Homogenization of the Robin problem in a thick multilevel junction
In the paper we consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ωε, which is the union of a domain Ω₀ and a large number 2N of thin rods with variable thickness of order ε = O(N⁻¹). The thin rods are divided into two levels depending on their length....
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| Cite this: | Homogenization of the Robin problem in a thick multilevel junction / U.De Maio, T.A. Mel'nyk, C. Perugia // Нелінійні коливання. — 2004. — Т. 7, № 3. — С. 336-355. — Бібліогр.: 6 назв. — англ. |
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| author | De Maio, U. Mel'nyk, T.A. Perugia, C. |
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| citation_txt | Homogenization of the Robin problem in a thick multilevel junction / U.De Maio, T.A. Mel'nyk, C. Perugia // Нелінійні коливання. — 2004. — Т. 7, № 3. — С. 336-355. — Бібліогр.: 6 назв. — англ. |
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| description | In the paper we consider a mixed boundary-value problem for the Poisson equation in a plane two-level
junction Ωε, which is the union of a domain Ω₀ and a large number 2N of thin rods with variable thickness
of order ε = O(N⁻¹). The thin rods are divided into two levels depending on their length. In addition,
the thin rods from each level are ε-periodically alternated. We investigate the asymptotic behaviour of the
solution as ε → 0 under the Robin conditions on the boundaries of the thin rods. By using some special
extension operators, the convergence theorem is proved.
Розглядається мiшана крайова задача для рiвняння Пуассона у плоскому дворiвневому з’єднаннi
Ωε, яке є об’єднанням деякої областi Ω₀ та великої кiлькостi 2N тонких стержнiв iз змiнною
товщиною порядку ε = O(N⁻¹) Тонкi стержнi роздiлено на два рiвнi в залежностi вiд їх довжини. Крiм того, тонкi стержнi з кожного рiвня ε-перiодично чергуються. Вивчено асимптотичну поведiнку розв’язку, коли ε → 0, при крайових умовах Робiна на межах тонких стержнiв.
Iз використанням спецiальних операторiв продовження доведено теорему збiжностi.
|
| first_indexed | 2025-12-02T09:28:58Z |
| format | Article |
| fulltext |
UDC 517.956
HOMOGENIZATION OF THE ROBIN PROBLEM
IN A THICK MULTILEVEL JUNCTION
УСЕРЕДНЕННЯ ЗАДАЧI РОБIНА В ГУСТОМУ БАГАТОРIВНЕВОМУ
З’ЄДНАННI
U. De Maio
Università degli Studi di Napoli Federico II
Complesso Monte S. Angelo-Edificio "T", Via Cintia, 80126 Napoli, Italia
e-mail: udemaio@unina.it
T. A. Mel’nyk
Kyiv Nat. Taras Shevchenko Univ.
Volodymyrs’ka Str. 64, 01033, Kyiv, Ukraine
e-mail: melnyk@imath.kiev.ua
C. Perugia
Università degli Studi di Napoli Federico II
Complesso Monte S. Angelo-Edificio "T", Via Cintia, 80126 Napoli, Italia
e-mail: perugia@unina.it
In the paper we consider a mixed boundary-value problem for the Poisson equation in a plane two-level
junction Ωε, which is the union of a domain Ω0 and a large number 2N of thin rods with variable thickness
of order ε = O(N−1). The thin rods are divided into two levels depending on their length. In addition,
the thin rods from each level are ε-periodically alternated. We investigate the asymptotic behaviour of the
solution as ε → 0 under the Robin conditions on the boundaries of the thin rods. By using some special
extension operators, the convergence theorem is proved.
Розглядається мiшана крайова задача для рiвняння Пуассона у плоскому дворiвневому з’єднаннi
Ωε, яке є об’єднанням деякої областi Ω0 та великої кiлькостi 2N тонких стержнiв iз змiнною
товщиною порядку ε = O(N−1). Тонкi стержнi роздiлено на два рiвнi в залежностi вiд їх дов-
жини. Крiм того, тонкi стержнi з кожного рiвня ε-перiодично чергуються. Вивчено асимпто-
тичну поведiнку розв’язку, коли ε → 0, при крайових умовах Робiна на межах тонких стержнiв.
Iз використанням спецiальних операторiв продовження доведено теорему збiжностi.
Introduction. In this paper we consider a new type of thick junctions, namely, thick multilevel
junctions. A thick multilevel junction is the union of some domain, which is called the junction’s
body, and a large number N = O(ε−1) of thin domains of thickness of order O(ε). Here ε is a
small parameter. The thin domains are divided into a finite number of levels depending on their
length. In addition, the thin domains from each levels are ε-periodically alternated along some
manifold on the boundary of the junction’s body. This manifold is called the joint zone.
The aim of researches is to develop rigorous asymptotic methods for boundary-value
problems in thick multilevel junctions when the parameter ε goes to 0, i.e., when the number
of the attached thin domains increases and their thickness decreases. The asymptotic methods,
c© U. De Maio, T. A. Mel’nyk, and C. Perugia, 2004
336 ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
HOMOGENIZATION OF THE ROBIN PROBLEM IN A THICK MULTILEVEL JUNCTION 337
which were developed in [1 – 3] are used. A spectral problem in a plane thick multilevel juncti-
on with the flat boundaries of the thin rods was considered in [4]. Here we consider a mixed
boundary-value problem for the Poisson equation in a plane thick two-level junction with vari-
able thickness of the thin rods.
1. Statement of the problem. Let a, d1, d2, b1, b2 be positive real numbers and let d1 ≤
≤ d2, 0 < b1 < b2 < 1. Consider two positive piecewise smooth functions h1 and h2 on
the the segments [−d1, 0] and [−d2, 0], respectively. Suppose the functions h1 and h2 satisfy the
following conditions:
∃ δ0 ∈ (b1, b2) ∀ x2 ∈ [−d1, 0] : 0 < b1 − h1(x2)/2, b1 + h1(x2)/2 < δ0;
∀ x2 ∈ [−d2, 0] : δ0 < b2 − h2(x2)/2, b2 + h2(x2)/2 < 1.
It follows from these assumptions that there exist positive constants m0, M0 such that
0 < m0 ≤ h1(x2) < δ0 and |h′1(x2)| ≤ M0 a. e. in [−d1, 0],
(1)
0 < m0 ≤ h2(x2) < 1− δ0 and |h′2(x2)| ≤ M0 a. e. in [−d2, 0].
Let us divide segment [0, a] into N equal segments [εj, ε(j + 1)], j = 0, . . . , N − 1. Here N
is a large integer, therefore, the value ε = a/N is a small discrete parameter.
