµ(A1)f1. In order to find u2 we introduce
the bounded intertwining operator Tµ : H2(Ia) → H2(Ib) that solves
the problem (κψ′)′ + µρψ = 0 in Ib, ψ(0) = g(0), ψ(b) = 0 for each
g ∈ H2(Ia). Note that Tµ is a well-defined operator for all µ ∈ %(Â2).
Then u2 = TµRµ(A1)f1 + Rµ(Â2)f2 and the resolvent of A can be
written in the form
Rµ(A) =
( Rµ(A1) 0
TµRµ(A1) Rµ(Â2)
)
. (18)
From the explicit representation of Rµ(A) it follows that sets σ(A)
and σ(A1) ∪ σ(Â2) coincide.
(ii) We suppose that µ ∈ σ(A1) \ σ(Â2). Then there exists an
eigenvector Uµ = (u1, Tµu1), where u1 is an eigenvector of A1 and,
that is the same, one is an eigenfunction of problem (kφ′)′ +µrφ = 0
in Ia, φ(a) = φ′(0) = 0. Note that µ is a simple eigenvalue of the
problem. Indeed, (A− µI)Uµ = 0 follows from the evident equality
(A2 − µI)Tµ = 0 for all µ ∈ %(Â2).
Suppose now that µ ∈ σ(Â2) \ σ(A1). Then operator A has the
eigenvector Vµ = (0, u2), where u2 is an eigenvector of Â2. In other
words, u2 is an eigenfunction of the Dirichlet problem (13). Note
that each point of σ(Â2) is a simple eigenvalue. Furthermore, the
first component u1 must be zero, since µ 6∈ σ(A1).
Finally we shall show that each point of intersection σ(A1) ∩
σ(Â2) is an eigenvalue of algebraic multiplicity 2. Obviously, vector
Vµ = (0, u2), which appears above, is an eigenvector of A in this case
too. Next we consider the system
A1v1 − µ v1 = 0, A2v2 − µ v2 = u2 (19)
202 N. Babych, Yu. Golovaty
determining adjoined vectors. If v1 = 0, then v2 must be a solution of
the boundary value problem (κφ′)′+µρφ = −ρu2 in Ib, φ(0) = φ(b) =
0, which is unsolvable. Actually, since µ ∈ σ(Â2), by the Fredholm
alternative the problem admits a solution iff
∫ b
0
ρ|u2|2 dx = 0. This
contradicts the fact that u2 is an eigenvector of Â2. Consequently
we have to assume that v1 is an eigenvector of A1 and examine the
problem (κv′2)
′ + µρv2 = −ρu2 in Ib, v2(0) = v1(0), v2(b) = 0. Here
the Fredholm alternative gives the solvability condition
κ(0)u′2(0)v1(0) = −
∫ b
0
ρ u2
2 dx. (20)
We satisfy one by normalization of v1, because u′2(0) is different from
zero. This condition assures the existence of v2 and a solution V ∗
µ =
(v1, v2) of system (19). Vector V ∗
µ is the adjoined vector of A. Pair
{Vµ, V
∗
µ } forms a basis in the root space that corresponds to µ.
The last statement of the theorem follows from the Keldysh
theorem [3]. 2
We investigate the limit behaviour of eigenfunctions uε,n nor-
malized by conditions
∫ b
a
R(x) u2
ε,j(x) dx = 1, u′ε,j(b) > 0. (21)
Let us enumerate the eigenvalues of operator A in increasing order
and repeat each eigenvalue according to its multiplicity: µ1 ≤ µ2 ≤
· · · ≤ µj ≤ · · · . The next statement improves the conditional results
of Proposition 2.
Theorem 2. There exists a one-to-one correspondence between the
set of eigenvalues {λε
j}∞j=1 of perturbed problem (2)-(5) and the spect-
rum of operator A. Namely, ε−1λε
j → µj as ε → 0, for each j ∈
N. Furthermore, a sequence of the corresponding eigenfunctions uε,j
converges in H1(a, b) towards the eigenfunction u with eigenvalue µj.
Proof.
For the perturbed problem (2)-(5) we assign the matrix operator
in L
Aε =
(
Â1 0
0 A2
)
, D(Aε) =
{
(u1, u2) ∈ D(Â1) ⊕D(A2) :
u1(0) = u2(0), (ku′1)(0) = ε(κu′2)(0)
}
.
Asymptotic analysis of vibrating system 203
Clearly, if µε belongs to σ(Aε), then εµε is an eigenvalue of (2)-(5).
