CUBO A Mathematical Journal
Vol.11, No¯ 05, (39–49). December 2009

Boundary Stabilization of the Transmission Problem

for the Bernoulli-Euler Plate Equation

Kaïs Ammari

Département de Mathématiques, Faculté des Sciences de Monastir,

5019 Monastir, Tunisie

email: kais.ammari@fsm.rnu.tn

and

Georgi Vodev

Département de Mathématiques, UMR 6629 du CNRS,

Université de Nantes, 2 rue de la Houssinière, BP 92208,

FR-44322 Nantes Cedex 03, France

email : vodev@math.univ-nantes.fr

ABSTRACT

In this paper we consider a boundary stabilization problem for the transmission Bernoulli-

Euler plate equation. We prove uniform exponential energy decay under natural conditions.

RESUMEN

En este artículo consideramos un problema de estabilización en la frontera para la ecuación

de Bernoulli-Euler Plate. Nosotros probamos decaimiento exponencial uniforme de la energia

sobre condiciones naturales

Key words and phrases: Transmission problem, boundary stabilization, Bernoulli-Euler plate

equation.

Math. Subj. Class.: 35B05, 93D15, 93D20.



40 Kaïs Ammari and Georgi Vodev CUBO
11, 5 (2009)

1 Introduction and Statement of Results

Let Ω1 ⊂ Ω ⊂ R
n, n ≥ 2, be strictly convex, bounded domains with smooth boundaries Γ1 = ∂Ω1,

Γ = ∂Ω, Γ1 ∩Γ = ∅. Then O = Ω\Ω1 is a bounded, connected domain with boundary ∂O = Γ1 ∪Γ.

We are going to study the following mixed boundary value problem





(∂2t + c
2
∆

2
)u1(x,t) = 0 in Ω1 × (0, +∞),

(∂2t + ∆
2
)u2(x,t) = 0 in O × (0, +∞),

u1|Γ1 = u2|Γ1, ∂νu1|Γ1 = ∂νu2|Γ1, c∆u1|Γ1 = ∆u2|Γ1, c∂ν ∆u1|Γ1 = ∂ν ∆u2|Γ1,

u2|Γ = 0, ∆u2|Γ = −a∂ν∂tu2|Γ,

u1(x, 0) = u
0
1(x), ∂tu(x, 0) = u

1
1(x) in Ω1,

u2(x, 0) = u
0
2(x), ∂tu(x, 0) = u

1
2(x) in O,

(1.1)

where c > 1 is a constant, ν denotes the inner unit normal to the boundary and a is a non-negative

function on Γ. We suppose that there exists a constant a0 > 0 such that

a ≥ a0 on Γ. (1.2)

The controllability of the dynamical system determined by (1.1) without transmission (i.e. when

Ω1 = ∅) has been investigated by Krabs, Leugering and Seidman [10], Leugering [9], Lasiecka

and Triggiani [5], Ammari and Khenissi [1]. With transmission the exact controllability has been

established by Liu and Williams [11] in the case when the control is active on the part of boundary

whereas the controlled part of the boundary is supposed to satisfy the Lions geometric condition.

In this paper we prove that, under (1.2), the solutions of (1.1) are exponentially stable in the

energy space. The energy of a solution

u =

{
u1 in Ω1,

u2 in O,

of (1.1) at the time instant t is defined by

E(t) =
1

2

∫

Ω

(
|∂tu(x,t)|

2
+ α2(x) |∆u(x,t)|

2
)
α(x)−1 dx,

where

α(x) =

{
c in Ω1,

1 in O.

The solution of (1.1) satisfies the energy identity

E(t2) − E(t1) = −

∫ t2

t1

∫

Γ

a |∂ν∂tu(x,t)|
2
dΓ dt



CUBO
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Boundary Stabilization of the Transmission ... 41

for all t2 > t1 ≥ 0, and therefore the energy is a nonincreasing function of the time variable t.

