12_323_khelifi.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 2, 2010, 282-294

ISSN 1307-5543 – www.ejpam.com

Estimates for Resonant Frequencies Under Boundary Deforma-

tion in Multi-dimensional Space

Abdessatar Khelifi∗ and M. Nour. Shamma

Department of mathematics & Faculty of Sciences & Arts, Al Gassim University, Al-Rass Province,

Kingdom of Saudi Arabia

Abstract. In multi-dimensional space, we address the integral equation method to investigate the

interplay between the geometry, boundary conditions and the properties of the resonant frequencies

and their associated eigenfunctions under boundary variations of domain. We provide a rigorous

derivation of asymptotic expansions for eigenfunctions and we establish error estimations for both

resonant frequencies and eigenfunctions of the Helmholtz eigenvalue problem.

2000 Mathematics Subject Classifications: 35C15, 35P05, 45C05

Key Words and Phrases: Resonant frequencies, Boundary integral equation, Error estimates

1. Introduction

The resonant frequencies may evolve under shape deformation, as separated, distinct

eigenvalues of the Helmholtz eigenvalue problem. But, the main difficulty in solving eigen-

value problems relates to the continuation of multiple eigenvalues of the unperturbed con-

figuration. The properties of eigenvalue problems under shape deformation have been the

subject of comprehensive studies [9, 22] and the area continues to carry great importance

to this day [4, 7, 6, 8, 12, 13, 14, 15]. A substantial portion of these investigations relate

to properties of smoothness and analyticity of eigenvalues and eigenfunctions with respect

to perturbations. Bruno and Reitich have presented in [4, Theorem 2, p.172 and Section 3,

pp.180-183] some explicit constructions of high-order boundary perturbation expansions for

eigenelements in two dimensions. Their algorithm is based on certain properties of joint ana-

lytic dependence on the boundary perturbations and spatial variables of the eigenfunctions. In

a series of papers [19]-[21], Ozawa derived the leading-order term in the asymptotic expan-

sions of simple eigenvalues in domain with a specific geometry. Nevertheless, in our paper we

remove the condition that eigenvalue is simple and provide more accurate asymptotic expan-

sions for eigenfunctions in domain with more general shape. Recently, Lanza de Cristoforis

∗Corresponding author.

Email addresses: abdessatar.khelifi�fsb.rnu.tn (A. Khelifi), shamman01�yahoo.
om. (M. Shamma)

http://www.ejpam.com 282 c© 2010 EJPAM All rights reserved.



A. Khelifi, M. Shamma / Eur. J. Pure Appl. Math, 3 (2010), 282-294 283

and Lamberti have developed in [13] some preliminary abstract results for the dependence of

the eigenvalues upon perturbation. Their applications to the Dirichlet eigenvalue problem for

the Laplace operator appear clearly in Section 3 of their paper and in Theorem 3.21 when they

justify the analyticity result for some symmetric functions of eigenvalues. Our analysis and

uniform asymptotic formulas of the eigenfunctions, which are represented by the single-layer

potential involving the Green function, are considerably different from those in [12, 13, 10].

Next, Our method differ, essentially, from the classical methods used to study the analytic

dependence of the eigenfunctions of a real or complex parameter and used to give the asymp-

totic formulae for the eigenvalues.

The main goal of this paper is to justify and to give formulae for the convergence estimates

for both resonant frequencies and eigenfunctions associated to Helmholtz eigenvalue orob-

lem. Compared to papers in this fields [15, 17, 18], one can notice that our results in section

4, are important and give an idea to evaluate the speed of convergence.

The paper is organized as follows. In Section 2 we describe the central problem in this work,

and we remember some well-known results concerning the analyticity of the eigenvalues with

respect to ε. In Section 3 we develop a boundary integral formulation for solving the eigen-

value problem (2). From results found in [11] in two dimensional space, we end this Section

by presenting the main theorem which gives the analyticity and the uniform asymptotic expan-

sion for the eigenfunctions. Section 4 contains the main results of our paper which are deeply

based on the Osborn’s theorem. We then prove some error estimates for the convergence of

resonant frequencies.