Let us solve the equation (Aε − µI)u = f for f = (f1, f2) ∈ L
and µ ∈ %(Aε). Similarly to the previous theorem we obtain u1 =
Rµ(A1)f1 + εSµu2, u2 = Tµu1 + Rµ(Â2)f2, where Sµ : H2(Ib) →
H2(Ia) is a bounded intertwining operator that solves the problem
(kψ′)′ + µrψ = 0 in Ia, ψ(a) = 0 and (kψ′)(0) = (κg′)(0) for each
g ∈ H2(Ib). This yields that
(
I −εSµ
−Tµ I
) (
u1
u2
)
=
(Rµ(A1)f1
Rµ(Â2)f2
)
, (22)
where the matrix operator in the left-hand side is invertible as a small
perturbation of the invertible one. Letting ε→ 0 we can assert that
Rµ(Aε) =
(
I −εSµ
−Tµ I
)−1 (
Rµ(A1) 0
0 Rµ(Â2)
)
→
→
(
I 0
Tµ I
) (
Rµ(A1) 0
0 Rµ(Â2)
)
.
Hence, Rµ(Aε) → Rµ(A) in the uniform operator topology as ε→ 0,
which establishes a number-by-number convergence of the correspon-
ding eigenvalues [3, Th. 3.1].
Next we prove existence of the limit for the eigenfunctions under
normalization condition (21). We conclude from (17) that
∫ b
a
K(x)u′2ε (x) dx+ (1 − ε)κ(0)u′ε(+0)uε(+0) = µε.
For each ν there exists a twice differentiable solution ψ(x, ν) of
equation (κv′)′ + νρ v = 0 in Ib that satisfies conditions v(b) = 0,
v′(b) = 1. Moreover, ψ(x, ν) is an analytic function with respect to the
second argument for each fixed x [2, Th.1.5]. In particular, ψ(x, µε) →
ψ(x, µ) in C2(0, b) as µε → µ. Then there exits constant βε such that
uε(x) = βεψ(x, µε). Moreover, βε is bounded as ε→ 0, which is due to
condition (21). Therefore the values uε(+0) and u′ε(+0) are bounded
with respect to ε. Consequently we have
∫ b
a
K(x)u′2ε (x) dx ≤ µε +
(1 − ε)κ(0)|u′ε(+0)uε(+0)| ≤ M. Then finally the sequence {uε}ε>0
is precompact in the weak topology of H1(a, b). Let us consider a
subsequence uε′ such that uε′ → u in H1(a, b) weakly. We get uε′(x) =
βε′ψ(x, µε′) → βψ(x, µ) = u(x) in C2(0, b) for certain β. Note that
204 N. Babych, Yu. Golovaty
β > 0, which is due to (21). Moreover, u′
ε′(+0) → u′(+0) as ε′ → 0. A
passage to the limit in (17) implies that partial weak limit u satisfies
the identity
∫ b
a
K(x)u′φ′ dx + κ(0)u′(+0)φ(0) = µ
∫ b
a
R(x)uφ dx
for all φ ∈ C∞
0 (a, b). Moreover, u is different from zero, since
∫ b
a
R|u|2 dx = 1.
Consequently each weakly convergent subsequence of {uε}ε>0 tends
to u, where u is an eigenfunction of (15) that corresponds to the
eigenvalue µ and satisfies conditions ‖u‖L2(R,(a,b)) = 1 and u′(b) > 0.
Then the same conclusion can be drawn for the entire sequence. 2
Remark 1. In some cases value ε−1λε doesn’t actually depend on ε.
The latter takes place if and only if the three-points problem
{
(K(x)u′)′ + µR(x)u = 0 for x ∈ Ia ∪ Ib,
u(a) = u(b) = u′(−0) = u′(+0) = 0
(23)
has an eigenfunction u that is continuous at x = 0 (for a certain
eigenvalue µ). This situation is possible, for instance, in the case a =
−b when there exists even eigenfunction of the Dirichlet problem on
(−b, b). Then a trivial verification shows that λε = εµ is an eigenvalue
of (2)-(5) with the eigenfunction uε = u for all ε ∈ (0, 1].
Corollary 1. Restrictions of eigenfunction uε,j to the intervals Ia
and Ib converge towards the corresponding restrictions of eigenfunc-
tion u in H2(a, 0) and H2(0, b) respectively.
Proof. Set uε = uε,j. We consider equation (2) in the form u′′ε =
−k′k−1u′ε − µεrk
−1uε in Ia. Then from Theorem 2 we have
u′′ε → −k′k−1u′ − µrk−1u in L2(a, 0), (24)
where u is an eigenfunction of (15). From (15) it follows that the limit
(24) is exactly the second derivative of the limiting eigenfunction in
Ia. The proof for interval Ib is the same. 2
Asymptotic analysis of vibrating system 205
3. FormalAsymptotic Expansions of Eigenvalues
and Eigenfunctions.
