Homogenization of the Neumann-Fourier Problem in a Thick Two-Level Junction of Type 3:2:1
We consider a mixed boundary-value problem for the Poisson equation in a two-level junction " which is the union of a domain Ω₀ and a large number of thin cylinders with cross-section of order O(ε²): The thin cylinders are divided into two levels depending on their lengths. In addition, the t...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1066222016-10-06T00:24:50Z Homogenization of the Neumann-Fourier Problem in a Thick Two-Level Junction of Type 3:2:1 Mel'nyk, T.A. Vashchuk, P.S. We consider a mixed boundary-value problem for the Poisson equation in a two-level junction " which is the union of a domain Ω₀ and a large number of thin cylinders with cross-section of order O(ε²): The thin cylinders are divided into two levels depending on their lengths. In addition, the thin cylinders from each level are ε-periodically alternated. The nonuniform Neumann conditions are given on the lateral sides of the thin cylinders from the rst level and the uniform Fourier conditions are given on the lateral sides of the thin cylinders from the second level. We study the asymptotic behavior of the solution as ε → 0: The convergence theorem and the convergence of the energy integral are proved. 2006 Article Homogenization of the Neumann-Fourier Problem in a Thick Two-Level Junction of Type 3:2:1 / T.A. Mel`nyk, P.S. Vashchuk // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 3. — С. 318-337. — Бібліогр.: 37 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106622 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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| description |
We consider a mixed boundary-value problem for the Poisson equation in a two-level junction
" which is the union of a domain Ω₀ and a large number of thin cylinders with cross-section of order O(ε²): The thin cylinders are divided into two levels depending on their lengths. In addition, the thin cylinders from each level are ε-periodically alternated. The nonuniform Neumann conditions are given on the lateral sides of the thin cylinders from the rst level and the uniform Fourier conditions are given on the lateral sides of the thin cylinders from the second level. We study the asymptotic behavior of the solution as ε → 0: The convergence theorem and the convergence of the energy integral are proved. |
| format |
Article |
| author |
Mel'nyk, T.A. Vashchuk, P.S. |
| spellingShingle |
Mel'nyk, T.A. Vashchuk, P.S. Homogenization of the Neumann-Fourier Problem in a Thick Two-Level Junction of Type 3:2:1 Журнал математической физики, анализа, геометрии |
| author_facet |
Mel'nyk, T.A. Vashchuk, P.S. |
| author_sort |
Mel'nyk, T.A. |
| title |
Homogenization of the Neumann-Fourier Problem in a Thick Two-Level Junction of Type 3:2:1 |
| title_short |
Homogenization of the Neumann-Fourier Problem in a Thick Two-Level Junction of Type 3:2:1 |
| title_full |
Homogenization of the Neumann-Fourier Problem in a Thick Two-Level Junction of Type 3:2:1 |
| title_fullStr |
Homogenization of the Neumann-Fourier Problem in a Thick Two-Level Junction of Type 3:2:1 |
| title_full_unstemmed |
Homogenization of the Neumann-Fourier Problem in a Thick Two-Level Junction of Type 3:2:1 |
| title_sort |
homogenization of the neumann-fourier problem in a thick two-level junction of type 3:2:1 |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/106622 |
| citation_txt |
Homogenization of the Neumann-Fourier Problem in a Thick Two-Level Junction of Type 3:2:1 / T.A. Mel`nyk, P.S. Vashchuk // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 3. — С. 318-337. — Бібліогр.: 37 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT melnykta homogenizationoftheneumannfourierprobleminathicktwoleveljunctionoftype321 AT vashchukps homogenizationoftheneumannfourierprobleminathicktwoleveljunctionoftype321 |
| first_indexed |
2025-07-07T18:47:16Z |
| last_indexed |
2025-07-07T18:47:16Z |
| _version_ |
1837015015095271424 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2006, vol. 2, No. 3, pp. 318�337
Homogenization of the Neumann�Fourier Problem
in a Thick Two-Level Junction of Type 3:2:1
T.A. Mel'nyk, P.S. Vashchuk
Faculty of Mathematics and Mechanics, Taras Shevchenko Kyiv National University
64 Volodymyrska Str., Kyiv, 01033, Ukraine
E-mail:melnyk@imath.kiev.ua;
pavel_vaschuk@ukr.net
Received November 18, 2005
We consider a mixed boundary-value problem for the Poisson equation in
a two-level junction
" which is the union of a domain
0 and a large num-
ber of thin cylinders with cross-section of order O("2): The thin cylinders
are divided into two levels depending on their lengths. In addition, the thin
cylinders from each level are "-periodically alternated. The nonuniform Neu-
mann conditions are given on the lateral sides of the thin cylinders from the
�rst level and the uniform Fourier conditions are given on the lateral sides of
the thin cylinders from the second level. We study the asymptotic behavior
of the solution as " ! 0: The convergence theorem and the convergence of
the energy integral are proved.