A model plane thick two-level junction Ωε consists of the junction’s body
Ω0 = {x ∈ R2 : 0 < x1 < a, 0 < x2 < γ(x1) },
where γ ∈ C1([0, a]), γ(0) = γ(a), min[0,a] γ > 0, and a large number of the thin rods
G
(1)
j (ε) = {x ∈ R2 : |x1 − ε (j + b1)| < εh1(x2)/2, x2 ∈ (−d1, 0]}, j = 0, 1, . . . , N − 1,
G
(2)
j (ε) = {x ∈ R2 : |x1 − ε (j + b2)| < εh2(x2)/2, x2 ∈ (−d2, 0]}, j = 0, 1, . . . , N − 1,
i.e.,
Ωε = Ω0 ∪G(1)(ε) ∪G(2)(ε),
where G(1)(ε) = ∪N−1
j=0 G
(1)
j (ε), G(2)(ε) = ∪N−1
j=0 G
(2)
j (ε).
We see that the number of the thin rods is equal to 2N and they are divided into two levels
G(1)(ε) and G(2)(ε) depending on their length (we recall that d1 ≤ d2). The small parameter ε
characterizes the distance between the thin neighboring rods and their thickness. The thickness
of the rods from the first level is equal to εh1 and to εh2 for the rods from the second level.
These thin rods from each level are ε-periodically alternated along the segment I0 = {x : x1 ∈
∈ [0, a], x2 = 0}.
Denote by Υ(i,±)
j (ε) the lateral surfaces of the thin rodG(i)
j (ε); the signs "+"or "−" indicate
the right or left surface respectively. The base of G(i)
j (ε) will be denoted by Θ(i)
j (ε). Also we
introduce the following notations:
Υ(i,±)(ε) :=
N−1⋃
j=0
Υ(i,±)
j (ε), Θ(i)(ε) :=
N−1⋃
j=0
Θ(i)
j (ε),
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
338 U. DE MAIO, T. A. MEL’NYK, AND C. PERUGIA
Υ(i)(ε) := Υ(i,+)(ε) ∪Υ(i,−)(ε) ∪Θ(i)(ε),
for i = 1, 2.
In Ωε we consider the following mixed boundary-value problem:
−∆x uε(x) = fε(x), x ∈ Ωε,
∂νuε(x) = −ε k1 uε(x), x ∈ Υ(1)(ε),
∂νuε(x) = −ε k2 uε(x), x ∈ Υ(2)(ε),
∂p
x1uε(0, x2) = ∂p
x1uε(a, x2), x2 ∈ [0, γ(0)], p = 0, 1,
∂νuε(x) = 0, x ∈ Γε.
(2)
Here ∂ν = ∂/∂ν is the outward normal derivative, ∂x1 = ∂/∂x1, the constants k1 and k2 are
positive. Thus, we have the Robin conditions on the boundaries of the thin rods, the periodic
conditions on the vertical sides of Ω0 and the Neumann condition on the other part Γε of ∂Ωε.
We can regard without loss of generality that the right-hand side fε belongs toL2(Ω2),where
Ω2 is the interior of Ω0 ∪D2, D2 = (0, a) × (−d2, 0) is a rectangle that is filled up by the thin
rods from the second level in the limit passage as ε → 0. Similarly, D1 = (0, a) × (−d1, 0) and
Ω1 is the interior of Ω0 ∪D1.
We assume that
fε → f0 in L2(Ω2) as ε → 0. (3)
The aim of our research is to study the asymptotic behaviour of the solution to problem (2)
as ε → 0, i.e., when the number of attached thin rods infinitely increases and their thickness
tends to 0.
2. Auxiliary inequalities. First we recall that for every fixed value ε, in accordance with
the main results of the theory of boundary-value problems, there exists a unique weak solution
uε ∈ Hε to problem (2) such that the integral identity∫
Ωε
∇uε · ∇ϕdx + ε k1
∫
Υ(1)(ε)
uε ϕdlx + ε k2
∫
Υ(2)(ε)
uε ϕdlx =
∫
Ωε
fε(x)ϕ(x) dx (4)
holds for any function ϕ ∈ Hε, where
Hε = {u ∈ H1(Ωε) : ∂p
x1
u(0, x2) = ∂p
x1
u(a, x2), x2 ∈ [0, γ(0)], p = 0, 1}.
In addition, the solution uε satisfies the inequality
‖uε‖H1(Ωε) ≤ c1‖fε‖L2(Ωε). (5)
Let us show that the constant c1 in (5) is independent of the small parameter ε.
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
HOMOGENIZATION OF THE ROBIN PROBLEM IN A THICK MULTILEVEL JUNCTION 339
Lemma 1. For ε small enough, the usual norm ‖ · ‖H1(Ωε) in the Sobolev space H1(Ωε) and
the norm:
‖v‖ε,k1,k2 =
∫
Ωε
|∇v|2 dx + ε k1
∫
Υ(1)(ε)
v2 dlx + ε k2
∫
Υ(2)(ε)
v2 dlx
1/2
are uniformly equivalent, i.e., there exist constants C1 > 0, C2 > 0 and ε0 such that for all
ε ∈ (0, ε0) and any function v ∈ H1(Ωε) the inequalities
C1‖v‖H1(Ωε) ≤ ‖v‖ε,k1,k2 ≤ C2‖v‖H1(Ωε) (6)
are satisfied.
Proof. Let us defined the following function:
Y (t) =
{
−t+ b1, t ∈ [0, δ0),
−t+ b2, t ∈ [δ0, 1),
(7)
and then periodically extend it to R.