Introduce the Hilbert space H = V × H, where H = L2(Ω,α(x)−1dx) and the space V is definded

as follows. On the Hilbert space H consider the operator G defined by

G

(
u1

u2

)
=

(
− c∆u1

− ∆u2

)
, ∀

(
u1
u2

)
∈ D(G),

with domain

D(G) =
{
(u1,u2) ∈ H = L

2
(Ω1,c

−1dx) ⊕ L2(O) : u1 ∈ H
2
(Ω1),u2 ∈ H

2
(O),

u2|Γ = 0, u1|Γ1 = u2|Γ1, ∂νu1|Γ1 = ∂νu2|Γ1

}
.

The operator G is a strictly positive self-adjoint one with a compact resolvent. Set V = D(G) with

norm ‖f‖V := ‖Gf‖H . The solutions to (1.1) can be expressed by means of a semigroup on H as

follows (
u

ut

)
= eitA

(
u0

u1

)
,

where the operator A is defined by

A

(
u

v

)
= −i

(
v

− α2 ∆2u

)
, ∀

(
u
v

)
∈ D(A),

with domain

D(A) =
{
(u,v) ∈ H : (v, ∆2u) ∈ H, u|Γ = 0, ∆u|Γ = −a∂νv|Γ, u1|Γ1 = u2|Γ1,

∂νu1|Γ1 = ∂νu2|Γ1, c ∆u1|Γ1 = ∆u2|Γ1, c∂ν ∆u1|Γ1 = ∂ν ∆u2|Γ1

}
.

Using Green’s formula it is easy to see that

Im

〈
A

(
u
v

)
,

(
u
v

)〉

H

=

∫

Γ

a |∂νv|
2
dΓ ≥ 0, ∀

(
u
v

)
∈ D(A), (1.3)

which in turn implies that A generates a continuous semigroup (e.g. see Theorems 4.3 and 4.6

from [12, p.14-15]). Let ρ(A) denote the resolvent set of A. Since the resolvent of A is a compact

operator on H, C \ ρ(A) is a discrete set of eigenvalues of A. It follows from (1.3) that

C \ ρ(A) ⊂ {z ∈ C : Im z ≥ 0}.

Moreover, using the Carleman estimates of [3] one can conclude that, under the condition (1.2),

the operator A has no eigenvalues on the real axis. Our main result is the following

Theorem 1.1 Assume (1.2) fulfilled. Then there exist constants C,γ > 0 such that

E(t) ≤ C e−γt E(0). (1.4)



42 Kaïs Ammari and Georgi Vodev CUBO
11, 5 (2009)

When Ω1 = ∅ the estimate (1.4) follows from combining the results of [1] and [2] under

the more natural assumptions that (1.2) holds only on some non-empty part Γ0 of Γ and that

every generalized ray in Ω hits Γ0 at a non-diffractive point (see [2] for the definition and more

details). Therefore in the general case of transmission, the estimate (1.4) should hold true under

less restrictive conditions, but to our best knoweledge no such results have been proved so far.

One of the reasons for this is the fact that the classical flow for the transmission problem is quite

complex. Indeed, when a ray in O hits Γ1 it splits into two rays - a reflected one staying in O and

another one entering into Ω1. The picture is similar when a ray in Ω1 hits Γ1. In particular, there

are rays which never reach the boundary Γ where the dissipation is active. Note that it is crucial

for (1.4) to hold that c > 1. Roughly speaking, what happens in this case is that the rays staying

inside Ω1 carry a negligible amount of energy. Note that when c < 1 the estimate (1.4) does not

hold anymore. Indeed, in this case one can use the quasi-modes constructed in [13] (which are

due to the existence of the so-called interior totally reflected rays and which are concentrated on

Γ1) to get quasi-modes for our problem, too. Consequently, we conclude that when c < 1 there

exist infinitely many eigenvalues {λj} of A such that 0 < Im λj ≤ CN |λj|
−N , ∀N ≫ 1, which is

an obstruction to (1.4).

To prove Theorem 1.1 it suffices to show that (A − z)−1 : H → H is O(1) for z ∈ R, |z| ≫ 1,

which implies that (A − z)−1 is analytic in Im z ≤ C, C > 0. This is carried out in Section 3 (see

Theorem 3.1) using the a priori estimates established in Section 2. The advantage of this approach

is that the problem of proving (1.4) is reduced to obtaining a priori estimates for the solutions

to the corresponding stationary problem (see Theorem 2.1). This in turn is easier than studying

(1.1) directly because we can make use of the estimates for the stationary transmission problem

obtained in [4] (under the assumptions that Ω1 is strictly convex and c > 1).