2. Problem Description

Let Ω be a bounded domain in Rd , d ≥ 2, with a connected Lipschitz boundary ∂Ω and

ν denotes the unit outward normal to ∂Ω. Let Ti > 0, for all i ∈ {1, · · · , d − 1} and let

γ(t), β(t) : t = (t1, · · · , td−1) ∈ [0, T1]× · · · × [0, Td−1]→ Rd , be two analytic, Ti-periodic

(in each composite t i ) functions and satisfying the following assumption:

〈γ′(t),β(t)〉 = 0, for allt = (t1, · · · , td−1) ∈ [0, T1]× · · · × [0, Td−1]. (1)

where 〈., .〉 denotes the usual product scalar in Rd . We introduce,

γε(t) = γ(t) + εβ(t), ε ∈ R.

With this definition, (t,ε) 7→ γε(t) is an analytic function on [0, T1] × · · · × [0, Td−1] × R,

Ti-periodic in each composite t i .

We consider the bounded domain Ωε in Rd with smooth boundary ∂Ωε parameterized by the

function γε(t):

∂Ωε = {γε(t), t = (t1, · · · , td−1) ∈ [0, T1]× · · · × [0, Td−1]}.

Let µ0 and ǫ0 denote the permeability and the permittivity of the background medium Ω ≡
Ωε=0, and assume that µ0 > 0 and ǫ0 > 0 are positive constants. Let µ1 > 0 and ǫ1 > 0



A. Khelifi, M. Shamma / Eur. J. Pure Appl. Math, 3 (2010), 282-294 284

denote the permeability and the permittivity of Ωε\Ω0. Introduce the piecewise-constant

electric permittivity

ǫε(x) =

(

ǫ0, x ∈ Ω0,

ǫ1, x ∈ Ωε\Ω0.

If we allow the degenerate case ε = 0, then the function ǫ0(x) equals the constant ǫ0. The

piecewise constant magnetic permeability, µε(x) is defined analogously.

In this paper, we deal with the asymptotic behavior associated with the following Helmholtz

eigenvalue problem:

div(
1

µε
grad u(ε)) +ω2(ε)ǫεu(ε) = 0 in Ωε, and u(ε) = 0 on ∂Ωε, (2)

where the function u represents some electric field or magnetic field (or rather, the transversal

strength) and ω2(ε) is the perturbed resonant frequency (eigenfrequency) associated to the

above problem (2). The eigenfunction u0, in the absence of any deformation, satisfies the

following equations:

−∆u0(x) =ω
2
0(ǫ0µ0)u0(x), x ∈ Ω, and u0(x) = 0, x ∈ ∂Ω, (3)

It is well known that the operator −∆ on L2(Ωε) with domain H2(Ωε)∩H1
0(Ωε) is self-adjoint

with compact resolvent. Consequently, its spectrum consists entirely of isolated, real and pos-

itive eigenvalues with finite multiplicity, and there are corresponding eigenfunctions which

make up an orthonormal basis of L2(Ωε). By extending our approach for treating the eigen-

value problem (3) we will investigate the splitting of the eigenvalues and derive their asymp-

totic expansions under boundary perturbations.

Let ω2
0 > 0 denote an eigenfrequency of the eigenvalue problem (3) for ε= 0 with geometric

multiplicity m in the domain Ω ≡ Ω0. There exists a small constant r0 > 0 such that ω2
0 is the

unique eigenfrequency of (3) for ε= 0 in the set
�

ω2,ω ∈ Dr0
(ω0)
	

, where Dr0
(ω0) is a disk

of center ω0 and radius r0. Let us call the ω0-group the totality of the perturbed eigenfrequen-

cies of (3) for ε > 0 generated by splitting from ω2
0 and chosen to be an increasing family. The

following analyticity result is well-known [9]-[22].

Theorem 1. ( Kato [12, §VII.6], Rellich [25, §§II.2 and II.6]) There exits ε0 > 0 such that

for |ε| < ε0, the ω0-group consists of m− eigenfrequencies, ω2
j (ε), j = 1, . . . , m (repeated ac-

cording to their multiplicity). Moreover, they are analytic functions with respect to ε satisfying

ω2
j (0) =ω

2
0, j = 1, . . . , m. The normalized eigenfunctions associated to the ω0-group of eigenfre-

quencies are analytic and their values at 0 ({u j

0
}1≤ j≤m) are m linearly independent solutions of

the unperturbed eigenvalue problem.

Classical regularity results and the previous theorem imply that the eigenfunctions asso-

ciated to the ω0-group of eigenvalues are separately analytic in the small parameter ε and

the spatial variable x . By an integral equation technique we also established [13, Thm. 5.4,

p.1219] the joint analytic dependence of these functions with respect to (x ,ε).