Integrating by parts the integral ε
∫
G(1)(ε)∪G(2)(ε)
Y (x1/ε) ∂x1v dx and taking into account
that the outward normal to the lateral surfaces Υ(i,±)
j (ε) of the thin rodG(i)
j (ε), except for some
set of measure zero, has the form
ν
(i)
± (ε) =
1√
1 + ε24−1|h′i(x2)|2
(
±1 , −ε h
′
i(x2)
2
)
, i = 1, 2, j = 0, . . . , N − 1, (8)
we get the identity
ε
2∑
i=1
∫
Υ(i,±)(ε)
hi(x2)
2
√
1 + ε24−1|h′i(x2)|2
v dlx =
=
∫
G(1)(ε)∪G(2)(ε)
v dx− ε
∫
G(1)(ε)∪G(2)(ε)
Y
(x1
ε
)
∂x1v dx ∀ v ∈ H1(Ωε). (9)
Using the identity (9), the properties of the trace operator and taking into account that
maxR |Y | ≤ 1, we obtain
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
340 U. DE MAIO, T. A. MEL’NYK, AND C. PERUGIA
‖v‖2
ε,k1,k2
=
∫
Ωε
|∇v|2 dx+
+ ε
2∑
i=1
ki
∫
Υ(i,±)(ε)
2
√
1 + ε24−1|h′i(x2)|2
hi(x2)
hi(x2)
2
√
1 + ε24−1|h′i(x2)|2
v2 dlx+
+ ε
2∑
i=1
ki
∫
Θ(i)(ε)
v2 dx1 ≤
≤
∫
Ωε
|∇v|2 dx+ c1
2∑
i=1
∫
Υ(i,±)(ε)
εhi(x2)
2
√
1 + ε24−1|h′0(x2)|2
v2 dlx + εc2
2∑
i=1
‖v‖2
H1(G(i)(ε))
≤
≤ c3‖v‖2
H1(Ωε)
+ c1
( ∫
G(1)(ε)∪G(2)(ε)
v2 dx− ε
∫
G(1)(ε)∪G(2)(ε)
Y
(x1
ε
)
2v ∂x1v dx
)
≤
≤ c3‖v‖2
H1(Ωε)
+ c1
( ∫
G(1)(ε)∪G(2)(ε)
v2 dx+
∫
G(1)(ε)∪G(2)(ε)
ε
(
(∂x1v)
2 + v2
)
dx
)
≤
≤ C2‖u‖2
H1(Ωε)
. (10)
Similarly, we obtain
‖v‖2
H1(Ωε)
=
∫
Ωε
|∇v|2 dx+
∫
Ω0
v2 dx+
∫
G(1)(ε)∪G(2)(ε)
v2 dx =
∫
Ωε
|∇v|2 dx+
∫
Ω0
v2 dx+
+ ε
2∑
i=1
∫
Υ(i,±)(ε)
hi(x2)
2
√
1 + ε24−1|h′i(x2)|2
v2 dlx + ε
∫
G(1)(ε)∪G(2)(ε)
Y
(x1
ε
)
2v ∂x1v dx≤
≤ C3‖v‖2
ε,k1,k2
+
∫
Ω0
v2 dx+ ε
∫
G(1)(ε)∪G(2)(ε)
v2 dx,
whence
‖v‖2
H1(Ωε)
≤ C4
‖v‖2
ε,k1,k2
+
∫
Ω0
v2 dx
. (11)
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
HOMOGENIZATION OF THE ROBIN PROBLEM IN A THICK MULTILEVEL JUNCTION 341
Now let us show that there exists a positive constant C5 such that for ε small enough and
for any v ∈ H1(Ωε), ∫
Ω0
v2 dx ≤ C5‖v‖2
ε,k1,k2
. (12)
We argue by contradiction. If not, then there exist sequences {εm : m ∈ N} and {vm} ⊂
⊂ H1(Ωεm) such that limm→0 εm = 0, ∫
Ω0
v2
m dx = 1, (13)
∫
Ωεm
|∇vm|2 dx + εm
2∑
i=1
ki
∫
Υ(i)(εm)
v2
m dlx <
1
m
. (14)
Since the sequence {vm} is bounded in H1(Ω0), we may assume without loss of generality
that it is a Cauchy sequence in L2(Ω0). From inequality (14) it follows that {vm} is a Cauchy
sequence also in H1(Ω0),
‖vm − vn‖2
H1(Ω0) ≤ ‖vm − vn‖2
L2(Ω0) +
1
m
+
1
n
.
Hence, {vm} converges to some element v0 ∈ H1(Ω0). Obviously, v0 ≡ const in H1(Ω0) and,
due to (13), v0 = |Ω0|−1/2, where |Ω0| denotes the measure of the domain Ω0.
Then, the sequence of the traces of {vm} converges to v0 in L2(∂Ω0) as well and it is easy to
verify that
∫
I0(εm)
v2
m(x1, 0) dx1 =
2∑
i=1
∫
I0
χi(x1/εm) v2
m(x1, 0) dx1 →
→
2∑
i=1
hi(0)
∫
I0
v2
0(x1, 0) dx1 =
(
h1(0) + h2(0)
)
|Ω0|−1a 6= 0, m → ∞,
(15)
where I0(ε) := I0 ∩ Ωε and χi(·) is a 1-periodic function such that
χi(t) =
1, t ∈
(
bi −
hi(0)
2
, bi +
hi(0)
2
)
,
0, t ∈ [0, 1] \
(
bi −
hi(0)
2
, bi +
hi(0)
2
)
,
(16)
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
342 U. DE MAIO, T. A. MEL’NYK, AND C. PERUGIA
for i = 1, 2. Obviously,
χi(x1/ε) →
1∫
0
χi(t) dt = hi(0) weakly in L2(0, a) as ε → 0.
On the other hand, from (9) and (14) it follows that∫
G(1)(εm)∪G(2)(εm)
(
|∇vm|2 + v2
m
)
dx ≤ C6
m
and, therefore,∫
I0(εm)
v2
m(x1, 0) dx1 ≤ C7
∫
G(1)(εm)∪G(2)(εm))
(
|∇vm|2 + v2
m
)
dx ≤ C8
m
,
where the constants C6, C7, C8 are independent of m. This means that∫
I0(εm)
v2
m(x1, 0) dx1 → 0 as m → ∞. (17)
However (17) varies from (15). This contradiction establishes estimate (12).
Thus, by virtue of (11) and (12), we obtain the left inequality in (6).
The lemma is proved.
Remark 1. Hereafter all constants {ci, Ci} in asymptotic inequalities are independent of the
parameter ε.
3. Extension operator. Due to the a-periodic condition in problem (2), we can assume that
the function fε and the solution uε are a-periodic functions with respect to x1.
Theorem 1. Let condition (3) be satisfied and, in addition, there exist constants C1 and ε0
such that for all values ε ∈ (0, ε0) ∫
Ωε
(Fε (x))2 dx ≤ C1, (18)
where Fε (x) = ε−1( fε(x+ εē1)− fε(x) ) (ē1 = (1, 0)).
Then there exist extension operators
P(1)
ε : H1(Ω0 ∪G(1)(ε)) 7→ H1(Ω1) and P(2)
ε : H1(Ω0 ∪G(2)(ε)) 7→ H1(Ω2)
such that, for the solution uε,
‖ P(1)
ε uε ‖H1(Ω1) + ‖ P(2)
ε uε ‖H1(Ω2)≤ C2
(
‖Fε‖L2(Ωε) + ‖fε‖L2(Ωε)
)
≤ C3. (19)
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
HOMOGENIZATION OF THE ROBIN PROBLEM IN A THICK MULTILEVEL JUNCTION 343
Proof. The first step in the proof is to show that scattering of the values of the solution uε
on neighboring thin rods is small in some sense.
Here we assume for simplicity that γ ≡ const. In general case similarly as in the proof of
Theorem 4.1 [4], we should multiply the differential equation of problem (2) by a smooth cut-
off function χ0 such that χ0(x2) = 0 for x2 ≥ γ0, and χ0(x2) = 1 for x2 ≤ γ0/2, where γ0 =
= minx1∈[0,a] γ(x1); and consider the function vε = χ0 uε which is a solution to the correspondi-
ng boundary-value problem in a thick two-level junction whose junction’s body is the rectangle
[0, a]× [0, γ0].
Thus, the problem (2) is invariant under the ε-shift along the axis x1. This means that the
function
Uε(x) = ε−1(uε(x+ εē1)− uε(x)), ē1 = (1, 0), (20)
is a solution, a-periodic in x1, to the following problem:
−∆xUε(x) = Fε(x), x ∈ Ωε,
∂νUε(x) = −ε k1 Uε(x), x ∈ Υ(1)(ε),
∂νUε(x) = −ε k2 Uε(x), x ∈ Υ(2)(ε),
∂p
x1Uε(0, x2) = ∂p
x1Uε(a, x2), x2 ∈ [0, γ(0)], p = 0, 1,
∂νUε(x) = 0, x ∈ Γε.