2 A Priori Estimates

Let λ ≫ 1 and consider the problem





(−c2∆2 + λ4)u1 = v1 in Ω1,

(−∆2 + λ4)u2 = v2 in O,

u1|Γ1 = u2|Γ1, ∂νu1|Γ1 = ∂νu2|Γ1, c∆u1|Γ1 = ∆u2|Γ1, c∂ν ∆u1|Γ1 = ∂ν ∆u2|Γ1,

u2|Γ = 0, ∆u2|Γ = a
(
−iλ2∂νu2|Γ + F

)
,

(2.1)

with functions v1 ∈ L
2
(Ω1), v2 ∈ L

2
(O) and F ∈ L2(Γ). In what follows in this section all Sobolev

spaces Hs, s ≥ 0, will be equipped with the semi-classical norm (with a small parameter λ−1).

This means that the semi-classical norm in Hs is equivalent to the classical one times a factor λ−s.

Theorem 2.1 Assume (1.2) fulfilled. Then there exist constants C,λ0 > 0 so that for λ ≥ λ0 the

solution to (2.1) satisfies the estimate

‖u1‖H3(Ω1) + ‖u2‖H3(O) ≤ Cλ
−2 ‖v1‖L2(Ω1) + Cλ

−2 ‖v2‖L2(O) + Cλ
−2 ‖F‖L2(Γ) . (2.2)



CUBO
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Boundary Stabilization of the Transmission ... 43

Proof. We will first prove the following

Lemma 2.2 Assume (1.2) fulfilled. Then for every λ,µ > 0 we have the estimate

‖∂νu2|Γ‖L2(Γ) ≤ Cµ‖u1‖L2(Ω1) + Cµ‖u2‖L2(O)

+Cλ−2µ−1 ‖v1‖L2(Ω1) + Cλ
−2µ−1 ‖v2‖L2(O) + Cλ

−2 ‖F‖L2(Γ) . (2.3)

Proof. Applying Green’s formula in Ω1 and O, we get respectively

Im
〈
c−1v1,u1

〉
L2(Ω1)

= Im 〈c∆u1|Γ1,∂νu1|Γ1〉L2(Γ1) − Im 〈c∂ν ∆u1|Γ1,u1|Γ1〉L2(Γ1) , (2.4)

Im 〈v2,u2〉L2(O) = −Im 〈∆u2|Γ1,∂νu2|Γ1〉L2(Γ1) + Im 〈∂ν ∆u2|Γ1,u2|Γ1〉L2(Γ1)

+Im 〈∆u2|Γ,∂νu2|Γ〉L2(Γ) . (2.5)

Summing up (2.4) and (2.5) and using the boundary conditions, we obtain the identity

Im
〈
c−1v1,u1

〉
L2(Ω1)

+ Im 〈v2,u2〉L2(O)

= −λ2 〈a∂νu2|Γ,∂νu2|Γ〉L2(Γ) + 〈F,a∂νu2|Γ〉L2(Γ) . (2.6)

By (1.2) and (2.6), we conclude

a0λ
2 ‖∂νu2|Γ‖

2
L2(Γ) ≤ λ

2µ2 ‖u1‖
2
L2(Ω1)

+ λ2µ2 ‖u2‖
2
L2(O)

+λ−2µ−2 ‖v1‖
2
L2(Ω1)

+ λ−2µ−2 ‖v2‖
2
L2(O) + Cλ

−2 ‖F‖
2
L2(Γ) , (2.7)

which clearly implies (2.3). 2

It is easy to see that (2.2) follows from combining Lemma 2.2 with the following

Proposition 2.3 There exist constants C,λ0 > 0 so that for λ ≥ λ0 the solution to (2.1) satisfies

the estimate

‖u1‖H3(Ω1) + ‖u2‖H3(O) ≤ Cλ
−2 ‖v1‖L2(Ω1) + Cλ

−2 ‖v2‖L2(O)

+C ‖∂νu2|Γ‖L2(Γ) + Cλ
−2 ‖F‖L2(Γ) . (2.8)

Proof. Clearly, the functions ũ1 = (λ
2 − c∆)u1, ũ2 = (λ

2 − ∆)u2 satisfy the equation






(c∆ + λ2)ũ1 = v1 in Ω1,

(∆ + λ2)ũ2 = v2 in O,

ũ1|Γ1 = ũ2|Γ1, ∂νũ1|Γ1 = ∂νũ2|Γ1.

