Báo cáo hóa học: " On Friedrichs-type inequalities in domains rarely perforated along the boundary" ppt

RESEARCH Open Access
On Friedrichs-type inequalities in domains rarely
perforated along the boundary
Yulia Koroleva
1,2
, Lars-Erik Persson
2,3
and Peter Wall
2*
* Correspondence:
2
Department of Engineering
Sciences and Mathematics, Luleå
University of Technology, SE-971
87 Luleå, Sweden
Full list of author information is
available at the end of the article
Abstract
This article is devoted to the Friedrich s inequality, where the domain is periodically
perforated along the boundary. It is assumed that the functions satisfy homogeneous
Neumann boundary conditions on the outer boundary and that they vanish on the
perforation. In particular, it is proved that the best constant in the inequality
converges to the best constant in a Friedrichs-type inequality as the size of the
perforation goes to zero much faster than the period of perforation. The limit
Friedrichs-type inequality is valid for functions in the Sobolev space H
1
.
AMS 2010 Subject Classification: 39A10; 39A11; 39A70; 39B62; 41A44; 45A05.
Keywords: Friedrichs-type inequ alities, homogenization, perforated along the
boundary
1 Introduction

This article deals with Friedrichs-type inequalities f or functions defined on domains
which have a periodic perforation along the boundary. The size, shape and distribution
of the perforation are described by a small parameter. It is assumed that the perfora-
tion is “rare”, i.e., the size of the local perforation is much smaller than the period of
perforation.Moreover,weconsiderthecasewherethefunctionssatisfyahomoge-
neous Neumann condition on the part of the boundary corresponding to the domain
without perfo ration and vanish on the perforation. The questions we are interested in
are; how does the best constant in the Friedrichs-type inequality depend on the small
parameter and what happens in the limit case where the parameter tends to zero? In
particular, we will prove that the best constant converges to th e best constant in a dif-
ferent type of Friedrichs inequality. The limit inequality is valid for all functions in the
Sobolev space H
1
.
Similar questions, with different types of microheterogeneities in a neighborhood of
the boundary, were studied in [1-5]. In [1] (see also [2]), domains with a periodical
perforation along the boundary were considered and the precise asymptotics of the
best constant in a Friedrichs-type inequality was established. It was assumed that the
size of perforation and the period were of the same order. Two different cases with
non-periodical perforation were considere d in [4,5]. The convergence of the constant,
as the size of perforation tends to zero, to the constant in the limit inequality was
proved. In [3], a Friedrichs-type inequality was proved for functions vanishing on small
periodically alternating pieces of t he boundary. The length of the pieces where the
Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129
/>© 2011 Koroleva et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativec ommons.or g/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
any medium, pro vided the original work is properly cited.
functions vanish was assumed to be o f the same order as the length of the period. In
particular, the precise asymptotics of the best constant, with respect to the small para-
meter describing the heterogeneous boundary condition, was derived.

2 Preliminaries and statement of the problem
Let Ω ⊂ ℝ
2
be a bounded domain such that the boundary, ∂Ω, is Lipschitz continuous.
Suppose that coordinates (x
1
, x
2
)areusedinΩ. Introduce local coordinates (s, t)ina
small neighborhood of ∂Ω in the following way: choose the origin O Î ∂Ω and a point
P(s, t) in a neighborhood of ∂Ω.Then,t is the distance from P to the boundary and s
is the counter clock-wise length of the boundary from O to the point P
1
(s, t), where
P
1
is the point for which
t =
|
PP
1
|
(see Figure 1).
Consider a semi-strip B ={ξ Î ℝ
2
:0<ξ
1
<1,ξ
2
>0}andaclosedsetT

μ
⊂ B
depending on a small parameter μ Î (0, 1] which characterizes the size and the shape
of the perforation (see Figure 2).
We study the case when T
μ
is shrinking in a uniform way as μ goes to zero. More-
over, we assume that T
μ
is uniformly bounded with respect to μ, i.e., there exists r Î
ℝ, r > 0 such that T
μ
⊂ {ξ Î ℝ
2
:0<ξ
1
<1,0<ξ
2
<r} for all μ Î (0, 1].
Let
T
1
μ
be 1-periodic extension of T
μ
with respect to ξ
1
and
T
ε