(21)
By virtue of Lemma 1 and condition (18), we get the following estimate:
‖Uε‖H1(Ωε) ≤ C2 ‖Fε‖L2(Ωε) ≤ C3. (22)
At first we extend the solution uε the domain Ω1 by using the "linear matching"
P̂ (1)
ε (uε)(x) =
uε, x ∈ G(1)(ε),
Bε
j (x2) + Sε
j (x2)
(
x1 − ε
(
j + b1 +
h1(x2)
2
))
, x ∈ Q̃
(1)
j (ε),
(23)
in the domain Ω0 ∪G(1)(ε) ∪ Q̃(1)(ε). Here
Bε
j (x2) = uε
(
ε
(
j + b1 +
h1(x2)
2
)
, x2
)
,
Sε
j (x2) =
1
ε(1− h1(x2))
(
uε
(
ε
(
j + 1 + b1 −
h1(x2)
2
)
, x2
)
−Bε
j (x2)
)
,
Q̃(1) (ε) =
N⋃
j=−1
Q̃
(1)
j (ε) ,
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
344 U. DE MAIO, T. A. MEL’NYK, AND C. PERUGIA
and the domain
Q̃
(1)
j (ε) =
{
x : x2 ∈ (−d1,−ε), x1 ∈
(
ε
(
j + b1 +
h1(x2)
2
)
, ε
(
j + 1 + b1 −
h1(x2)
2
))}
is situated between two rods G(1)
j (ε) and G(1)
j+1(ε). In the case of the extreme rods, we perform
the a-periodic extension of problem (2) with respect to the axis Ox1.
Without the loss of generality, we can assume here that h1 is smooth on [−d1, 0]. It is easy
to calculate that
‖P̂ (1)
ε (uε)‖2
H1( eQ(1)
j (ε))
=
=
∫
eQ(1)
j (ε)
∣∣∣∣∣Bε
j (x2) + Sε
j (x2)
[
x1 − ε
(
j + b1 +
h1 (x2)
2
)]∣∣∣∣∣
2
dx+
+
∫
eQ(1)
j (ε)
∣∣∣∣∣(Bε
j (x2)
)′ + Sε
j (x2) +
(
Sε
j (x2)
)′×
×
[
x1 − ε
(
j + b1 +
h1 (x2)
2
)]∣∣∣∣∣
2
dx. (24)
Further, we will not indicate that functions Bε
j , Sε
j and h1 are depending on x2 if it doesn’t
lead to a confusion. By using the inequality (a+ b)2 ≤ 2a2 + 2b2 and properties (1) for h1, we
get
‖P̂ (1)
ε (uε)‖2
H1( eQ(1)
j (ε))
≤ 2
∫
eQ(1)
j (ε)
(
Bε
j
)2 +
+ 2
∫
eQ(1)
j (ε)
(
Sε
j
)2 [
x1 − ε
(
j + b1 +
h1
2
)]2
dx+
+ 2
∫
eQ(1)
j (ε)
((
Bε
j
)′)2
dx+ 4
∫
eQ(1)
j (ε)
(
Sε
j
)2
dx+
+ 4
∫
eQ(1)
j (ε)
((
Sε
j
)′)2
[
x1 − ε
(
j + b1 +
h1
2
)]2
dx.
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HOMOGENIZATION OF THE ROBIN PROBLEM IN A THICK MULTILEVEL JUNCTION 345
Now, taking into account the geometry of the domain Q̃(1)
j (ε), we deduce
∥∥∥P̂ (1)
ε (uε)
∥∥∥2
H1( eQ(1)
j (ε))
≤ 4ε (1−m0)
−ε∫
−d1
[(
Bε
j
)2 +
((
Bε
j
)′)2
+
(
Sε
j
)2]
dx2 + 4
−ε∫
−d1
[(
Sε
j
)2 +
+
((
Sε
j
)′)2
] ε
�
j+1+b1−h1
2
�∫
ε
�
j+b1+
h1
2
�
[
x1 − ε
(
j + b1 +
h1
2
)]2
dx1 dx2 ≤
≤ 4ε (1−m0)
−ε∫
−d1
[(
Bε
j
)2 +
((
Bε
j
)′)2
+
(
Sε
j
)2]
dx2+
+ 4
ε3 (1−m0)
3
3
−ε∫
−d1
[(
Sε
j
)2 +
((
Sε
j
)′)2
]
dx2 ≤
≤ C1
ε
−ε∫
−d1
[(
Bε
j
)2 +
((
Bε
j
)′)2
+
(
Sε
j
)2]
dx2 +
+ ε3
−ε∫
−d1
[(
Sε
j
)2 +
((
Sε
j
)′)2
]
dx2
. (25)
Now, let us estimate each term in the right-hand side of (25) by using the following two
inequalities:
u2(0) ≤ 2ε−1
ε∫
0
u2(t) dt+ 2ε
ε∫
0
(
u′(t)
)2
dt, (26)
(u(0)− u(ε))2 ≤ ε
ε∫
0
(
u′
)2 (t) dt (27)
that hold for every u ∈ H1 ([0, ε]) .
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
346 U. DE MAIO, T. A. MEL’NYK, AND C. PERUGIA
By adapting (26) to our case, we obtain
u2
ε
(
ε
(
j + b1 +
h1
2
)
, x2
)
≤ 2ε−1
ε
�
j+b1+
h1
2
�∫
ε
�
j+b1−h1
2
� u
2
ε(x) dx1,+2ε
ε
�
j+b1+
h1
2
�∫
ε
�
j+b1−h1
2
� (∂x1uε(x))
2 dx1,
and integrating over (−d1,−ε) , we have
ε
−ε∫
−d1
(
Bε
j
)2
dx2 ≤ 2
{
‖uε‖2
L2
� eG(1)
j (ε)
� + ε2 ‖∂x1uε‖2
L2
� eG(1)
j (ε)
�
}
≤ c1 ‖uε‖2
H1
�
G
(1)
j (ε)
� , (28)
where G̃(1)
j (ε) = G
(1)
j (ε) ∩ {x : −d1 < x2 < −ε}.