(2.9)

Using the results of [4] we will prove the following



44 Kaïs Ammari and Georgi Vodev CUBO
11, 5 (2009)

Proposition 2.4 There exist constants C,λ0 > 0 so that for λ ≥ λ0 the solution to (2.9) satisfies

the estimate

‖ũ1‖H1(Ω1) + ‖ũ2‖H1(O) ≤ Cλ
−1 ‖v1‖L2(Ω1) + Cλ

−1 ‖v2‖L2(O)

+C ‖ũ2|Γ‖L2(Γ) + Cλ
−1 ‖∂νũ2|Γ‖L2(Γ) . (2.10)

Proof. Denote by d(x) the distance between x ∈ O and Γ. Choose a function χ ∈ C∞(Rn), χ = 1

on Ω1, χ = 0 on R
n \ Ω, so that supp [∆,χ] ⊂ {x ∈ O : d(x) ≤ δ}, where 0 < δ ≪ 1. Choose also

a function ϕ(t) ∈ C∞0 (R), 0 ≤ ϕ ≤ 1, ϕ(t) = 1 for |t| ≤ δ, ϕ(t) = 0 for |t| ≥ 2δ, dϕ(t)/dt ≤ 0 for

t ≥ 0, and set ψ(x) = ϕ(d(x)). Clearly, ψ = 1 on {x ∈ O : d(x) ≤ δ}. We have





(c∆ + λ2)ũ1 = v1 in Ω1,

(∆ + λ2)χũ2 = ṽ2 = χv2 + [∆,χ]ũ2 in R
n \ Ω1,

ũ1|Γ1 = χũ2|Γ1, ∂νũ1|Γ1 = ∂νχũ2|Γ1.

(2.11)

By the results of [4] (see (4.2)) we have

‖ũ1‖H1(Ω1) + ‖χũ2‖H1(Rn\Ω1) ≤ Cλ
−1 ‖v1‖L2(Ω1) + Cλ

−1 ‖ṽ2‖L2(Rn\Ω1)

≤ Cλ−1 ‖v1‖L2(Ω1) + Cλ
−1 ‖v2‖L2(O) + C ‖ψũ2‖H1(O) . (2.12)

Since Γ is strictly concave viewed from the interior, we have the following (see Proposition 2.2 of

[4])

Proposition 2.5 There exist constants C,δ0 > 0 so that for 0 < δ ≤ δ0 we have the estimate

‖ψũ2‖H1(O) ≤ Cλ
−1
∥∥(∆ + λ2)ũ2

∥∥
L2(O)

+ C ‖ũ2|Γ‖L2(Γ)

+Cλ−1 ‖∂νũ2|Γ‖L2(Γ) + Oδ(λ
−1/2

) ‖ũ2‖H1(O) . (2.13)

Combining (2.12) and (2.13), we get

‖ũ1‖H1(Ω1) + ‖ũ2‖H1(O)

≤ Cλ−1 ‖v1‖L2(Ω1) + Cλ
−1 ‖v2‖L2(O) + Cλ

−1/2 ‖ũ2‖H1(O)

+C ‖ũ2|Γ‖L2(Γ) + Cλ
−1 ‖∂νũ2|Γ‖L2(Γ) , (2.14)

which clearly implies (2.10). 2

By Green’s formula we have

Re
〈
c−1ũ1,u1

〉
L2(Ω1)

= c−1λ2 ‖u1‖
2
L2(Ω1)

+ ‖∇u1‖
2
L2(Ω1)

+ Re 〈∂νu1|Γ1,u1|Γ1〉L2(Γ1) ,

Re 〈ũ2,u2〉L2(O) = λ
2 ‖u2‖

2
L2(O) + ‖∇u2‖

2
L2(O) − Re 〈∂νu2|Γ1,u2|Γ1〉L2(Γ1)



CUBO
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Boundary Stabilization of the Transmission ... 45

+Re 〈∂νu2|Γ,u2|Γ〉L2(Γ) .