μ
is the image of
T
1
μ
under the mapping s = εξ
1
, t = εξ
2
,whereε is a small parameter,
0 <ε 1,
1
ε
∈
N
.
Define the domain

ε
= \T
ε
μ
(see Figure 2). Further, we assume that μ = μ(ε) and that
μ = μ
(
ε
)
→ 0asε → 0
.
(1)

Hence, ε > 0 is a parameter which describes both the size of the perforation and the
length of the period.
Consider the following spectral problem:
⎧
⎪
⎨
⎪
⎩
−u
ε
= λ
ε
u
ε
i
n 
ε
,
u
ε
=0 onT
ε
μ
,
∂u
ε
∂
ν
=0 on∂
.

(2)
Here ν denotes the unit outward normal to Ω. The limit problem for (2) depends on
how fast the size of the perforation goes to zero relative the length of the period. It
was proved in [6] (see also [7]) that if the perforation is “rare”, i.e., the size of the local
perforation goes to zero much faster than the period of perforation, then the limit pro-
blem for (2) is the Robin boundary value problem
P
(
s; t
)
t
s
P
1
(s;t)
O
:
:
W
¶
W
:
Figure 1 The local coordinate system.
Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129
/>Page 2 of 12

−u
0
= λ
0

u
0
in ,
∂u
0
∂
ν
+ pu
0
=0on∂
,
(3)
where 0 <p < ∞. The precise meaning of that the perforation is rare is given in (14)
later on. The faster the size of the local perforation goes to zero the smaller p will be.
There is a critical speed which gives that p is equal to zero, see [6,7]. In such situations
the limit problem is of N eumann type. The limit problem is of Dirichlet type (p = ∞)
when the size of the local perforation does not go to zero “fast enough” relative the
period of perforation.
According to the general theory o f elliptic operators, there exist countable sets
{λ
k
ε
}
and
{λ
k
0
}
of eigenvalues of (2) and (3) which satisfy
0 <λ

1
ε
≤ λ
2
ε
≤···≤λ
k
ε
≤···,0<λ
1
0
≤ λ
2
0
≤···≤λ
k
0
≤···
.
Using the same arguments as in [4], it follows that
λ
1
ε
> 0
.Thistogetherwiththe
variational formulation of the smallest eigenvalue of (2) lead to the following Frie-
drichs-type inequality for functions u Î H
1
(Ω) which vanish on
T

ε
μ
:


ε
u
2
dx ≤ K
ε


ε
|∇u|
2
dx
,
(4)
where K
ε
is the best constant and is given by
K
ε
=
1
λ
1
ε
.
(5)

In the case with p = ∞ (Dirichlet boundary conditions in the limit problem) the
smallest eigenvalue
λ
1
0
for the limit problem is related to the best constant in the Frie-
drichs inequality f or functions in
H
1
0
(
)
. Indeed, via the variational formulation of
λ
1
0
we have that


u
2
dx ≤ K
0


|∇u|
2
dx
,
(6)

where the best constant is given by
K
0
=1/λ
1
0
.
»
2
»
1
§
B
T
¹
1
e
W
0
Figure 2 Geometry of the perforated domain.
Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129
/>Page 3 of 12
A geometrical proof of that K
ε
® K
0
was presented in [4] for the case p = ∞.The
goal of this article is to answer the following questions, in the case 0 <p < ∞:(1)Is
there a Friedrichs-type inequality related to the limit Robin boundary value problem?
(2) If the answer on the first questions is yes, how is then K

ε
related to the best con-
stant, K
0
, in this Friedrichs-type inequality. We will see that there is such a Friedrichs-
type inequality and we p resent a result describing the asymptotic relation between K
ε
and K
0
. Moreover, as a result of our analysis we also obtain the convergence o f the
eigenvalues,
λ
k
ε
→ λ
k
0
, for the case 0 <p < ∞.
3 The main results
The following Friedrichs-type inequality holds for functions in H
1
(Ω):
Proposition 1 There exists a constant K
0
>0such that


u
2
dx ≤ K

0
⎛
⎝


|∇u|
2
dx + p

∂
u
2
ds
⎞
⎠
(7)
for any u Î H
1
(Ω). Moreover, the best constant is
K
0
=1/λ
1
0
,where
λ
1
0
is the smallest
eigenvalue in the limit problem (3).