Moreover,
−ε∫
−d1
(
Sε
j
)2
dx2 =
=
−ε∫
−d1
{
1
ε(1− h1)
[
uε
(
ε
(
j + 1 + b1 −
h1
2
)
, x2
)
− uε
(
ε
(
j + b1 +
h1
2
)
, x2
)]}2
dx2 ≤
≤ c2ε
−2
−ε∫
−d1
[
uε
(
ε
(
j + 1 + b1 −
h1
2
)
, x2
)
− uε
(
ε
(
j + b1 −
h1
2
)
, x2
)
+
+ uε
(
ε
(
j + b1 −
h1
2
)
, x2
)
− uε
(
ε
(
j + b1 +
h1
2
)
, x2
)]2
dx2 =
= c2ε
−2
−ε∫
−d1
[
εUε
(
ε
(
j + b1 −
h1
2
)
, x2
)
+
+ uε
(
ε
(
j + b1 −
h1
2
)
, x2
)
− uε
(
ε
(
j + b1 +
h1
2
)
, x2
)]2
dx2 ≤
≤ 2c2
−ε∫
−d1
[
Uε
(
ε
(
j + b1 −
h1
2
)
, x2
)]2
dx2+
+ 2c2ε−2
−ε∫
−d1
[
uε
(
ε
(
j + b1 +
h1
2
)
, x2
)
− uε
(
ε
(
j + b1 −
h1
2
)
, x2
)]2
dx2.
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
HOMOGENIZATION OF THE ROBIN PROBLEM IN A THICK MULTILEVEL JUNCTION 347
By (26) and (27), we have
−ε∫
−d1
(
Sε
j
)2
dx2 ≤ 4c2ε−1 ‖Uε‖2
L2
� eG(1)
j (ε)
� + 4c2ε ‖∂x1Uε‖2
L2
� eG(1)
j (ε)
� +
+ 2c2ε−1M0
−ε∫
−d1
ε
(
j+b1+
h1
2
)
∫
ε
(
j+b1−
h1
2
) (∂x1uε(x))
2 dx1dx2 ≤
≤ c4
(
ε−1 ‖Uε‖2
L2
� eG(1)
j (ε)
� + ε ‖∂x1Uε‖2
L2
� eG(1)
j (ε)
� + ε−1 ‖∂x1uε‖2
L2
� eG(1)
j (ε)
�
)
.
(29)
Thus
ε3
−ε∫
−d1
(
Sε
j
)2
dx2 ≤ ε
−ε∫
−d1
(
Sε
j
)2
dx2 ≤
≤ c4
(
‖Uε‖2
L2
� eG(1)
j (ε)
� + ε2 ‖∂x1Uε‖2
L2
� eG(1)
j (ε)
� + ‖∂x1uε‖2
L2
� eG(1)
j (ε)
�
)
≤
≤ c5
(
‖Uε‖2
H1
�
G
(1)
j (ε)
� + ‖uε‖2
H1
�
G
(1)
j (ε)
�
)
. (30)
Now we are going to estimate the other terms in (25),
−ε∫
−d1
((
Bε
j (x2)
)′)2
dx2 =
=
−ε∫
−d1
[
ε
h′1 (x2)
2
∂x1uε
(
ε
(
j + b1 +
h1
2
)
, x2
)
+ ∂x2uε
(
ε
(
j + b1 +
h1
2
)
, x2
)]2
dx2 ≤
≤ M2
0
2
ε2
−ε∫
−d1
[
∂x1uε
(
ε
(
j + b1 +
h1
2
)
, x2
)]2
dx2+
+ 2
−ε∫
−d1
(
∂x2uε
(
ε
(
j + b1 +
h1
2
)
, x2
))2
dx2. (31)
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
348 U. DE MAIO, T. A. MEL’NYK, AND C. PERUGIA
If we apply to the last two integrals in (31) the same calculation of (28) we obtain
ε
−ε∫
−d1
((
Bε
j (x2)
)′)2
dx2 ≤ c7
(
ε2 ‖∂x1uε‖2
L2
� eG(1)
j (ε)
� + ε4
∥∥∂2
x1
uε
∥∥2
L2
� eG(1)
j (ε)
� +
+ ‖∂x2uε‖2
L2
� eG(1)
j (ε)
� + ε2
∥∥∂2
x1,x2
uε
∥∥2
L2
� eG(1)
j (ε)
�) ≤
≤ c7
(
‖∇uε‖2
L2
� eG(1)
j (ε)
� + ε2 ‖uε‖2
H2
� eG(1)
j (ε)
�
)
.
To estimate the norm ‖uε‖2
H2
� eG(1)
j (ε)
�, we use the second energy inequality [5] with the smooth
cut-off function
χε (x2) =
{
0, x2 ≥ −ε
2
,
1, x2 ≤ −ε.
We obtain that ε2 ‖uε‖2
H2
� eG(1)
j (ε)
� ≤ c
(
‖uε‖2
H1
�
G
(1)
j (ε)
� + ‖fε‖2
L2
�
G
(1)
j (ε)
�
)
. So,
ε
−ε∫
−d1
((
Bε
j (x2)
)′)2
dx2 ≤ c8
(
‖uε‖2
H1
�
G
(1)
j (ε)
� + ‖fε‖2
L2
�
G
(1)
j (ε)
�
)
. (32)
Now, let us estimate the integral of square of the derivative,
(
Sε
j (x2)
)′ =
h′1 (x2)
1− h1(x2)
Sε
j (x2) +
1
ε (1− h1(x2))
{
ε ∂x2Uε
(
ε
(
j + b1 −
h1
2
)
, x2
)
+
+
[
∂x2uε
(
ε
(
j + b1 −
h1
2
)
, x2
)
− ∂x2uε
(
ε
(
j + b1 +
h1
2
)
, x2
)]}
−
− h′1(x2)
2 (1− h1(x2))
[
∂x1uε
(
ε
(
j + 1 + b1 −
h1
2
)
, x2
)
+ ∂x1uε
(
ε
(
j + b1 +
h1
2
)
, x2
)]
.
Taking into account the properties of h1, its derivative (see (1)), estimate (29) and applying (26)
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
HOMOGENIZATION OF THE ROBIN PROBLEM IN A THICK MULTILEVEL JUNCTION 349
and (27), we deduce
−ε∫
−d1
((
Sε
j
)′)2
dx2 ≤
≤ c9
[
c4
(
ε−1 ‖Uε‖2
L2
� eG(1)
j (ε)
� + ε ‖∂x1Uε‖2
L2
� eG(1)
j (ε)
� + ε−1 ‖∂x1uε‖2
L2
� eG(1)
j (ε)
�
)
+
+ c10
(
ε−1 ‖∂x2Uε‖2
L2
� eG(1)
j (ε)
� + ε
∥∥∂2
x1,x2
Uε
∥∥2
L2
� eG(1)
j (ε)
�)+
+ c11ε
−1
∥∥∂2
x1,x2
uε
∥∥2
L2
� eG(1)
j (ε)
� + c12
(
ε−1 ‖∂x1uε‖2
L2
� eG(1)
j+1(ε)
� + ε
∥∥∂2
x1
uε
∥∥2
L2
� eG(1)
j+1(ε)
�)+
+ c13
(
ε−1 ‖∂x1uε‖2
L2
� eG(1)
j (ε)
� + ε
∥∥∂2
x1
uε
∥∥2
L2
� eG(1)
j (ε)
�)] ≤
≤ c14
(
ε−1 ‖Uε‖2
H1
� eG(1)
j (ε)
� + ε ‖∇Uε‖2
L2
� eG(1)
j (ε)
� + ε ‖Uε‖2
H2
� eG(1)
j (ε)
� +
+ ε−1 ‖uε‖2
H2
� eG(1)
j (ε)∪ eG(1)
j+1(ε)
� + ε−1 ‖∇uε‖2
L2
� eG(1)
j (ε)∪ eG(1)
j+1(ε)
�
)
.