Summing up these identities and using that u2|Γ = 0, we get

Re
〈
c−1ũ1,u1

〉
L2(Ω1)

+ Re 〈ũ2,u2〉L2(O)

= c−1λ2 ‖u1‖
2
L2(Ω1)

+ ‖∇u1‖
2
L2(Ω1)

+ λ2 ‖u2‖
2
L2(O) + ‖∇u2‖

2
L2(O) . (2.15)

It is easy to see that (2.15) implies the estimate

‖u1‖H2(Ω1) + ‖u2‖H2(O) ≤ Cλ
−2 ‖ũ1‖L2(Ω1) + Cλ

−2 ‖ũ2‖L2(O) . (2.16)

Applying the same arguments to the functions ∇u1 and ∇u2 we also get

∥∥λ−1∇u1
∥∥

H2(Ω1)
+
∥∥λ−1∇u2

∥∥
H2(O)

≤ Cλ−2 ‖ũ1‖H1(Ω1) + Cλ
−2 ‖ũ2‖H1(O)

+ελ−1/2
∥∥(λ−1∂ν )2u2|Γ

∥∥
L2(Γ)

+ Cε−1λ−1/2
∥∥λ−1∂νu2|Γ

∥∥
L2(Γ)

, (2.17)

for any ε > 0. On the other hand, by the trace theorem we have

∥∥λ−1∂νu2|Γ
∥∥

L2(Γ)
≤ Cλ1/2 ‖u2‖H2(O) , (2.18)

∥∥(λ−1∂ν )2u2|Γ
∥∥

L2(Γ)
≤ Cλ1/2 ‖u2‖H3(O) . (2.19)

Thus, combining (2.16)-(2.19) and taking ε small enough, independent of λ, we conclude

‖u1‖H3(Ω1) + ‖u2‖H3(O) ≤ Cλ
−2 ‖ũ1‖H1(Ω1) + Cλ

−2 ‖ũ2‖H1(O) . (2.20)

By (2.10) and (2.20) we arrive at the estimate

‖u1‖H3(Ω1) + ‖u2‖H3(O) ≤ Cλ
−3 ‖v1‖L2(Ω1) + Cλ

−5/2 ‖v2‖L2(O)

+Cλ−1 ‖∂νu2|Γ‖L2(Γ) + Cλ
−2 ‖∆u2|Γ‖L2(Γ) + Cλ

−3 ‖∂ν ∆u2|Γ‖L2(Γ) . (2.21)

It is easy to see now that (2.8) follows from combining (2.21) and the following

Lemma 2.6 We have the estimate

λ−3 ‖∂ν ∆u2|Γ‖L2(Γ) ≤ Cλ
−1 ‖∂νu2|Γ‖L2(Γ)

+Cλ−2 ‖∆u2|Γ‖L2(Γ) + Cλ
−5/2 ‖v2‖L2(O) + Cλ

−1/2 ‖u2‖H3(O) . (2.22)

Proof. Choose a function ψ ∈ C∞(Rn) such that ψ = 1 on a small neighbourhood of Rn \ Ω,

ψ = 0 outside another small neighbourhood of Rn \Ω. Then the function w = (∆+λ2)ψu2 satisfies

the equation 




(−∆ + λ2)w = w̃ = −ψv2 − [∆
2,ψ]u2 in Ω,

w|Γ = ∆u2|Γ,

∂νw|Γ = ∂ν ∆u2|Γ + λ
2∂νu2|Γ.