Proof. The variational formulation of the smallest eigenvalue of the limit problem (3)
is
λ
1
0
= min
u∈H
1
()\{0}
⎧
⎪
⎨
⎪
⎩


|∇u|
2
dx + p

∂
u
2
ds


u
2
dx
⎫

⎪
⎬
⎪
⎭
.
For details, see paragraph 2.5 in [8]. From this, it is clear that the inequality (7)
holds. It also follows that the best constant is
1/λ
1
0
.
Let us define the following set of functions:
W =

u ∈ H
1
():
∂u
∂ν
+ pu =0 on∂,0< p < ∞

.
Note that solutions of the limit problem (3) belong to W. We remark that an
inequality of the form (6) cannot be valid for functions in W. Indeed,
Proposition 2 There is no C >0such that the inequality


u
2
dx ≤ C



|∇u|
2
d
x
(8)
holds for all functions in W.
Proof. We prove the statement by a counter example. Let
K
m
= {x ∈  :dist
(
x, ∂
)
≥ 1/m, m ∈ N}
.
Define the function u
m
such that u
m
= m + p on K
m
and u
m
= m on ∂Ω.Itispossi-
ble to construct a s mooth transition from K
m
to ∂Ω such that ∂u
m

/∂ν + pu
m
=0on
∂Ω and
|
∇u
m
|
2
≤ k

m + p −m
1/m

2
= kp
2
m
2
Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129
/>Page 4 of 12
in Ω\K
m
for s ome constant k.Notethat∇u
m
=0onK
m
and t hat |Ω\K
m
|~1/m (~

means asymptotically equal to). We get that


|∇u
m
|
2
dx


u
2
m
dx
≤
kp
2
m
2
|\K
m
|


u
2
m
dx
≤
kp

2
m
2
|\K
m
|
m
2
||
→
0
as m ® ∞. Thus, (8) cannot hold.
We will now consider how to estimate the difference between the best constants, K
0
and K
ε
, in t he inequalities (7) and (4). First, observe that by Proposition 1 and (5) we
have that
|
K
ε
− K
0
| =




1
λ

1
ε
−
1
λ
1
0




.
(9)
To estimate
|1/λ
1
ε
− 1/λ
1
0
|
we will use the i deas in the general method for estimating
the difference between eigenvalues and eigenvectors of two operators defined on differ-
ent spaces, which was introduced by Oleinik et al. [9], see also [10].
For the readers convenience, we review the main ideas in the method mentioned
above. Indeed, let H
ε
and H
0
be separable Hilbert spaces with the inner products

(u
ε
, v
ε
)
H
ε
,(u, v)
H
0
and norms
|
|u
ε
||
H
ε
, ||u||
H
0
, respectively; assume that A
ε
Î L(H
ε
)and
A
0
Î L(H
0
) are linear continuous operators and Im A

0
⊆ V ⊆ H
0
,whereV is a linear
subspace of H
0
. The following conditions are supposed to hold:
C1 There exist linear contin uous operators R
ε
: H
ε
® H
0
and a constant c >0such
that
(R
ε
f , R
ε
f )
H
ε
→ c(f , f )
H
0
as ε →
0
for any f Î V.
C2 The operators A
ε

: H
ε
® H
ε
and A
0
: H
0
® H
0
are positive, compact and self-
adjoint. Moreover,
sup
ε
||
A
ε
||
L
(
H
ε
)
< +
∞
.
C3 For all f ÎV it holds that
|
|A
ε

R
ε
f −R
ε
A
0
f ||
H
ε
→ 0asε → 0
.
C4 The sequence of operators A
ε
is uniformly compact in the following sense: if we
take a sequence {f
ε
}, where f
ε
Î H
ε
, such that
sup
ε
||f
ε
||
H
ε
< +
∞

, then there exist a sub-
sequence
{f
ε
k
}
and vector w
0
Î V such that
|
|A
ε
k
f
ε
k
− R
ε
k
w
0
||
H
ε
k
→
0
as ε
k
® 0.