Again applying the second energy inequality as above, we have
ε3
−ε∫
−d1
((
Sε
j
)′)2
≤ c15
(
ε2 ‖Uε‖2
H1
�
G
(1)
j (ε)
� + ε4 ‖Uε‖2
H1
� eG(1)
j (ε)
� + ε2 ‖Fε‖2
L2
�
G
(1)
j (ε)
� +
+ ‖uε‖2
H1
�
G
(1)
j (ε)∪G
(1)
j+1(ε)
� + ‖fε‖2
L2
�
G
(1)
j (ε)∪G
(1)
j+1(ε)
�
)
. (33)
Thus, by (28), (30), (32) and (33), the right-hand side of (25) is estimated in the following
way:
‖P̂ (1)
ε (uε)‖2
H1
( eQ(1)
j (ε)
) ≤ c16
(
‖uε‖2
H1
�
G
(1)
j (ε)
� + ‖Uε‖2
H1
�
G
(1)
j (ε)
� + ‖fε‖2
L2
�
G
(1)
j (ε)
� +
+ ε2 ‖Fε‖2
L2
�
G
(1)
j (ε)
� + ‖uε‖2
H1
�
G
(1)
j (ε)∪G
(1)
j+1(ε)
� + ‖fε‖2
L2
�
G
(1)
j (ε)∪G
(1)
j+1(ε)
�
)
.
(34)
Summing (33) over j from −1 to N, using Lemma 1 and (3), we get
‖P̂ (1)
ε (uε)‖2
H1
(
G(1)(ε)∪ eQ(1)(ε)
) ≤ c17
(
‖fε‖2
L2(Ωε)
+ ‖Fε‖2
L2(Ωε)
)
. (35)
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
350 U. DE MAIO, T. A. MEL’NYK, AND C. PERUGIA
Now it remains to extend P̂ (1)
ε (uε) to
T
(1)
j (ε) =
{
x : x2 ∈ (−ε, 0) , x1 ∈
(
ε(j + b1 + 2−1h1(x2)), ε(j + 1 + b1 − 2−1h1(x2))
)}
,
j = −1, 0, 1, . . . , N.
Since the domains T (1)
j (ε), j = −1, 0, 1, .., N , are equal (each of this domain can be obtained
from T
(1)
0 (ε) by a parallel shift along the axis Ox1), we use results about extension operators
in perforated domains [6]. It follows from these results that there exist an extension operator
P
(1)
ε : H1
(
G(1)(ε) ∪ Q̃(1)(ε)
)
7→ H1(Ω1), uniformly bounded in ε.
Thus, the extension operator P(1)
ε := P
(1)
ε ◦ P̂ (1)
ε is constructed and it satisfies the uniform
estimate (19). Similarly we can construct the operator P(2)
ε : H1(Ω0∪G(2)(ε)) 7→ H1(Ω2) which
also satisfies (19).
The theorem is proved.
4. Convergence theorem. To prove this theorem we should pass to the limit in the integral
identity (4) as ε → 0. For this we will use identity (9), the extension operators constructed in
Section 3, and the following characteristic functions
χ(i)
ε (x) := χ(i)
(x1
ε
, x2
)
=
0, x ∈ Ω0,
1, x ∈ G(i)(ε),
0, x ∈ Di \G(i)(ε),
i = 1, 2.
We can assume that these functions are ε-periodic with respect to x1. Similarly as in Section 4
[1], we can prove that χ(i)
ε → hi weakly in L2(Di) as ε → 0.
Theorem 2. Let uε be a weak solution to problem (2). Then
(
uε
)
|Ω0 → v+
0 ,
(
P(1)
ε uε
)∣∣∣
D1
→ v1,−
0 ,
(
P(2)
ε uε
)∣∣∣
D2
→ v2,−
0
weakly in H1(Ω0), H1(D1), H1(D2) respectively as ε → 0, where the vector-valued function
v0(x) =
v+
0 (x), x ∈ Ω0,
v1,−
0 (x), x ∈ D1,
v2,−
0 (x), x ∈ D2,
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
HOMOGENIZATION OF THE ROBIN PROBLEM IN A THICK MULTILEVEL JUNCTION 351
is the unique weak solution to the following problem:
−∆x v
+
0 (x) = f0(x), x ∈ Ω0,
∂p
x1v
+
0 (0, x2) = ∂p
x1v
+
0 (a, x2), p = 0, 1, x2 ∈ [0, γ(0)],
∂νv
+
0 (x) = 0, x ∈ Γγ ,
−∂x2
(
h1(x2) ∂x2v
1,−
0 (x)
)
+ 2k1 v
1,−
0 (x) = h1(x2)f0(x), x ∈ D1,
∂x2v
1,−
0 (x1,−d1) = 0, x1 ∈ I0,
−∂x2
(
h2(x2) ∂x2v
2,−
0 (x)
)
+ 2k2 v
2,−
0 (x) = h2(x2)f0(x), x ∈ D2,
∂x2v
2,−
0 (x1,−d2) = 0, x1 ∈ I0,
v+
0 (x1, 0) = v1,−
0 (x1, 0) = v2,−
0 (x1, 0), x1 ∈ I0,
∂x2v
+
0 (x1, 0) = h1(0)∂x2v
1,−
0 (x1, 0)+
+ h2(0)∂x2v
2,−
0 (x1, 0), x1 ∈ I0,
(36)
where Γγ = {x : x2 = γ(x1), x1 ∈ I0}.
Proof. With the help of (9), the extension operators P(i)
ε and the functions χ(i)
ε , i = 1, 2, we
rewrite identity (4) in the following way:
∫
Ω0
∇uε · ∇ϕdx+
2∑
i=1
(∫
Di
χ(i)
ε (x)∇
(
P(i)
ε uε
)
· ∇
(
ϕ
)
dx + εki
a∫
0
(
χ(i)
ε P(i)
ε uε ϕ
)∣∣∣
x2=−di
dx1︸ ︷︷ ︸
A1
+
+ 2ki
∫
Di
√
1 + ε24−1|h′i(x2)|2
hi(x2)
χ(i)
ε (x)
(
P(i)
ε uε
)
(x)ϕ(x) dx
)
=
= 2ε
2∑
i=1
ki
∫
G(i)(ε)
Y
(x1
ε
)√1 + ε24−1|h′i(x2)|2
hi(x2)
∂x1
(
uε ϕ
)
dx
︸ ︷︷ ︸
A2
+
+
∫
Ω0
fε(x)ϕ(x) dx+
2∑
i=1
∫
Di
χ(i)
ε (x)fε(x)ϕ(x) dx ∀ ϕ ∈ H1
q,x1
(Ω2), (37)
where H1
q,x1
(Ω2) = {ϕ ∈ H1(Ω2) : ϕ(0, x2) = ϕ(a, x2) for x2 ∈ (0, γ(0))}.