A. Khelifi, M. Shamma / Eur. J. Pure Appl. Math, 3 (2010), 282-294 285

We will develop a boundary integral formulation for solving the eigenvalue problem (2). The

integral equations applying to this problem will be obtained from a study of the layer poten-

tials for the Helmholtz equation. 2 For λ > 0, a fundamental solution Γλ(x) to the Helmholtz

operator ∆+λ2 in Rd , d = 2,3, is given by

Γλ(x) =







− i

4
H
(1)
0 (λ‖x‖), d = 2,

− eiλ‖x‖

4π‖x‖ , d = 3,

for x 6= 0, where H
(1)
0

is the Hankel function of the first kind of order 0.

Suppose that G(x , y) = Γω(ε)pµεǫε(x − y). The singularity of this function has the form:

G(x , y) ∼
(

1

2π
log‖x − y‖+ · · · as , x → y d = 2,
1

4π‖x−y‖ + · · · as x → y d = 3.

The following operator is well defined [2, 16]

S(ω) : H−1/2(∂Ωε)→ H1/2(∂Ωε)

where

S(ω) : g→
∫

∂Ωε

G(·, y)g(y)dσ(y).

For such g and every x ∈ ∂Ωε, we denote by g+(x) and g−(x) the limits of g(y) as y → x ,

from y ∈ Ωε and y ∈ Rd \Ωε, respectively, when these limits exist. It is a well-known classical

result that, for x ∈ ∂Ωε,
S(ω)g(x) = (Sl(ω)g)+(x) = (Sl(ω)g)−(x)

where the operator Sl(ω) called single-layer potential (see [5], [16]) and S(ω) is pseudo-

differential operator of order −1.

Throughout this paper, we use for simplicity the notation

H
ς

♯
(]0, T1[×· · ·×]0, Td−1[) = Hς(Rd−1/]0, T1[×· · ·×]0, Td−1[), for ς ∈ R, where

Hς(Rd−1/]0, T1[×· · ·×]0, Td−1[) denotes the classical Sobolev Hς-space on the quotient

R
d−1/]0, T1[×· · ·×]0, Td−1[ (Adams [1]).

Using change of variables and integral equations, the following result immediately holds (see

[23]).

Proposition 1. Let Aε(ω) : H
−1/2
♯
(]0, T1[×· · ·×]0, Td−1[)→ H

1/2
♯
(]0, T1[×· · ·×]0, Td−1[) be

defined as follows:

Aε(ω) f (t) =
�

S(ω) f (γ−1
ε )
�

(γε(t))

=

∫

]0,T1[×···×]0,Td−1[

G(γε(t),γε(s))|∇γε(s)| f (s)ds for f ∈ H
−1/2

♯
(]0, T1[×· · ·×]0, Td−1[).



A. Khelifi, M. Shamma / Eur. J. Pure Appl. Math, 3 (2010), 282-294 286

Then the operator-valued function Aε(ω) is Fredholm analytic with index 0 in C \ iR−. More-

over, A−1
ε (ω) is a meromorphic function and its poles are in

�ℑ(z) ≤ 0
	

, where ℑ(z) means the

imaginary part of z and ℜ(z) is the real part.

Using the proprieties of the operator-valued function Aε given by Proposition 1 and using

the Lemma 5.3 found in [11], we can easily prove the following results.

Theorem 2. Let K 0 be a bounded neighborhood of Ω0 in Rd . Then there exists a constant

ε1 > 0 smaller than ε0 such that an orthonormal basis of eigenfunctions (u j(ε)) j corresponding

to the ω0 − group, (ω2
j (ε)) j, in H1

0(Ωε) can be chosen to depend holomorphically in (x ,ε) ∈
K0×] − ε1,ε1[. Moreover these eigenfunctions satisfy the following uniform expansion: for

x ∈ K0,

u j(ε) = u
j

0 +
∑

n≥1

u( j)n ε
n,

where the family u
j

0
builds a basis of eigenfunctions of (3) associated to ω2

0 and normalized in

L2(Ω0). The terms u
( j)
n are computed from the Taylor coefficients of the normal derivatives.

3. Convergence Estimate

In this Section we are in a position to use the Theorems 1 and 2 in order to establish cer-

tain estimates for the convergence of the eigenfunctions u j(ε) and the corresponding eigen-

frequencies ω2
j (ε), for all j = 1, · · · , m. Let α1 > 0 be the smaller positive constant such

that Ω0 ⊂ Ωε and ∂Ωε ∩ ∂Ω0 = ;, for 0 < ε < α1 and define the open, bounded domain

Ω̃ε ≡ Ωε\Ω0.