Let us also introduce the spectral problems for operators A
ε
, A
0
:
A
ε
u
k
ε
= μ
k
ε
u
k
ε
, μ
1
ε
≥ μ
2
ε
≥···, k =1,2, (u
l
ε
, u
m
ε
)=δ
lm

,
(10)
A
0
u
k
0
= μ
k
0
u
k
0
, μ
1
0
≥ μ
2
0
≥···, k =1,2, (u
l
0
, u
m
0
)=δ
lm
,
(11)
where δ

lm
is the Kronecker symbol, the eigenvalues
μ
k
ε
, μ
k
0
are repeated according to
their multiplicities. The following lemma holds true (see [9, Chapter III]).
Lemma 3 Suppose that the conditions C1-C4 are valid. Then, there is a sequence
{β
k
ε
}
such that
β
k
ε
→ 0
as
ε → 0, 0 <β
k
ε
<μ
k
0
and the following estimate:
|
μ

k
ε
− μ
k
0
|≤
μ
k
0
c
−
1
2
μ
k
0
− β
k
ε
sup
v∈N(μ
k
0
,A
0
),||v||
H
0
=1
||A

ε
R
ε
v −R
ε
A
0
v||
H
ε
(12)
Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129
/>Page 5 of 12
holds, where
N( μ
k
0
, A
0
)={v ∈ H
0
: A
0
v = μ
k
0
v
}
.
Let us now give a more precise definition of that the perforation is rare. Indeed,

introduce the space V
μ(ε)
as the closure of the set of functions in v Î C
∞
(ℝ
2
∩ {ξ
2
>
0}) which are 1-periodic with respect to ξ
1
,vanishingonT
μ(ε )
and with finite

B
|∇v|
2
d
ξ
. The closure is with respect to the norm

v

=

B
|∇v|
2
dξ +



v
2
dξ
1
.More-
over, define the value
η
μ(ε)
=inf
v∈V
μ(ε)
\{0}

B
|∇v|
2
dξ


v
2
dξ
1
,
(13)
where Σ:=∂B ∩ {ξ
2
= 0} (see Figure 2). Moreover, we define the number p as

lim
ε→0
η
μ(ε)
ε
= p
.
(14)
In fact, the number p corresponds to the ratio between measure of small set T
μ
and
the length of p eriod, i.e., it describes how much of the Dirichlet condition pe r cell of
periodicity we have.
We will now prove the following estimate for |K
ε
- K
0
|:
Theorem 4 Let K
ε
and K
0
be the constants in (4) and (7). If 0<p < ∞ is defined by
(14), then there exists a constant C, independent of ε, such that
|
K
ε
− K
0
|≤C


√
η
μ(ε)
+



η
μ(ε)
ε
− p



+
√
εη
μ(ε)

.
(15)
Proof. By (9) we will have an estimate of |K
ε
- K
0
| if we have an estimate of
|1/λ
1
ε

− 1/λ
1
0
|
. In order to obtain such an estimate we will use the result in Lemma 3.
Indeed, we introduce two auxiliary problems:
⎧
⎪
⎨
⎪
⎩
−u
ε
= f in 
ε
,
u
ε
=0 onT
ε
μ
,
∂u
ε
∂ν
=0 on∂
(16)
and the corresponding limit problem

−u

0
= f in ,
∂u
0
∂
ν
+ pu
0
=0on∂
,
(17)
where f Î L
2
(Ω)andp satisfies (14). The fact that (17) is the limit problem for (16)
for any f was established in [6]. More precisely, it was proved that if u
ε
Î H
1
(Ω
ε
)and
u
0
Î H
1
(Ω) are weak solutions of (16) and (17), then u
ε
⇀ u
0
weakly in H