Because of (3), (5), (19), the sequences
{
χ(i)
ε ∂xj
(
P(i)
ε uε
)}
ε>0
, j = 1, 2, (38)
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
352 U. DE MAIO, T. A. MEL’NYK, AND C. PERUGIA
are bounded in L2(Di), i = 1, 2. Therefore, we can choose a subsequence of {ε} ( still denoted
by {ε}) such that χ(i)
ε ∂xj
(
P(i)
ε uε
)
→ σ
(i)
j weakly in L2(Di), j = 1, 2, i = 1, 2, and
(
uε
)
|Ω0 → v+
0 ,
(
P(1)
ε uε
)∣∣∣
D1
→ v1,−
0 ,
(
P(2)
ε uε
)∣∣∣
D2
→ v2,−
0
weakly in H1(Ω0), H1(D1), H1(D2) and strongly in L2(Ω0), L2(D1), L2(D2) respectively as
ε → 0. Since
(
uε
)
| I0 =
(
P(1)
ε uε
)
| I0 =
(
P(2)
ε uε
)
| I0 , the traces of limit functions are equal as
well, i.e., v+
0 (x1, 0) = v1,−
0 (x1, 0) = v2,−
0 (x1, 0), x1 ∈ I0.
Obviously, the summands A1 and A2 in (37) vanishe as ε → 0. Now, passing to the limit in
(37) and taking (3), (38) into account, we obtain
∫
Ω0
∇v+
0 · ∇ϕdx+
2∑
i=1
∫
Di
2∑
j=1
σ
(i)
j (x) ∂xjϕ(x) dx + 2ki
∫
Di
vi,−
0 ϕdx
=
=
∫
Ω0
f0(x)ϕ(x) dx +
2∑
i=1
∫
Di
hi(x2) f0(x)ϕ(x) dx, ϕ ∈ H1
q,x1
(Ω2). (39)
Next we should find σ(i)
j , j = 1, 2, i = 1, 2.
In order to determine σ(i)
1 , i = 1, 2, we consider the integral identity (4) with the following
test functions :
ψ1(x) = ε
0, x ∈ Ω0,
Y (x1/ε)φ1(x), x ∈ G(1)(ε),
0, x ∈ G(2)(ε),
ψ2(x) = ε
0, x ∈ Ω0,
0, x ∈ G(1)(ε),
Y (x1/ε)φ2(x), x ∈ G(2)(ε),
where φ1 and φ2 are arbitrary functions from C∞0 (D1) and C∞0 (D2) respectively. It is obvious
that ψ1 and ψ2 belong to H1
q,x1
(Ωε). As a result, we get
∫
D1
χ(1)
ε (x) ∂x1P
(1)
ε (uε)φ1 dx = O(ε),
∫
D2
χ(2)
ε (x) ∂x1P
(2)
ε (uε)φ2 dx = O(ε), ε → 0,
whence σ(1)
1 ≡ 0 and σ(2)
1 ≡ 0.
Next let us define σ(1)
2 . Take any function φ ∈ C∞0 (D1) and perform the following calculati-
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
HOMOGENIZATION OF THE ROBIN PROBLEM IN A THICK MULTILEVEL JUNCTION 353
ons:∫
D1
χ(1)
ε (x)∂x2
(
P(1)
ε uε(x)
)
φ(x) dx =
N−1∑
j=0
∫
G
(1)
j (ε)
∂x2
(
uε(x)
)
φ(x) dx =
=
N−1∑
j=0
( ∫
Υ
(1,±)
j (ε)
uε φα
(1)
2 (x2, ε) dlx −
∫
G
(1)
j (ε)
uε ∂x1φdx
)
=
= −2−1ε
0∫
−d1
h′1(x2)
N−1∑
j=0
(
uεφ
)∣∣
x1=ε(j+b1±h1(x2)/2)
dx2−
−
∫
D1
χ(1)
ε (x)
(
P(1)
ε uε
)
∂x2φdx =: B1(ε) +B2(ε). (40)
Hereα(1)
2 (x2, ε) = −εh′1(x2)
(
2
√
1 + ε24−1(h′1(x2))2
)−1
is the second coordinate of the outward
unit normal ν(1)
± (see (8) to the lateral surfaces Υ(1,±)
j (ε) of the thin rodG(1)
j (ε). It is easy to veri-
fy that
lim
ε→0
B2(ε) = −
∫
D1
h1(x2)v
1,−
0 (x)∂x2φ(x) dx. (41)
To find the limit of B1(ε) we rewrite this value in the following way:
B1(ε) = −2−1ε
0∫
−d1
h′1(x2)
N−1∑
j=0
ε(j+b1+h0(x2)/2)∫
ε(j+b1−h0(x2)/2)
∂x1
(
uεφ
)
dx1
dx2−
− ε
0∫
−d1
h′1(x2)
N−1∑
j=0
(
(uε − v1,−
0 )φ
)
|x1=ε(j+b1−h0(x2)/2)
dx2−
−
0∫
−d1
h′1(x2)
N−1∑
j=0
(
v1,−
0 φ
)
|x1=ε(j+b1−h0(x2)/2)
(
ε(j + 1)− εj
) dx2. (42)
The first term in (42) is bounded by ε‖uε‖H1(G(1)(ε))‖φ‖H1(D1).Due to estimate (26), the second
term in (42) is estimated by the value
c1
(∥∥P(1)
ε uε − v1,−
0
∥∥
L2(G(1)(ε))
+ ε2
∥∥∂x1(P
(1)
ε uε − v1,−
0 )
∥∥
L2(G(1)(ε))
)
‖φ‖H1(D1). (43)
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
354 U. DE MAIO, T. A. MEL’NYK, AND C. PERUGIA
Since for almost all points x2 ∈ (−d1, 0) the function v1,−
0 ∈ H1(0, a), the inner sum in the
third term in (42) is the Riemann sum for the integral
∫ a
0
v1,−
0 φdx1. Then in view of Lebesgue’s
Theorem and Fubini’s Theorem, the limit of the third term is equal to
−
∫
D1
h′1(x2) v
1,−
0 (x)φ(x) dx. (44)
Passing to the limit in (40) and taking (41) – (44) into account, we get
σ
(1)
2 (x) = h1(x2) ∂x2v
1,−
0 (x), x ∈ D1.
Similarly we deduce that σ(2)
2 (x) = h2(x2) ∂x2v
2,−
0 (x), x ∈ D2.