3.1. Asymptotics of Simple Eigenvalues. In this section we
construct the complete asymptotic expansions of eigenvalues λε and
eigenfunctions uε. We begin with the examination of eigenvalues λε
j
for which the limit µ = lim
ε→0
λε
j/ε is a simple eigenvalue of operator
A. Clearly, µ depends on j, which we do not indicate for the sake
of notation simplicity. The asymptotic expansions of the eigenvalues
and the corresponding eigenfunctions are represented by
λε ∼ ε (µ+ εν1 + · · ·+ εnνn + · · · ), (25)
uε(x) ∼
{
y0(x) + εy1(x) + · · · + εnyn(x) + · · · for x ∈ Ia,
z0(x) + εz1(x) + · · ·+ εnzn(x) + · · · for x ∈ Ib,
(26)
where µ is an arbitrary eigenvalue of limit problem (15). Then
u(x) =
{
y0(x) for x ∈ Ia,
z0(x) for x ∈ Ib
(27)
is the corresponding eigenfunction of (15) as it follows from Th. 2.
Since in this section we treat only the simple eigenvalues µ, according
to Th. 1weonlyconsiderhere twopossible situations: µ ∈ σ(A1)\σ(Â2)
and µ ∈ σ(Â2)\σ(A1).
3.1.1. Case µ ∈ σ(A1)\σ(Â2). We fix the corresponding eigen-
function y0 of operator A1 such that
0
∫
a
ry2
0 dx = 1 and y0(0) > 0.
Since µ doesn’t belong to the spectrum of Â2 there exists a unique
solution z0 to the problem
(κz′0)
′ + µρz0 = 0 in Ib, z0(0) = y0(0), z0(b) = 0. (28)
An easy computation shows that the next terms of the expansions
are unique solutions to the recurrent sequence of problems
(ky′n)′ + µryn = −νnry0 − r
n−1
∑
j=1
νj yn−j in Ia,
yn(a) = 0, (ky′n)(0) = (κz′n−1)(0),
∫ 0
a
ryny0 dx = 0,
(29)
(κz′n)′ + µρzn = −ρ
n
∑
j=1
νj zn−j in Ib,
zn(0) = yn(0), zn(b) = 0
(30)
206 N. Babych, Yu. Golovaty
with νn = −(κz′n−1)(0)y0(0) for n = 1, 2, . . . . The last formula for νn
is obtained as the solvability condition of (29). Note that all solutions
yn, zn are smooth functions.
Remark 2. It might happened that z′0(0) = 0 (cf. the proof of Th.
2). In this case function u defined by (27) is exactly an eigenfunction
of the perturbed problem for each ε ∈ (0, 1]. Then the construction
of asymptotics is interrupted and we can state that there exists an
eigenvalue λε = εµ for all ε > 0. The corresponding eigenfunction
uε(x) =
{
y0(x) for x ∈ Ia,
z0(x) for x ∈ Ib
doesn’t depend on ε.
3.1.2. Case µ ∈ σ(Â2)\σ(A1) This situation immediately implies
y0 = 0 (cf. the proof of Th. 1, part (ii)). We fix the corresponding
eigenfunction z0 of Â2 such that
b
∫
0
ρz2
0 dx = 1 and z′0(0) > 0. A
trivial verification shows that the next terms of expansions (26) are
the unique smooth solutions to the problems
(ky′n)′ + µryn = −r
n−1
∑
j=1
νj yn−j in Ia,
yn(a) = 0, (ky′n)(0) = (κz′n−1)(0),
(κz′n)′ + µρzn = −νnρz0 − ρ
n−1
∑
j=1
νj zn−j in Ib,
zn(0) = yn(0), zn(b) = 0,
∫ b
0
ρznz0 dx = 0,
(31)
with νn = −(κz′0)(0)yn(0) for n = 1, 2, . . . . Such choice of νn assures
the solvability of (31).
3.2. Asymptotics of Double Eigenvalues. In this subsection we
treat the case when for two successive eigenvalues λε
j and λε
j+1 the
corresponding ratios ε−1λε
j and ε−1λε
j+1 converge to the same limit µ.
It is obvious that µ must belong to the intersection σ(A1)∪σ(Â2). Let
us assume that the eigenvalues and the corresponding eigenfunctions
admit expansions
λε ∼ ε (µ+
√
εν1 + εν2 + · · · ), (32)
uε(x) ∼
{ √
εw1(x) + εw2(x) + · · · for x ∈ (a, 0),
v0(x) +
√
ε v1(x) + ε v2(x) + · · · for x ∈ (0, b),
(33)
Asymptotic analysis of vibrating system 207
because the eigenvectors of operator A that correspond to double
eigenvalues µ have the form Vµ = (0, v0) (see Th. 1). Substituting
(32), (33) into the perturbed problem we obtain
(κv′0)
′ + µρv0 = 0 in Ib, v0(0) = v0(b) = 0, (34)
(kw′
1)
′ + µrw1 = 0 in Ia, w1(a) = w′
1(0) = 0. (35)
We fix µ ∈ σ(A1) ∪ σ(Â2) and introduce the functions
U(x) =
{
0 for x ∈ Ia
v(x) for x ∈ Ib
, U∗(x) =
{
w∗(x) for x ∈ Ia
v∗(x) for x ∈ Ib
(36)
that correspond to the eigenvector and adjoined vector of A (cf.