1
(Ω)asε ®
0 which implies the convergence
||
u
ε
− u
0
||
L
2
(

)
→ 0asε → 0
.
(18)
Note that here and from now on, u
ε
is assumed to be defined in whole Ω and van-
ishing on
T
ε
μ
.
Let us now prove the following estimates for the solutions of (16) and (17):

u
ε


H
1
(
ε
)
≤ k
1


f


L
2
(

ε
)
(19)
Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129
/>Page 6 of 12
and
|
|u
0
||
H
1
(


)
≤ k
2
||f ||
L
2
(

)
,
(20)
where k
1
and k
2
are independent of ε. First, we recall that using the technique devel-
oped in [4] (see also [5]), one can prove the following Friedrichs-type inequality for
functions w belonging to H
1
(Ω) and vanishing on
T
ε
μ
:


w
2
dx ≤ K



|∇w|
2
dx
,
(21)
where K does not depend on ε. In particular, the inequality (21) implies that the
solution of (16) satisfies the estimate


ε
u
2
ε
dx ≤ K


ε
|∇u
ε
|
2
dx
.
(22)
By choosing u
ε
as the test function in the weak formulation of (16), we have



ε
|∇u
ε
|
2
dx =


ε
fu
ε
dx
.
Using the Hölder inequality and (22), we obtain that
|
|∇u
ε
||
L
2
(

ε
)
≤
√
K||f ||
L
2
(


ε
)
.
From this and (22) the estimate (19) follows, with
k
1
=

K(1 + K)
. Let us now prove
the estimate (20). Indeed, we start by showing that for any w Î H
1
(Ω)\{0} there exists
a constant M which does not depend on w such that


|∇w|
2
dx|p

∂

w
2
ds ≥ M||w||
2
H
1
()

.
(23)
Suppose that the contradiction holds: i.e., that for any m there exists w
m
Î H
1
(Ω)\{0}
such that


|∇w
m
|
2
dx + p

∂

w
2
m
ds <
1
m
||w
m
||
2
H
1

()
.
Denote
v
m
= w
m
/||w
m
||
H
1
(

)
. Then,
||
v
m
||
H
1
(

)
=1
(24)
and



|∇v
m
|
2
dx + p

∂

v
2
m
ds <
1
m
.
(25)
By the inequalities (7) and (25), we have that


v
2
m
dx ≤ K
0
⎛
⎝


|∇v
m

|
2
dx + p

∂
v
2
m
ds
⎞
⎠
<
K
0
m
.
(26)
Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129
/>Page 7 of 12
From (25) and (26), it follows that v
m
® 0inH
1
( Ω), which contradicts to (24).
Thus, the estimate (23) is proved. Choosing u
0
as a test function in the weak formula-
tion of (17) leads to the identity



|∇u
0
|
2
dx + p

∂

u
2
0
ds =


fu
0
dx
.
(27)
By applying (23) to u
0
, using (27) and the Cauchy-Schwarz inequality, we get that
|
|u
0
||
H
1
()
≤

1
M
||f ||
L
2
()
,
(28)
which is estimate (20) with k
2
=1/M.
To estimate
|1/λ
1
ε
− 1/λ
1
0
|
we will now use the method, which was described above,
for estimating the difference between eigenvalues. Indeed, define the spaces H
ε
= L
2
(Ω
ε
), H
0
= V = L
2

(Ω) and the restriction operator R
ε
: H
0
® H
ε
. Define t he operators
A
ε
and A
0
in the following way: A
ε
f = u
ε
and A
0
f = u
0
,whereu
ε
and u
0
are the weak
solutions of problems (16) and (17), respectively. Let us verify the conditions C1-C4.
The condition C1 is valid with c = 1. Indeed, take f Î V. Then,
(R
ε
f , R
ε

f )
ε
=


ε
f
2
(x)dx →


f
2
(x)dx =(f , f)
0
as ε ® 0 due to the fact that measure of
T
ε
μ
→
0
as ε ® 0.
Let us verify the condition C2. First, we prove that the operator A
ε
is self-adjoint. Let
f and g be functions in L
2
(Ω
ε
)anddefineu