Lemma 1. Let the functions u j(ε) and u
j

0, for j = 1, · · · , m, be given by Theorem 2. Then, there

exist some positive constants ε2 < ε1 and C j, such that

‖∇(u j(ε)− u
j

0)‖L2(Ω̃ε)
≤ C j|Ωε\Ω0|1/2,

for 0< ε < ε2. The constant C j depends on ω0 and u
j

0, but is otherwise independent of ε.

Proof. Define the function U(ε) = u j(ε)− u
j

0, for 0 < ε < inf(ε1,α1) where ε1 is given by

Theorem 2 and combine the equations (2) and (3) and let for simplicity ω=ω j(ε)
p
µεǫε, we

compute that U(ε) solves:

−∆U =ω2U + (ω2 −ω2
0ǫ0µ0)u

j

0 in Ωε. (4)

For z ∈ R, we define the function ϑ by

ϑ(z) =ω2z + (ω2−ω2
0ǫ0µ0)‖u j

0‖L∞(Ω0)
.

Then, we trivially remark,

|ϑ(z)| ≤ |ϑ(0)|+ω2|z|, ∀z ∈ R,



A. Khelifi, M. Shamma / Eur. J. Pure Appl. Math, 3 (2010), 282-294 287

and consequently,

|ϑ(U(ε))| ≤ |ϑ(0)|+ω2|U(ε)|. (5)

The fact that u j(ε)→ u
j

0 implies that there exists 0< α2 < inf(ε1,α1) such that for 0< ε < α2,

|U(ε)(x)| ≤ 2‖u j

0‖L∞(Ω0)
, for x ∈ Ωε. (6)

Moreover, we remember that ω2(ε) → ω2
0ǫ0µ0, then there exists α3 ≥ 0 such that: ω2 ≤

ω2
0ǫ0µ0+

1

3
, for 0≤ ε≤ α3.

Now, it is useful to introduce the following function:

ϑ̃(U) =ω2U + (ω2 −ω2
0ǫ0µ0)u

j

0,

where U is the solution of (4). If we examine each term on the right hand side of (5) sepa-

rately, we find out that the first term is bounded by

|ϑ(0)| ≤ 1

3
‖u j

0‖L∞(Ω0)
, for 0< ε < α3.

The second term is bounded by

ω2|U(ε)| ≤ 2(ω2
0ǫ0µ0 +

1

3
)‖u j

0‖L∞(Ω0)
, for 0< ε < ε2 = inf(α2,α3).

These estimates give

‖ϑ̃(U(ε))‖L∞(Ω̃ε) ≤ (1+ 2ω2
0ǫ0µ0)‖u j

0‖L∞(Ω0)
, for 0< ε < ε2. (7)

Next, the relation (4) implies

−∆U(ε) = ϑ̃(U(ε)), inΩε.

By integrating by parts in Ω̃ε, we find that the function U(ε) is solution to the following

problem:

∀v ∈ H1(Ω̃ε),

∫

Ω̃ε

∇U(ε)∇̄vd x =

∫

Ω̃ε

ϑ̃(U(ε))v̄d x +

∫

∂ Ω̃ε

∂ U

∂ ν
v̄ds(x). (8)

On the other hand, it is not hard to see that (Trace Theorem),

�

�

∫

∂ Ω̃ε

∂ U

∂ ν
v̄ds(x)
�

�≤ |Ω̃ε|1/2 sup
z∈∂ Ω̃ε
|∂ U(z)

∂ ν
|.‖v‖L2(Ω̃ε)

.

But, the relation (6) implies that supz∈∂ Ω̃ε |
∂ U(z)

∂ ν
| is a positive constant c∗ independent of ε. If

we choose v = U(ε) and if we consider relations (7) and (8) we deduce that,

‖∇U‖2
L2(Ω̃ε)
≤ [c∗ + (1+ 2ω2

0ǫ0µ0)‖u j

0‖L∞(Ω0)
]|Ω̃ε|1/2‖U‖L2(Ω̃ε)

. (9)



A. Khelifi, M. Shamma / Eur. J. Pure Appl. Math, 3 (2010), 282-294 288

By Poincare’s inequality, there exists some positive constant C(Ω̃ε) such that

‖U‖L2(Ω̃ε)
≤ C(Ω̃ε)‖∇U‖L2(Ω̃ε)

.