vectors Vµ and V ∗
µ in Th. 1). Here v is an eigenfunction of (34) such
that
∫ b
0
ρv2 dx = 1, v′(0) > 0 and adjoined vector U∗ is chosen such
that (U, U∗)L2(R,(a,b)) = 0. We also introduce an eigenfunction w of
(35) such that
∫ 0
a
rw2 dx = 1 and w(0) > 0. It follows that v0 = αv
and w1 = βw with certain constants α and β. In addition, α must be
different from zero. The next problems to solve are
{
(κv′1)
′ + µρv1 = −ν1αρv in Ib,
v1(0) = βw(0), v1(b) = 0,
(37)
{
(kw′
2)
′ + µrw2 = −ν1βrw in Ia,
w2(a) = 0, k(0)w′
2(0) = ακ(0)v′(0).
(38)
In general case both problems (37) and (38) are unsolvable, since µ
belongs to the spectra σ(A1) and σ(Â2) at one time. Hence we have to
apply Fredholm’s alternative for both the problems. After multiplying
equations (38) and (37) by eigenfunctions v and w respectively and
integrating by parts, one yields the common solvability condition:
(
0 ω
ω 0
)(
α
β
)
= −ν1
(
α
β
)
, (39)
where ω = (κwv′)(0) is positive. Since the first component of vector
γ = (α, β) must be different from zero, −ν1 is an eigenvalue of the
matrix in (39). Therefore if either ν1 = ω and γ = (1,−1) or ν1 = −ω
and γ = (1, 1), then problems (37), (38) admit solutions. Moreover,
functions ν1w∗ and ν1v∗ solve problems (35) and (37) respectively
for both values of ν1. Actually these problems imply immediately
208 N. Babych, Yu. Golovaty
(A − µ)U∗ = ωU . In other words, the first corrector is an adjoined
vector of A that corresponds to the eigenvector ωU . It causes no
confusion that we use the same letters U , U∗ to designate a function
of L2(a, b) and a vector in L.
Summarizing, we formally demonstrate that there exists a pair
of closely adjacent eigenvalues λε
j and λε
j+1 that admit the asymptotic
expansions
λε
j = εµ− ε3/2ω +O(ε2), λε
j+1 = εµ+ ε3/2ω +O(ε2), as ε→ 0.
As of asymptotics of eigenfunctions we have
uε,j(x) = U(x) −√
ε ωU∗(x) +O(ε),
uε,j+1(x) = U(x) +
√
εωU∗(x) +O(ε).
These eigenfunctions subtend an infinitely small angle in L2-space as
ε → 0. Hence uε,j and uε,j+1 stick together at the limit. The latter
gives rise to the loss of completeness of the limit eigenfunction system.
Suppose that ν1 = ω and γ = (1,−1). Then we will denote by
V1 and W2 such solutions of the problems that
∫ b
0
ρV1v dx = 0 and
∫ 0
a
rW2w dx = 0. We see at once that −V1 and −W2 are solutions of
(37), (38) for ν1 = −ω and γ = (1, 1).
From now on we distinct two branches of expansions (32)
λε
− ∼ ε(µ−√
εω + εν−2 + · · ·+ εn/2ν−n + . . . ),
λε
+ ∼ ε(µ+
√
εω + εν+
2 + · · ·+ εn/2ν+
n + . . . ),
(40)
and the corresponding branches of (33) are
u±ε (x) ∼
∼
{
∓√
εw(x) ± εw±
2 (x) + · · ·+ εn/2w±
n (x) . . . , x ∈ Ia,
v(x) ±√
ε v±1 (x) + εv±2 (x) + · · · + εn/2v±n (x) . . . , x ∈ Ib.
(41)
All coefficients are endowed with indexes + or − if they depend on the
choice of the sign of the first corrector ν1 = ±ω. Note that the high
order correctors in (40), (41) have to be calculated separately for both
the branches. We now turn to the case ν1 = ω and find coefficients
ν+
n , w+
n and v+
n . To shorten notation, we omit upper index "+" for a
while. Next, we see that problems (37) and (38) admit many solutions
Asymptotic analysis of vibrating system 209
v1 = V1 +α1v and w2 = W2 +β1w, where α1, β1 are constants. These
constants can be obtained from the consistency of problems
{
(κv′2)
′ + µρv2 = −ν1ρ (V1 + α1v) − ν2ρv, x ∈ Ib
v2(0) = W2(0) + β1w(0), v2(b) = 0,
(42)
{
(kw′
3)
′ + µrw3 = −ν1r (W2 + β1w) − ν2rw1, x ∈ Ia
w3(a) = 0, k(0)w′
3(0) = κ(0) (V1 + α1v)
′ (0).