ε
= A
ε
f and v
ε
= A
ε
g.Ifwechosev
ε
as test
function in the weak formulation of (16) with f in the right-hand side and u
ε
as a test
function in the case when the right-hand side is g, then we obtain that


ε
fv
ε
dx =


ε
∇u
ε
·∇v
ε
dx =



ε
gu
ε
dx
.
Hence,
(A
ε
f , g)
L
2
(
ε
)
=(u
ε
, g)
L
2
(
ε
)
=


ε
∇v
ε
·∇u
ε

dx =(f, v
ε
)
L
2
(
ε
)
=(f , A
ε
g)
L
2
(
ε
)
.
Now, we prove the operator A
0
is self-adjoint. Define u
0
= A
0
f and v
0
= A
0
g, where f,
g Î L
2

(Ω). According to the weak formulation of (17), we find that
(A
0
f , g)
L
2
()
=


u
0
gdx =


∇v
0
·∇u
0
dx −

∂
u
0
∂v
0
∂ν
ds
=



∇v
0
·∇u
0
dx + p

∂
u
0
v
0
ds =


∇u
0
·∇v
0
dx −

∂
v
0
∂u
0
∂ν
d
s
=



fv
0
dx =(f, A
0
g)
L
2
()
.
Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129
/>Page 8 of 12
That the operator A
ε
is positive follows from
(A
ε
f , f )
L
2
()
=


u
ε
fdx =



|∇u
ε
|
2
dx >
0
if u
ε
≠ 0 (i.e., f ≠ 0). Similarly, we obtain that A
0
is positive. Indeed, the weak formu-
lation of (17) gives
(A
0
f , f )
L
2
()
=


u
0
fdx =


|∇u
0
|
2

dx −

∂
u
0
∂
u
0
∂ν
d
s
=


|∇u
0
|
2
dx + p

∂

u
2
0
ds > 0
if u
0
≠ 0. Next, we show that A
ε

and A
0
are compact operators. Let {f
n
} be a bounded
sequence in L
2
(Ω
ε
). Then, estimate (19) implies that there exists a constant c such that
|
|A
ε
f
n
||
L
2
(

ε
)
= ||u
ε,n
||
L
2
(

ε

)
≤||u
ε,n
||
H
1
(

ε
)
≤ k
1
||f
n
||
L
2
(

ε
)
≤ c
.
Hence, there exist a subsequence of {u
ε,n
}and
˜
u
ε
∈ H

1
(

ε
)
such that
u
ε,n
k

˜
u
ε
weakly in H
1
(Ω
ε
) and thus strongly in L
2
(Ω
ε
) which exactly means that A
ε
is compact.
Moreover, (19) implies that
|
|A
ε
f ||
L

2
(

ε
)
≤ k
1
||f ||
L
2
(

ε
)
for any f Î L
2
(Ω
ε
). Hence,
sup
ε

A
ε

L
(
H
ε
)

≤ k
1
. The compactness of A
0
can be proved analogously by applying esti-
mate (20) instead of (19).
Let us verify the condition C3 is fulfilled. Take f Î L
2
(Ω). It follows by (18) that
|
|A
ε
R
ε
f −R
ε
A
0
f ||
L
2
(

ε
)
= ||A
ε
f −A
0
f ||

L
2
(

ε
)
= ||u
ε
− u
0
||
L
2
(

)
→
0
as ε ® 0.
Let us verify that the condition C4 is satisfied. Consider a sequence { f
ε
}, where f
ε
Î
L
2
(Ω
ε
) such that
sup

ε
||f
ε
||
L
2
(

ε
)
< +
∞
. Then,
|
|A
ε
f
ε
||
H
−1
(

ε
)
= ||u
ε
||
H
−1

(

)
≤ k
1
||f
ε
||
L
2
(

ε
)
< +∞
,
due to (19). The Rellich imbedding theorem implies that the sequence {A
ε
f
ε
}iscom-
pact in L
2
(Ω). Thus, there exists a subsequence {ε
k
} and w
0
Î L
2
(Ω) such that