The fact U and ∇U are uniformly bounded on Ωε implies there exists some constant C0

independent of ε (e.g.[13, p.33]) such that

C(Ω̃ε)≤ C0, (10)

and therefore the relation (9) becomes

‖∇U‖L2(Ω̃ε)
≤ C0[c∗ + (1+ 2ω2

0ǫ0µ0)‖u j

0‖L∞(Ω0)
]|Ω̃ε|1/2.

We take C j = C0[c∗ + (1+ 2ω2
0ǫ0µ0)‖u j

0‖L∞(Ω0)
] which concludes the proof.

The following main result holds.

Theorem 3. Let γ, β and Ωε be defined as in Section 2 and let the functions u j(ε) and u
j

0,

for j = 1, · · · , m, be given by Theorem 2. Then, there exist some constant 0 < ε3 ≤ 1/M,

M = maxt∈[0,T1]×···×[0,Td−1]
|β(t)| and some positive constant κ j dependent on ω0, u

j

0, |γ| and

M but otherwise independent of ε such that,

‖u j(ε)− u
j

0‖L2(Ω̃ε)
≤ κ jε

1/2,

for 0< ε < ε3.

Proof. For simplicity we can suppose that Ω0 is a disk with radius ̺0 > 0 in R2. It then

follows that |γ(t)| = ̺0. The proof is simple if we calculate the area |Ω̃ε| of the domain

Ω̃ε. But it is not hard to see that, in polar coordinates (̺,θ), there exists a regular function

Υ : [0,2π]→ R+; Υ(θ) = |β(θ)| such that the boundary ∂Ωε can be re-parameterized by

̺ = ̺(ε,θ) = ̺0+ εΥ(θ); θ ∈ [0,2π].

Therefore

|Ω̃ε| =
∫ 2π

0

[

∫ ̺0+εΥ(θ )

̺0

̺d̺]dθ =
1

2

∫ 2π

0

[ε2Υ2(θ) + 2ε̺0Υ(θ)]dθ . (11)

For ε < ε3 = inf(ε2, 1/M), we can write: ε2 ≤ ε/M ; which implies

ε2M2 ≤ εM .

Consequently the equality (11) gives

|Ω̃ε| ≤ πM(1+ 2̺0)ε. (12)

By Poincare’s inequality and Lemma 1 we write,

‖u j(ε)− u
j

0‖L2(Ω̃ε)
≤ C(Ω̃ε)C j|Ω̃ε|1/2.



A. Khelifi, M. Shamma / Eur. J. Pure Appl. Math, 3 (2010), 282-294 289

Finally, we obtain the desired result if we consider the relations (10) and (12) and if we

choose the constant κ j = C jC0

p

πM(1+ 2̺0).

To derive the corresponding formulae for the eigenvalues we will use an idea close to the

theorem of Osborn [18] which gives estimates for the convergence of the eigenvalues of a

sequence of compact operators. For our case we consider the Hilbert space L2(Ωε) with the

standard inner product 〈., .〉.
For any ϕ ∈ L2(Ωε), define the operator Tεϕ = vε, where vε is the solution to the problem

( −∆vε = ϕ in Ωε,

vε = 0 on ∂Ωε.
(13)

and we define the operator T0ϕ = v0, where v0 is the solution to the problem

( −∆v0 = ϕ in Ω0,

v0 = 0 on ∂Ω0.
(14)

The function ϕ 7→ (−∆)−1ϕ is continuous from L2(Ωε) to H1
0(Ωε). Clearly Tε and T0 are

compact operators from L2(Ω0) to L2(Ω0). From the standard H1 estimates for vε which are

independent of ε, we see that the set {Tε} is collectively compact. Hence all hypotheses hold

for the theorem of Osborn. Now if we set,

λ0 =
1

ω2
0

and λ j(ε) =
1

ω2
j
(ε)

,

then according to the problem (13)(resp. (14)) we can see that (λ j(ε),u j(ε)) (resp.(λ0,u
j

0))

is eigenpairs of Tε (resp. of T0) with ϕ =
1

ω2
j
(ε)

u j(ε). We remember that ω2
0 is an eigen-

frequency of multiplicity m with a corresponding set of orthonormal eigenfunctions {u j

0
} and

then R(P(0)) is just the m−dimensional subspace generated by {u j

0}(where P(0) means the

spectral projection associated with T0 and means the projection onto the space associated to

{u j

0} ).