(2.23)



46 Kaïs Ammari and Georgi Vodev CUBO
11, 5 (2009)

Clearly, to prove (2.22) it suffices to show the estimate

λ−1 ‖∂νw|Γ‖L2(Γ) ≤ C ‖w|Γ‖L2(Γ) + Cλ
−3/2 ‖w̃‖L2(Ω) . (2.24)

By the trace theorem we have

λ−1 ‖∂νw|Γ‖L2(Γ) + ‖w|Γ‖L2(Γ) ≤ Cλ
1/2 ‖w‖H2(Ω) . (2.25)

On the other hand, by Green’s formula

Re 〈w̃,w〉L2(Ω) = λ
2 ‖w‖

2
L2(Ω) + ‖∇w‖

2
L2(Ω) + Re 〈∂νw|Γ,w|Γ〉L2(Γ) ,

we get

‖w‖H2(Ω) ≤ Cλ
−2 ‖w̃‖L2(Ω) + ελ

−1/2
∥∥λ−1∂νw|Γ

∥∥
L2(Γ)

+ Cε−1λ−1/2 ‖w|Γ‖L2(Γ) (2.26)

for any ε > 0. Combining (2.25) and (2.26), and taking ε small enough, independent of λ, we

obtain (2.24). 2

3 Resolvent Estimates

We will show that Theorem 2.1 implies the following

Theorem 3.1 Under (1.2), we have the bound

∥∥(A − z)−1
∥∥
H→H

≤ Const, ∀z ∈ R. (3.1)

Proof. We will derive (3.1) from (2.2). Clearly, it suffices to prove (3.1) for |z| ≫ 1. Without loss

of generality we may suppose that z > 0. Let

(
u
v

)
∈ D(A)

satisfy the equation

(A − z)

(
u

v

)
=

(
f

g

)
∈ H.

Clearly, this equation can be rewritten as follows

(
−iv − zu

iα2∆2u − zv

)
=

(
f

g

)
.

Hence the function

u =

{
u1 in Ω1,

u2 in O,



CUBO
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Boundary Stabilization of the Transmission ... 47

satisfies the equation





(−c2∆2 + z2)u1 = ig1 − izf1 in Ω1,

(−∆2 + z2)u2 = ig2 − izf2 in O,

u1|Γ1 = u2|Γ1, ∂νu1|Γ1 = ∂νu2|Γ1, c∆u1|Γ1 = ∆u2|Γ1, c∂ν ∆u1|Γ1 = ∂ν ∆u2|Γ1,

u2|Γ = 0, ∆u2|Γ = a (−iz∂νu2|Γ + ∂νf2|Γ) .

(3.2)

Clearly, (3.1) is equivalent to the estimate

‖∆u‖L2(Ω) + ‖v‖L2(Ω) ≤ C ‖∆f‖L2(Ω) + C ‖g‖L2(Ω) . (3.3)

To prove (3.3) we write u = ug + uf , v = vg + vf , where

(A − z)

(
ug

vg

)
=

(
0

g

)
, (A − z)

(
uf

vf

)
=

(
f

0

)
.

Clearly, ug = (u
g
1,u

g
2) (resp. u

f
= (u

f
1,u

f
2 )) satisfies (3.2) with f ≡ 0 (resp. g ≡ 0). Applying

Theorem 2.1 with λ = z1/2 and F ≡ 0 and recalling that vg = zug, we get

‖∆ug‖L2(Ω) + ‖v
g‖L2(Ω) ≤ C ‖g‖L2(Ω) , (3.4)

with a constant C > 0 independent of z. It is easy to see that (3.3) would follow from (3.4) and

the estimate ∥∥∆uf
∥∥

L2(Ω)
+
∥∥vf

∥∥
L2(Ω)

≤ C ‖Gf‖H , ∀f ∈ D(G), (3.5)

with a constant C > 0 independent of z. In what follows we will derive (3.5) from Theorem 2.1.

Choose a function φ ∈ C∞0 (R), φ = 1 on [1/2, 2], φ = 0 on (−∞, 1/3]∪[3, +∞). Write f = f
♭
+f♮,

where

f♭ = φ(G/z)f, f♮ = (1 − φ)(G/z)f.

We have uf = u♭ + u♮, vf = v♭ + v♮, where

(A − z)

(
u♭

v♭

)
=

(
f♭

0

)
, (A − z)

(
u♮

v♮

)
=

(
f♮

0

)
.

Clearly, u♭ = (u♭1,u
♭
2) (resp. u

♮
= (u

♮
1,u

♮
2)) satisfies (3.2) with g ≡ 0 and f = f

♭ (resp. f = f♮).