A
ε
k
f
ε
k
→ w
0
as ε
k
→ 0
.
From this, we deduce that
|
|A
ε
k
f
ε
k
− R
ε
k
w
0
||
L
2
(
ε

k
)
→
0
as ε
k
® 0. Hence, all the con-
ditions C1-C4 are valid.
Let l
ε
be an eigenvalue of the -Δ operator with the boundary conditions given in
(16) and v
ε
the corresponding eigenvector. In this notation, we have that -Δv
ε
= l
ε
v
ε
and thus A
ε
(l
ε
v
ε
)=v
ε
. From this, it is evident that A
ε
v

ε
= (1/l
ε
)v
ε
. From this, it follows
that
μ
k
ε
=1/λ
k
ε
(
μ
k
ε
is defined in (10)). In the same way, we can deduce that
μ
k
0
=1/λ
k
0
.
Using the estimate (12), we have






1
λ
k
ε
−
1
λ
k
0





≤
1
1 −λ
k
0
β
k
ε
sup
v∈N(μ
k
0
,A
0
),||v||

L
2
(

)
=1
||A
ε
R
ε
v −R
ε
A
0
v||
L
2
(
ε
)
.
(29)
Recall that
N( μ
k
0
, A
0
)={v ∈ H
0

: A
0
v = μ
k
0
v
}
.Let
v ∈ N(μ
k
0
, A
0
)
.Ifwechoosef = v in
the problem (17), then the solution, denoted by
u
v
0
, can be expressed as
Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129
/>Page 9 of 12
u
v
0
= A
0
v = μ
k
0

v
.
Similarly, if we choose f = R
ε
v in the problem (16), then the solution, denoted by
u
v
ε
,
is of the form
u
v
ε
= A
ε
R
ε
v
.
In this notation, (29) reads





1
λ
k
ε
−

1
λ
k
0





≤
1
1 −λ
k
0
β
k
ε
sup
v∈N(μ
k
0
,A
0
),||v||
L
2
(

)
=1

||u
v
ε
− R
ε
u
v
0
||
L
2
(
e
)
.
(30)
In [6], it was proved that (17) is the limit problem corresponding to (16). By the
results in [6], it follows that there exists a constant C
1
such that
||u
ε
− u
0
||
L
2
()
≤ C
1


||f ||
L
∞
()

√
η
μ
+



η
μ
ε
− p




+ ||u
0
||
L
∞
()
√
εη
μ


(31)
for any f Î L
∞
(Ω). In particular, for the present choice of f, f = v, we have
|
|u
v
ε
− u
v
0
||
L
2
()
≤ C
1

||v||
L
∞
()

√
η
μ
+




η
μ
ε
− p




+ μ
k
0
||v||
L
∞
()
√
εη
μ

.
This together with the fact t hat eigenfunctions belong to
C
∞
(
¯

)
gives that there is a
constant C

2
(which depends on k) such that
sup
v∈N(λ
k
0
,A
0
),||v||
L
2
(

)
=1
||u
v
ε
− u
v
0
||
L
2
(
ε
)
≤ C
2


√
η
μ
+



η
μ
ε
− p



+
√
εη
μ

.
(32)
From this and (30), we obtain





1
λ
k

ε
−
1
λ
k
0





≤
1
1 −λ
k
0
β
k
ε
C
2

√
η
μ
+



η

μ
ε
− p



+
√
εη
μ

.
By Lemma 3, we have that
1 −λ
k
0
β
k
ε
>
0
for sufficiently small values of ε (as
β
k
ε
→ 0
).
Hence, there exists a constant C, independent of ε, such that