(43)
The solvability conditions for problems (42) and (43), which arrive
from Fredholm’s alternatives, can be represented as a linear algebraic
system
(
ν1 ω
ω ν1
)(
α1
β1
)
=
(
(κW2v
′)(0) + ν2
(κwV1)
′(0) − ν2
)
. (44)
The system has solution if and only if ν2 = 1
2
(κwV1
′ − κW2v
′) (0).
After the solvability condition is satisfied, system (44) has a partial
solution α1 = β1 = 1
2ω
(κwV ′
1 + κW2v
′) (0) and problems (42) and
(43) admit solutions V2 and W3 such that
b
∫
0
ρV2v dx = 0 and
0
∫
a
rW3w dx = 0
. Therefore, all other solutions of (42) and (43) allow the representation
v2 = V2 + α2v and w3 = W3 + β2w with real constants α2, β2.
We construct the general terms of expansions (40) and (41) as
solutions to the problems
(κv′n)′ + µρvn = −ρ
n
∑
j=1
νjvn−j, x ∈ Ib,
vn(0) = wn(0), vn(b) = 0,
(45)
(kw′
n+1)
′ + µrwn+1 = −r
n
∑
j=1
νjwn+1−j, x ∈ Ia,
wn+1(a) = 0, (kwn+1)
′(0) = (κvn−1)
′(0),
(46)
with
vn−1 = Vn−1 + αn−1v and wn = Wn + βn−1w, (47)
210 N. Babych, Yu. Golovaty
where Vn−1 and Wn are solutions of the previous problems chosen
accordingly to the orthogonality conditions
b
∫
0
ρVn−1v dx = 0 and
0
∫
a
rWnw dx = 0, n ≥ 2. Constants αn−1 and βn−1 we find from the
solvability conditions for (45) and (46) given by
(
ν1 ω
ω ν1
)(
αn−1
βn−1
)
=
(κWnv
′) (0) +
n−1
∑
j=2
νjαn−j + νn
(
κwV ′
n−1
)
(0) +
n−1
∑
j=2
νjβn+1−j − νn
. (48)
The latter has a solution if and only if νn = 1
2
(
κwV ′
n−1 − κWnv
′) (0).
Then system (48) has a partial solution
αn−1 = βn−1 =
1
2ω
(
κwV ′
n−1 + κWnv
′) (0) +
1
ω
n−1
∑
j=2
νjαn−j.
Substituting the constants into (47) we finish the general step of
recurrent algorithm. Hence, after coming back our natation we obtain
all coefficients ν+
n , v+
n and w+
n of series (40) and (41).
Similarly, we can construct the coefficients ν−n , v−n and w−
n of
series (40) and (41). Then, by induction we get that for any natural
n the coefficients satisfy relations ν−n = (−1)n ν+
n , v−n = (−1)n v+
n and
w−
n = (−1)n w+
n .
4. Justification of Asymptotic Expansions.
Let Lε be he weighted L2-space with the scalar product and
norm given by (6). We also introduce space Hε as the Sobolev space
H1
0 (a, b) with scalar product and norm
〈φ, ψ〉ε =
∫ 0
a
kφ′ ψ′ dx+ε
∫ b
0
κφ′ ψ′ dx, ‖φ‖Hε
=
√
〈φ, φ〉ε. (49)
It is easily seen that
c‖φ‖ ≤ ‖φ‖ε ≤ Cε−1/2‖φ‖, cε1/2‖φ‖1 ≤ ‖φ‖Hε
≤ C‖φ‖1, (50)
where ‖ · ‖ and ‖ · ‖1 are standard norms in L2(a, b) and H1
0 (a, b)
respectively.
For the sake of completeness, we introduce here below the clas-
sical result on quasimodes. Let A be a self-adjoint operator in Hilbert
space H with domain D(A) and σ > 0.
Asymptotic analysis of vibrating system 211
Definition. We will say that pair (µ, u) ∈ R ×D(A) is a quasimode
with accuracy to σ for operator A if ‖(A−µI)u‖H ≤ σ and ‖u‖H = 1.