Thus, we obtain that the vector-valued function v0 satisfies the following identity:
∫
Ω0
∇xv
+
0 · ∇xϕdx+
2∑
i=1
∫
Di
(
hi(x2) ∂x2v
i,−
0 (x) ∂x2ϕ(x) + 2ki v
i,−
0 (x)ϕ(x)
)
dx =
=
∫
Ω0
f0(x)ϕ(x) dx +
2∑
i=1
∫
Di
hi(x2) f0(x)ϕ(x) dx ∀ ϕ ∈ H1
q,x1
(Ω2). (45)
Identity (45) is the corresponding integral identity for problem (36) in the following ani-
zotropic Sobolev vector-space:
H0 =
{
u = (u0, u1, u2) ∈ V0 := L2(Ω0)× L2(D1)× L2(D2) |
u0 ∈ H1(Ω0), u0(0, x2) = u0(a, x2) for x2 ∈ (0, γ(0));
∃ ∂x2u1 ∈ L2(D1); ∃ ∂x2u2 ∈ L2(D2);
u0(x1, 0) = u1(x1, 0) = u2(x1, 0), x1 ∈ I0
}
with the scalar product
(
u,v
)
H0
=
∫
Ω0
∇u0 · ∇v0 dx +
2∑
i=1
∫
Di
(
hi(x2)∂x2ui ∂x2vi + 2kiuivi
)
dx.
Obviously, the space H0 continuously embeds in V0.
By using standard Hilbert space methods, we can state that there exists a unique weak
solution v0 ∈ H to problem (36), which is called the limit problem for problem (2). It should
be noted that in the rectangles D1 and D2 we have ordinary differential equations with respect
to x2 and there are no boundary conditions on the vertical sides of Di, i = 1, 2.
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
HOMOGENIZATION OF THE ROBIN PROBLEM IN A THICK MULTILEVEL JUNCTION 355
Due to the uniqueness of the solution to problem (36), the above reasoning holds for any
subsequence of {ε} chosen at the beginning of the proof. Therefore, the theorem is proved.
Example. In the case h1 ≡ const and h2 ≡ const, we can reduce problem (36) to some
boundary-value problem in the junction’s body. By solving these ordinary differential equations
with regard to the Neumann conditions and the first transmission condition in I0, we find
vi,−
0 (x) = − 1
ρi
x2∫
−di
sinh
(
ρi(x2 − t)
)
f0(x1, t) dt+
+
cosh
(
ρi(x2 + di)
)
cosh(ρidi)
v+
0 (x1, 0)− 1
ρi
0∫
−di
sinh(ρit) f0(x1, t) dt
, x ∈ Di, (46)
where ρi =
√
2kih
−1
i , i = 1, 2. Putting these functions in the second transmission condition,
we obtain the following problem:
−∆x v
+
0 (x) = f0(x), x ∈ Ω0,
∂p
x1v
+
0 (0, x2) = ∂p
x1v
+
0 (a, x2), p = 0, 1, x2 ∈ [0, γ(0)],
∂νv
+
0 (x) = 0, x ∈ Γγ ,
∂x2v
+
0 (x1, 0) =
(∑2
i=1 hiρi tanh(ρidi)
)
v+
0 (x1, 0),+f̂0(x1), x1 ∈ I0,
(47)
where
f̂0(x1) = −
2∑
i=1
hi
0∫
−di
(
cosh(ρit) + tanh(ρidi) sinh(ρit)
)
f0(x1, t) dt.
Problem (47) is a classical boundary-value problem with the Robin condition on I0. Obviously,
it has a unique weak solution from H1(Ω0).
1. De Maio U., Mel’nyk T. A. Homogenization of the Robin problem in a thick multi-structures of type 3:2:2 //
Asymptotic Analysis (to appear).
2. Mel’nyk T. A., Nazarov S. A. Asymptotics of the Neumann spectral problem solution in a domain of "thick
comb"type // Trudy Seminara imeni I.G. Petrovskogo. — 1996. — 19. — P. 138 – 173 (in Russian), and English
transl.: J. Math. Sci. — 1997. — 85, № 6. — P. 2326 – 2346.
3. Mel’nyk T. A. Homogenization of the Poisson equation in a thick periodic junction // Z. Anal. und ihre
Anwend. — 1999. — 18, № 4. — P. 953 – 975.
4. Mel’nyk T. A. Eigenmodes and pseudo-eigenmodes of thick multi-level junctions // Proc. Int. Conf. ”Days on
Diffraction-2004” (St. Petersburg, June 29 – July 2, 2004). — P. 51 – 52.
5. Ladyzhenskaya O. A. The boundary value problems of mathematical physics. — Berlin: Springer, 1985. —
295 p.
6. Oleinik O. A., Yosifian G. A., and Shamaev A. S. Mathematical problems in elasticity and homogenization.
— Amsterdam: North-Holland, 1992. — 310 p.
Received 09.07.2004
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
|
| id | nasplib_isofts_kiev_ua-123456789-177021 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-3076 |
| language | English |
| last_indexed | 2025-12-02T09:28:58Z |
| publishDate | 2004 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | De Maio, U. Mel'nyk, T.A. Perugia, C. 2021-02-09T20:42:38Z 2021-02-09T20:42:38Z 2004 Homogenization of the Robin problem in a thick multilevel junction / U.De Maio, T.A. Mel'nyk, C. Perugia // Нелінійні коливання. — 2004. — Т. 7, № 3. — С. 336-355. — Бібліогр.: 6 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/177021 517.956 In the paper we consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ωε, which is the union of a domain Ω₀ and a large number 2N of thin rods with variable thickness of order ε = O(N⁻¹). The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. We investigate the asymptotic behaviour of the solution as ε → 0 under the Robin conditions on the boundaries of the thin rods. By using some special extension operators, the convergence theorem is proved. Розглядається мiшана крайова задача для рiвняння Пуассона у плоскому дворiвневому з’єднаннi Ωε, яке є об’єднанням деякої областi Ω₀ та великої кiлькостi 2N тонких стержнiв iз змiнною товщиною порядку ε = O(N⁻¹) Тонкi стержнi роздiлено на два рiвнi в залежностi вiд їх довжини. Крiм того, тонкi стержнi з кожного рiвня ε-перiодично чергуються. Вивчено асимптотичну поведiнку розв’язку, коли ε → 0, при крайових умовах Робiна на межах тонких стержнiв. Iз використанням спецiальних операторiв продовження доведено теорему збiжностi. en Інститут математики НАН України Нелінійні коливання Homogenization of the Robin problem in a thick multilevel junction Усереднення задачі Робіна в густому багаторівневому з'єднанні Усреднение задачи Робина в густом многоуровневом соединении Article published earlier |
| spellingShingle | Homogenization of the Robin problem in a thick multilevel junction De Maio, U. Mel'nyk, T.A. Perugia, C. |
| title | Homogenization of the Robin problem in a thick multilevel junction |
| title_alt | Усереднення задачі Робіна в густому багаторівневому з'єднанні Усреднение задачи Робина в густом многоуровневом соединении |
| title_full | Homogenization of the Robin problem in a thick multilevel junction |
| title_fullStr | Homogenization of the Robin problem in a thick multilevel junction |
| title_full_unstemmed | Homogenization of the Robin problem in a thick multilevel junction |
| title_short | Homogenization of the Robin problem in a thick multilevel junction |
| title_sort | homogenization of the robin problem in a thick multilevel junction |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/177021 |
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