1
λ
k
ε
−
1
λ
k
0





≤ C

√
η
μ
+



η
μ
ε
− p




+
√
εη
μ

(33)
and the proof is complete.
As a consequence of the proof above we have the following result:
Corollary 5 The eigenvalues
λ
k
ε
of (2) converge to the corresponding eigenvalue
λ
k
0
of
(3).
Proof. We note that by (33)
|
λ
k
0
− λ
k
ε
| = λ

k
ε
λ
k
0





1
λ
k
ε
−
1
λ
k
0





≤ λ
k
ε
λ
k
0

C

√
η
μ
+



η
μ
ε
− p



+
√
εη
μ

.
It follows from (33) that
{λ
k
ε
}
is bounded. Hence,
|
λ

k
0
− λ
k
ε
|≤λ
k
ε
λ
k
0
C

√
η
μ
+



η
μ
ε
− p



+
√
εη

μ

→
0
as ε ® 0.
Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129
/>Page 10 of 12
Remark 6 The result established in Theorem 4 is valid for a wide class o f domains
with perforation along the boundary. It is possible to estimate the difference |K
ε
- K
0
|
more precisely in some partial cases. For some particular cases, it is in fact possible to
construct an asymptotic expansion for K
ε
with respect to ε as ε ® 0 via the method of
matching of asymptotic expansion (see, e.g., [11]).
We end this work by giving an example where the result in Theorem 4 is applicable.
Example 7 Consider a semi-strip B ={ξ Î ℝ
2
:0<ξ
1
<1,ξ
2
>0}and a bounded
domain T ⊂ B with smooth boundary ∂T. Let
T
μ
= {ξ ∈ B : μ

−1
(ξ − ξ
0
) ∈
¯
T, ξ
0
∈ B}, μ ∈ (0, 1]
. For this case, the following estimate
was derived in [6] for sufficiently small μ <μ
0
, μ
0
Î (0, 1]:
2π
(
1+ε
)
ln μ
(
ε
)
≤ η
μ(ε)
≤
2π
ln μ
0
− ln μ
(

ε
)
.
(34)
Let us choose μ(ε)=e
-A/ε
, where A >0.Then,lnμ(ε)=-A/ε. In this case, the estimate
(34) leads to
2π
(
1+ε
)
A
≤
η
μ(ε)
ε
≤
2π
ε ln μ
0
+ A
.
(35)
Hence, h
μ(ε)
/ε ® 2π/A = p. By (15) and (35), we have that
|
K
ε

− K
0
|
≤ C
⎛
⎝

2πε
ε ln μ
0
+ A
+

2πε
2
ε ln μ
0
+ A
⎞
⎠
+ C max

2π
ε ln μ
0
+ A
−
2π
A
,

2π
A
−
2π
(
1+ε
)
A

.
Itcanbeobservedthatifμ (ε)=ε
a
, a > 0, then (34) gives that h
μ(ε)
/ε ® ∞.This
means that the perforation is vanishing too slow in order to have Robin boundar y con-
ditions in the limit problems (3) and (17).
Remark 8 The result obtained in this article can be generalized to higher dimensional
domain. The crucial step is to prove the estimate similar to (31). This is a good future
research problem.
Acknowledgements
The authors thank Professor Gregory A. Chechkin for valuable remarks. The article was completed during the stay of
Yulia Koroleva as Post Doc at Luleå University of Technology in 2010-2011 and was supported by the Luleå University
of Technology (Sweden) and partially by the RFBR (Project 12-01-00214). The study of the third author was supported
by a grant from the Swedish Research Council (Project 621-2008-5186). We also thank both referees for several
suggestions, which have improved the final version of this article.
Author details
1
Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow Lomonosov State University,
Moscow 119991, Russia

2
Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-
971 87 Luleå, Sweden
3
Narvik University College, Postboks 385, 8505 Narvik, Norway
Authors’ contributions
All authors carried out the proofs. All authors conceived of the study, and participated in its design and coordination.
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 10 March 2011 Accepted: 5 December 2011 Published: 5 December 2011
Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129
/>Page 11 of 12
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doi:10.1186/1029-242X-2011-129
Cite this article as: Koroleva et al.: On Friedrichs-type inequalities in domains rarely perforated along the
boundary. Journal of Inequalities and Applications 2011 2011:129.
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Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129
/>Page 12 of 12