Lemma 1 (Vishik and Lyusternik). Suppose that the spectrum of
A is discrete. If (µ, u) is a quasimode of A with accuracy to σ, then
interval [µ − σ, µ + σ] contains an eigenvalue of A. Furthermore, if
segment [µ−d, µ+d], d > 0, contains one and only one eigenvalue λ
of A, then ‖u− v‖H ≤ 2d−1σ, where v is an eigenfunction of A with
eigenvalue λ, ‖v‖H = 1. [4, 5]
4.1. Simple Spectrum. We will denote by Λε,n = ε (µ+ εν1 + · · ·+
εnνn) and
Uε,n(x) =
{
y0(x) + εy1(x) + · · ·+ εnyn(x) for x ∈ Ia
z0(x) + εz1(x) + · · · + εnzn(x) for x ∈ Ib
the partial sums of series (25), (26). The perturbed problem is asso-
ciated with self-adjoint operator Aε = − 1
rε
d
dx
kε
d
dx
in Lε with the
domain D(Aε) = {f ∈ H : (kf ′)(−0) = ε(κf ′)(+0)}, where coeffi-
cients kε, rε are given by (1) for m = 1.
Theorem 3. If µj ∈ σ(A1)\σ(Â2), then eigenfunction uε,j of (2)-(5)
with eigenvalue λε
j converges in H1(a, b) towards the function
u(x) =
{
y(x) for x ∈ Ia
z(x) for x ∈ Ib,
where y is an eigenfunction of the problem (ky ′)′ + µry = 0 in Ia,
y(a) = y′(0) = 0 with eigenvalue µj, and z is a unique solution of the
problem (κz′)′ + µj ρz = 0 in Ib, z(0) = y(0), z(b) = 0.
If z′(0) = 0, then λε
j = εµj and uε,j = u for all ε > 0. Otherwise
λε
j and uε,j admit asymptotics expansions (25), (26) obtained in 3.1.1
for µ = µj. Moreover, the estimates of remainder terms hold
∣
∣ε−1λε
j − (µj + εν1 + · · · + εnνn)
∣
∣ ≤ cnε
n+1, (51)
‖uε,j − ϑεUε,n‖H1(a,b) ≤ Cnε
n+1, (52)
where ϑε is a normalizing multiplier with strictly positive limit as
ε→ 0.
212 N. Babych, Yu. Golovaty
Proof. The case z′(0) = 0 was considered in Remarks 1 and
2. Suppose that z′(0) 6= 0. We first check that the the series being
constructed in give us the quasimodes with accuracy to an arbitrary
order. It follows from (29), (30) that
∣
∣r−1
ε (kεU
′
ε,n)
′ + Λε,nUε,n
∣
∣ ≤ cnε
n+2 (53)
in [a, b] uniformly, Uε,n(a) = Uε,n(b) = 0, Uε,n(−0) = Uε,n(+0) and
βε,n = (kU ′
ε,n)(−0) − ε(κU ′
ε,n)(+0) = O(εn+1), ε→ 0. (54)
Note that Uε,n doesn’t belong to the domain of Aε since βε,n is
different from zero in the general case. Set φ(x) = x(x
a
− 1) for
x ∈ (a, 0) and φ(x) = 0 elsewhere. Then Vε,n = Uε,n + βε,nφ belongs
to D(Aε) and a simple computation gives ‖AεVε,n − Λε,nVε,n‖ε ≤
cnε
n+3/2. Hence (Λε,n, Vε,n/‖Vε,n‖ε) is a quasimode of operator Aε
with accuracy to cnε
n+2 because ‖Vε,n‖ε = O(ε−1/2). According to
the Vishik-Lyusternik Lemma there exists an eigenvalue λε of Aε
such that |λε − Λε,n| ≤ cnε
n+2, which establishes (51). Moreover,
there exists an unique eigenvalue λε = λε
j with such asymptotics by
Theorem 2. Next, for a certain d > 0 segment [Λε,n − dε,Λε,n + dε]
contains one and only one eigenvalue of Aε. Repeated application
of Lemma 1 enables us to write
∥
∥‖uε‖−1
ε · uε − ‖Vε,n‖−1
ε · Vε,n
∥
∥
ε
≤
2cnd
−1εn+1, where uε = uε,j. Hence, by (50)
∥
∥
∥
∥
uε −
‖uε‖ε
‖Vε,n‖ε
Vε,n
∥
∥
∥
∥
ε
≤ 2cn
d
‖uε‖εε
n+1 ≤ Cnε
n+1/2
and ϑε = ‖uε‖ε
‖Vε,n‖ε
converges to 1 by Theorem 2.