Applying Theorem 2.1 with λ = z1/2 and F = ∂νf
♭
2|Γ leads to the estimate

∥∥∥∆u♭
∥∥∥

L2(Ω)
+

∥∥∥v♭
∥∥∥

L2(Ω)
≤
∥∥∥∆u♭

∥∥∥
L2(Ω)

+ z
∥∥∥u♭
∥∥∥

L2(Ω)
+

∥∥∥f♭
∥∥∥

L2(Ω)

≤ Cz
∥∥∥f♭
∥∥∥

L2(Ω)
+ C

∥∥∥∂νf♭2|Γ
∥∥∥

L2(Γ)
≤ Cz

∥∥∥f♭
∥∥∥

L2(Ω)
+ C

∥∥∥f♭2
∥∥∥

H2(O)

≤ Cz
∥∥∥f♭
∥∥∥

H
+ C

∥∥∥Gf♭2
∥∥∥

H
≤ C

∥∥zG−1φ(G/z)
∥∥

H→H
‖Gf‖H

+C ‖φ(G/z)‖H→H ‖Gf‖H ≤ C ‖Gf‖H , (3.6)



48 Kaïs Ammari and Georgi Vodev CUBO
11, 5 (2009)

with a constant C > 0 independent of z, where the Sobolev space H2(O) is equipped with the

classical norm and we have used the trace theorem together with the fact that the norms on H

and L2(Ω) are equivalent. We would like to get a similar estimate for the functions u♮ and v♮. Set

U = (z − G)−1(z + G)−1(−izf♮) = −iz(z − G)−1(z + G)−1(1 − φ)(G/z)f.

Since the operator G−1 is bounded on H, we have

‖U‖H ≤ C ‖GU‖H ≤ Cz
−1 ‖Gf‖H (3.7)

with a constant C > 0 independent of z. It is easy to see that the function U = (U1,U2) satisfies

the equation





(−c2∆2 + z2)U1 = −izf
♮
1 in Ω1,

(−∆2 + z2)U2 = −izf
♮
2 in O,

U1|Γ1 = U2|Γ1, ∂νU1|Γ1 = ∂νU2|Γ1, c∆U1|Γ1 = ∆U2|Γ1, c∂ν ∆U1|Γ1 = ∂ν ∆U2|Γ1,

U2|Γ = 0, ∆U2|Γ = 0.

Hence the function w = u♮ − U satisfies the equation





(−c2∆2 + z2)w1 = 0 in Ω1,

(−∆2 + z2)w2 = 0 in O,

w1|Γ1 = w2|Γ1, ∂νw1|Γ1 = ∂νw2|Γ1, c∆w1|Γ1 = ∆w2|Γ1, c∂ν ∆w1|Γ1 = ∂ν ∆w2|Γ1,

w2|Γ = 0, ∆w2|Γ = a
(
−iz∂νw2|Γ + ∂νf

♮
2|Γ − iz∂νU2|Γ

)
.

Therefore, by Theorem 2.1 we obtain

∥∥∆u♮ − ∆U
∥∥

L2(Ω)
+ z

∥∥u♮ − U
∥∥

L2(Ω)
≤ C

∥∥∥∂νf♮2|Γ
∥∥∥

L2(Γ)
+ Cz ‖∂νU2|Γ‖L2(Γ)

≤ C
∥∥∥f♮2
∥∥∥

H2(O)
+ Cz ‖U2‖H2(O) ≤ C

∥∥Gf♮
∥∥

H
+ Cz ‖GU‖H . (3.8)

By (3.7) and (3.8) we conclude

∥∥∆u♮
∥∥

L2(Ω)
+
∥∥v♮
∥∥

L2(Ω)
≤
∥∥∆u♮

∥∥
L2(Ω)

+ z
∥∥u♮
∥∥

L2(Ω)
+
∥∥f♮
∥∥

L2(Ω)
≤ C ‖Gf‖H . (3.9)

Clearly, (3.5) follows from (3.6) and (3.9). 2

Received: March, 2009. Revised: April, 2009.



CUBO
11, 5 (2009)

Boundary Stabilization of the Transmission ... 49

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