Although each of the eigenvalues λ1(ε), · · · ,λm(ε) are close to λ0, their arithmetic mean is

generally a closer approximation [3]. Thus we define

λ̂(ε) =
1

m

m
∑

j=1

1

ω2
j
(ε)

. (15)

In the terminology of [9] this is the weighted mean of the λ0−group. The next Lemma gives

an estimate for λ0 − λ̂(ε) which will be useful to prove our main result.

Lemma 2. Let ε3 be the positive constant given by Theorem 3. Then there exists a positive

constant K1 such that for ε < ε3,

|λ0− λ̂(ε)| ≤ K1ε
1/2.



A. Khelifi, M. Shamma / Eur. J. Pure Appl. Math, 3 (2010), 282-294 290

Proof. We write,

‖Tεu j

0 − T0u
j

0‖L2(Ω̃ε)
= ‖Tεu j(ε) + Tεu

j

0 − Tεu j(ε)− T0u
j

0‖L2(Ω̃ε)

≤ ‖ 1

ω2
j
(ε)

u j(ε)−
1

ω2
0

u0‖L2(Ω̃ε)
+ ‖Tεu j

0 − Tεu j(ε)‖L2(Ω̃ε)
. (16)

If we set z j(ε) =
1

ω2
j
(ε)

u j(ε) and z
j

0 =
1

ω2
0

u
j

0, we see that z j(ε) and z
j

0 are solutions to the

problems (3) and (2) respectively and z j(ε)→ z
j

0 as ε tends to 0. Therefore, Theorem 3 gives

for ε < ε3,

‖z j(ε)− z
j

0‖L2(Ω̃ε)
≤ κ jε

1/2.

In other words, for reasons of compactness of Tε and according to Theorem 3 we have

‖Tεu j

0 − Tεu j(ε)‖L2(Ω̃ε)
= ‖Tε(u j

0 − u j(ε))‖L2(Ω̃ε)

≤ K‖u j

0 − u j(ε)‖L2(Ω̃ε)
≤ K .κ jε

1/2.

Then, the relation (16) becomes

‖Tεu j

0
− T0u

j

0
‖L2(Ω̃ε)

≤ κ j(1+ K)ε1/2. (17)

Inserting all this information into the theorem of Osborn [21, Thm.3], we obtain

1

ω2
0

− 1

m

m
∑

j=1

1

ω2
j
(ε)
=

1

m

m
∑

j=1

〈(T0 − Tε)u
j

0,u
j

0〉+ ε1/2O(1).

The proof follows by reconsidering again relation (17).

Next, to estimating |ω2
0−ω2

j (ε)| we may, firstly, estimate |λ0−λ j(ε)| for each j.

Lemma 3. There exist some constants 0< ε4 ≤ ε3 and k
(1)

j
such that for all j = 1, · · · , m,

|λ0−λ j(ε)|m ≤ k
(1)

j
ε5/2, for 0≤ ε≤ ε4.

Proof. Let Tε(yε) = λ j(ε)yε such that ‖yε‖= 1. We can then choose̟∗ ∈ Ker((λ0−T ∗0 )
m)

in such a way that 〈yε,̟∗〉 = 1. Then,

〈(λ0 − T ∗0 )
m̟∗, yε〉 = 0

and therefore,

|〈(λ0−λ j(ε))
m̟∗, yε〉| = |〈(λ0−λ j(ε))

m̟∗, yε〉 − 〈(λ0 − T ∗0 )
m̟∗, yε〉|

= | −
m−1
∑

l=0

(λ0−λ j(ε))
l〈(λ0− T ∗0 )

m−1−l(λ j(ε)− T ∗0 )̟
∗, yε〉|. (18)



A. Khelifi, M. Shamma / Eur. J. Pure Appl. Math, 3 (2010), 282-294 291

Now we prove that |λ0−λp(ε)| ≤ |λ0− λ̂(ε)|, for each positive integer p. But, the relation,