Pair (λε, uε) satisfies identity 〈uε, ψ〉ε = λε(uε, ψ)ε for all ψ ∈
H1
0 (a, b). Similarly,
〈Vε,n, ψ〉ε = Λε,n(Vε,n, ψ)ε + αε(ψ),
where |αε(ψ)| ≤ cεn+1/2‖ψ‖Hε
. The latter gives
‖uε − ϑεVε,n‖Hε
≤
≤ Λε,n‖uε − ϑεVε,n‖ε + |λε − Λε,n| ‖uε‖ε + |αε(uε − ϑεVε,n)| ≤
≤ 2µj Cnε
n+3/2 + cn‖uε‖ εn+3/2 + cεn+1/2‖uε − ϑεVε,n‖Hε
Asymptotic analysis of vibrating system 213
and consequently ‖uε −ϑεVε,n‖Hε
≤ Cnε
n+3/2. From this and (50) we
thus get estimate (52). 2
The same proof works for the rest part of the simple spectrum
of A.
Theorem 4. If µj ∈ σ(Â2)\σ(A1), then eigenfunction uε,j of (2)-(5)
with eigenvalue λε
j converges towards function
u(x) =
{
0 for x ∈ Ia,
z(x) for x ∈ Ib
in H1(a, b), where z is an eigenfunction of the problem (κz ′)′+µ ρz =
0 in Ib, z(0) = 0, z(a) = 0 with eigenvalue µj. Moreover λε
j and uε,j
admit asymptotic expansions (25), (26) obtained in 3.1.2 for µ = µj
with the estimates of remainder terms
∣
∣ε−1λε
j − (µj + εν1 + · · · + εnνn)
∣
∣ ≤ cnε
n+1,
‖uε,j − ϑεUε,n‖H1(a,b) ≤ Cnε
n+1.
Here ϑε is a normalizing multiplier that converges to a positive con-
stant as ε→ 0.
4.2. Double Spectrum. We introduce the partial sums of (40), (41)
Λ±
ε,n = ε(µj ± ε1/2ω + εν±2 + · · ·+ εn/2ν±n ), (55)
U±
ε,n =
{
∓ε1/2w + εw±
2 + · · ·+ εn/2w±
n for x ∈ Ia
v + ε1/2v±1 + · · · + εn/2v±n for x ∈ Ib
(56)
with all coefficients constructed in Section for certain double eigen-
value µ = µj = µj+1. Set V ±
ε,n = U±
ε,n + β±
ε,nφ, where β−
ε,n and β+
ε,n
are residuals in condition (4) for U−
ε,n and U+
ε,n respectively defined
similarly as in (54). Moreover, β±
ε,n = O(ε(n+1)/2) as ε→ 0.
Analysis similar to that in the proof of Theorem 3 leads to the
following result.
Proposition 3. The pairs (Λ−
ε,n, V
−
ε,n/‖V −
ε,n‖ε) and (Λ+
ε,n, V
+
ε,n/‖V +
ε,n‖ε)
are quasimodes of operator Aε with accuracy to cnε
n/2.
Proposition 4. There exist two closely adjacent eigenvalues λ−
ε and
λ+
ε of (2)-(5) with the asymptotics
λ±ε
ε
= µj ±
√
εω + εν±2 + · · ·+ εn/2ν±n +O(ε(n+1)/2), (57)
214 N. Babych, Yu. Golovaty
where µj is a double eigenvalue of operator A and ω, ν±k were defined
in Sec. .
Proof. From Proposition 3 and the Vishik-Lyusternik Lemma
it follows that there exists at least one eigenvalue of Aε in each εn/2-
vicinity of Λ−
ε,n and Λ+
ε,n. Moreover, |λ±
ε − Λ±
ε,n| ≤ cnε
n/2. Evidently,
eigenvalues λ−
ε , λ+
ε are different, because Λ+
ε,n − Λ−
ε,n ≥ ωε3/2 and
εn/2-vicinities of Λ−
ε,n and Λ+
ε,n don’t intersect for n > 3 and sufficient
small ε. In fact, |λ+
ε −λ−ε | ≥ cε3/2 for certain positive c. We conclude
from |λ±ε − Λ±
ε,n+3| ≤ cn+3ε
(n+3)/2 that
∣
∣
∣
λ±
ε
ε
− (µj ±
√
εω + · · · + ε
n
2 ν±n )
∣
∣
∣
≤
≤ cn+3ε
n+1
2 +
∑3
k=1 ε
n+k
2 |ν±n+k| ≤ Cnε
n+1
2 ,
which establishes (57). 2
We consider two planes in L2(a, b). Let π be the root subspace
that corresponds to double eigenvalue µi and π(ε) be the linear span
of two eigenfunctions u−ε and u+
ε that correspond to eigenvalues λ−
ε
and λ+
ε . These eigenfunctions as above are normalized by (21).
Theorem 5. The root subspace π is the limit position of plane π(ε)
as ε→ 0 that is to say ‖Pπ(ε) − Pπ‖ → 0, where Pπ(ε) and Pπ are the
orthogonal projectors onto planes π(ε) and π.
Proof. Nevertheless both eigenfunction u−
ε and u+
ε converge to
the same limit being the eigenfunction of A with eigenvalue µj,
the πε has regular asymptotic behaviour as ε → 0. We choose new
L2(R, (a, b))-orthogonal basis in π(ε): fε = 1
2
(u+
ε +u−ε ), gε = 1
2ω
√
ε
(u+
ε −
u−ε ).
By Theorem 2 the first vector fε converges in L2 towards eigen-
function U ∈ π given by (36). Next, function gε solves the problem
(kg′ε)
′ +
λ+
ε
ε
rgε =
λ−ε − λ+
ε
2ωε
√
ε
ru−ε in Ia,
(κg′ε)
′ +
λ+
ε
ε
ρgε =
λ−ε − λ+
ε
2ωε
√
ε
ρu−ε in Ib,
gε(a) = 0, gε(b) = 0,
gε(−0) = gε(+0), (kg′ε)(−0) = ε(κg′ε)(+0).
Asymptotic analysis of vibrating system 215
Since ε−1λ+
ε → µj, ε
−3/2(λ+
ε − λ−ε ) → 2ω by (57) and the right-hand
side is orthogonal to the eigenfunction u+
ε in Lε, one obtains that
norms ‖gε‖H2(a,0) and ‖gε‖H2(0,b) are bounded as ε → 0. Taking into
account Corollary 1 we can assert that each converging subsequence
gε′ converges as ε→ 0 towards a solution of the problem
{
(kg′)′ + µj rg = 0 in Ia, (κg′)′ + µj ρg = −ρv in Ib,
g(a) = 0, g(b) = 0, g(−0) = g(+0), g′(−0) = 0,
because u−ε converges to eigenfunction U , which equals v in Ib and
vanishes in Ia. Hence, all partial limits of the second basis vector gε
have to be the adjoined vectors corresponding to the eigenvalue µj.
In fact, by orthogonality of fε and gε these limits belong to the line
{αU∗ |α ∈ R} ⊂ π , which is orthogonal to U (see (36) for definition
of U∗). 2
Indeed, in previous statements λ−
ε = λε
j, λ
+
ε = λε
j+1 and u−ε =
uε,j, u
+
ε = uε,j+1, by Theorem 2. Next theorem summarizes all infor-
mation on bifurcation of the double spectrum.
Theorem 6. Let µj ∈ σ(A1) ∩ σ(Â2) be a double eigenvalue with
eigenfunction U and adjoined function U∗ given by (36), µj = µj+1.
Then both eigenfunction uε,j and uε,j+1 converge to the same eigen-
function U and the difference 1√
ε
(uε,j+1 − uε,j) converges to adjoined
function γU∗ for certain γ 6= 0. Besides, λ−
ε = λε
j, λ
+
ε = λε
j+1 and
uε,j, uε,j+1 admit asymptotic expansions (40), (41) derived in Section
3.2 for µ = µj. The estimates of remainder terms hold
∣
∣ε−1λ±ε −
(
µj ±
√
εω + εν±2 + · · ·+ εn/2ν±n
)∣
∣ ≤ c±n ε
(n+1)/2, (58)
‖uε,j − ϑ−ε U
−
ε,n‖H1(a,b) ≤ C−
n ε
n+1
2 , ‖uε,j+1 − ϑ+
ε U
+
ε,n‖H1(a,b) ≤ C+
n ε
n+1
2 ,
(59)
where ϑ±
ε are normalizing multipliers with strictly positive limit as
ε→ 0.
Proof. It remains to prove estimates (59). From (58) and Theorem
2 it may be concluded that for certain d > 0 and n ≥ 2 interval
[Λ−
ε,n − dε2,Λ−
ε,n + dε2] contains eigenvalue λε
j only. In view of Prop.
3 and the Vishik-Lyusternik Lemma, we have
∥
∥
∥
∥
uε,j −
‖uε,j‖ε
‖V −
ε,n‖ε
V −
ε,n
∥
∥
∥
∥
ε
≤ 2cn
dε2
‖uε‖εε
n/2 ≤ Cnε
n−5
2 .
216 N. Babych, Yu. Golovaty
As in the proof of Theorem 3 we can obtain ‖uε,j − ϑ−ε U
−
ε,n‖H1(a,b) ≤
Cnε
n−4
2 . Since all the coefficients of sum U−
ε,n are bounded in H1(a, b),
the first estimate (59) follows from the last inequality with n replaced
by n+ 5. The same proof works for uε,j+1. 2
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University of Bath, United Kingdom,
Lviv National University, Ukraine
n.babych@bath.ac.uk
yu_holovaty@franko.lviv.ua
Received 1.03.07
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