λ0− λ̂(ε)− (λ0−λp(ε)) = λp(ε)−
1

m

m
∑

j=1

λ j(ε)

implies that, for p and q the integers such that λp = sup1≤ j≤m |λ j(ε)| and λq = inf1≤ j≤m |λ j(ε)|,
the following relation holds(the family (ω2

j ) j is increasing):

|λ0− λ̂(ε)| ≥ sup(|λ0−λp(ε)|, |λ0−λq(ε)|)
which gives that |λ0−λ j(ε)| ≤ |λ0−λ̂(ε)| for 1≤ j ≤ m. Therefore, the fact that λ0−λ̂(ε)→ 0

implies that there exists δ1 > 0 such that for all 0≤ l ≤ m− 1,

|λ0−λ j(ε)|l ≤ |λ0− λ̂(ε)|, for 0< ε < δ1. (19)

Next, if we insert the relation (19) into (18) we obtain the following inequality

|λ0−λ j(ε)|m ≤ |λ0− λ̂(ε)|
m−1
∑

l=0

‖λ0−T ∗0 ‖m−1−l max
‖ψ∗‖=1

|〈(λ j(ε)−T0)yε,ψ
∗〉|, for 0< ε < δ1.

(20)

For any ψ∗ ∈ R(P(0)∗), with ‖ψ∗‖ = 1, and the fact that Pj(ε)
−1Pj(ε) is the identity on

R(P(0)) (where Pj(ε) means the spectral projection associated with Tε and is a projection

onto the direct sum of the spaces of the eigenvectors corresponding to Tε ) we write,

|〈(λ j(ε)− T0)yε,ψ
∗〉| = |〈Pj(ε)

−1Pj(ε)(Tε− T0)yε,ψ
∗〉| = |〈(Tε− T0)yε, (Pj(ε)

−1Pj(ε))
∗ψ∗〉|.

Due to the estimate (4.10) found in [21, p.722], we obtain

|〈(Tε− T0)yε, (Pj(ε)
−1Pj(ε))

∗ψ∗〉| ≤ c.ε2‖yε‖L2 .‖(Pj(ε)
−1Pj(ε))

∗ψ∗‖L2 , for 0< ε < δ1,

where c is a positive constant. Then,

|〈(Tε − T0)yε, (Pj(ε)
−1Pj(ε))

∗ψ∗〉| ≤ c.ε2‖yε‖L2 .‖Pj(ε)
−1‖.‖Pj(ε)‖.‖ψ∗‖L2 .

The norm ‖Pj(ε)‖ is bounded in ε since Pj(ε)→ P(0) pointwise. In other words, for ε small

enough and for f ∈ R(P(0)) with ‖ f ‖ = 1, we have

1−‖Pj(ε) f ‖= ‖P(0) f ‖ − ‖Pj(ε) f ‖ ≤ ‖(P(0)− Pj(ε)) f ‖ ≤
1

2

and hence ‖P(ε) f ‖ ≥ 1

2
which implies ‖P(ε)−1‖ ≤ 2 for ε small enough, say for 0 ≤ ε ≤ δ2

where δ2 is a positive constant. Then the relation (20) becomes,

|λ0−λ j(ε)|m ≤ c′.ε2|λ0− λ̂(ε)|. (21)

The proof is achieved if we use Lemma 2 and we take ε4 = inf(ε3,δ1,δ2) and k
(1)

j
= c′K1.

The main estimate will be given in the following theorem which its proof follows easily by

Lemma 3.



REFERENCES 292

Theorem 4. Let ω2
0 > 0 be the eigenfrequency of the problem (2) with geometric multiplicity

m, and ω2
j (ε), for j = 1, · · · , m given by Theorem 1. Then, there exist some positive constants

0< ε5 ≤ ε4 and k
(2)

j
such that

|ω2
0 −ω2

j (ε)| ≤ k
(2)

j
ε

5

2m ,

for 0< ε≤ ε5.

Proof. It is not hard to see that for all j = 1, · · · , m,

|λ0−λ j(ε)|=
�

�

�

ω2
j (ε)−ω2

0
�

ω0ω j(ε)
�2

�

�

� (22)

But the fact ω2
j (ε)→ ω2

0 as ε tends to 0 implies that there exists some constant δ3 > 0 such

that |�ω j(ε)ω0

�2| < 3

2
ω4

0, for 0< ε < δ3.

The relation (22) implies,

|ω2
j (ε)−ω2

0| ≤
3

2
ω4

0|λ0−λ j(ε)|, for 0< ε < δ3.

The theorem follows immediately by considering Lemma 3, we take ε5 = inf(δ3,ε4) and we

choose k
(2)

j
=

3

2
ω4

0(k
(1)

j
)1/m.

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