Key words: homogenization, multi-level junctions, asymptotic behavior
of solutions.
Mathematics Subject Classi�cation 2000: 35B27, 35J25, 35C20, 35B25.
1. Introduction and Statement of the Main Result
Asymptotic methods for the investigation of boundary-value problems in do-
mains with complex dependence on a small parameter (perforated domains, par-
tially perforated domains, skeleton structures, and thin domains) were considered
in numerous papers (see, e.g., [1]�[14]) and the references therein). Boundary-
value problems in thick singularly degenerating junctions (the number of compo-
nents of such junctions increases in�nitely if the perturbation parameter " tends
to zero) have speci�c di�culties and deserve special attention. As shown in [15],
boundary-value problems in thick singularly degenerating junctions lose coerci-
tivity as "! 0, that essentially complicates asymptotic researches.
It is necessary to note that boundary-value problems in domains with quickly
oscillating boundaries, when ratio of the amplitude to the period of the oscillation
c
T.A. Mel'nyk, P.S. Vashchuk, 2006
Homogenization of the Neumann�Fourier Problem
is bounded or in�nitesimal quantity as the period of the oscillation tends to zero,
have no such asymptotic di�culties and properties (see, e.g., [13, 16]). For thick
junctions this ratio tends to in�nity.
The �rst works in this direction were papers [17]�[19] in which the asymptotic
behavior of the Green function of the Neumann problem for the Helmholtz equa-
tion in an unbounded thick junction was studied. In [20]�[30] thick singularly
degenerating junctions were classi�ed, asymptotic methods for the investigation
of the main boundary-value problems of mathematical physics in thick junctions
of di�erent types were developed, the convergence theorems were proved, the �rst
terms of asymptotic expansions were constructed, the corresponding estimates
were proved, and the in�uence of boundary conditions given at the boundaries of
thick junctions and the geometric con�guration of thick junctions on the asymp-
totic behavior of solutions was investigated.
A thick junction
" of type k : p : d is a domain in Rn which consists
of some domain
0 and a large number of "-periodically situated thin domains
along some manifold on the boundary of
0: This manifold is called the joint
zone and the domain
0 is called the junction's body. Here " is a small parameter
which characterizes the distance between the neighboring thin domains and their
thicknesses. In general, the junction's body and the joint zone can depend on "
as well. The type k : p : d of a thick junction refers to the limiting dimensions of
the body, the joint zone, and each of the attached thin domains respectively.
These thick junctions are the prototypes of widely used engineering construc-
tions, industrial installations, spaceship grids as well as of other physical and
biological systems with very distinct characteristic scales.
The aim of researches is to develop rigorous asymptotic methods for boundary-
value problems in thick junctions as the parameter " goes to 0; i.e., when the
number of the attached thin domains in�nitely increases and their thicknesses
tend to zero.
In the present paper we consider a new kind of thick junctions, namely, thick
multi-level junctions. A thick multi-level junction is a thick junction in which the
thin domains are divided into �nitely many levels depending on their lengths. In
addition the thin domains from each level are "-periodically alternated along the
joint zone.
For the �st time the problem in a plane two-level junction was considered in
[31] where the asymptotic behavior of eigenvalues and eigenfunctions of the spec-
tral problem was studied (the full proofs were published in [32]). In [33], with the
help of special extension operators, a convergence theorem was proved for a so-
lution to the Poisson equation in a plane two-level junction with homogeneous
Fourier boundary conditions at the boundaries of thin rods. In [34] the authors
proved the convergence theorem and the convergence of the energy integral for
a solution to the Poisson equation in a plane two-level junction with "-periodically
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 319
T.A. Mel'nyk, P.S. Vashchuk
alternated boundary Neumann and Dirichlet conditions at the boundaries of thin
rods from the �rst and the second levels respectively. In [35], with the method
of matched asymptotic expansions being used, the �rst terms of the asymptotic
expansion of a solution to a boundary-value problem with minimum smoothness
conditions imposed on the right-hand side were constructed and asymptotic esti-
mates in the Sobolev space H1(
") as " ! 0 were proved. It should be noted
that these plain thick multi-level junctions have type 2 : 1 : 1 according to the
classi�cation given in [20]�[30].
In the present paper we study the asymptotic behavior of a solution to a mixed
boundary-value problem in the three-dimensional thick two-level junction of type
3 : 2 : 1 and investigate the in�uence of boundary conditions on the asymptotic
behavior. In particular, the inhomogeneous Neumann boundary conditions are
given on the lateral sides of the thin cylinders from the �rst level and the ho-
mogeneous Fourier boundary conditions are given on the lateral sides of the thin
cylinders from the second level. Besides, the thin cylinders from the �rst and
from the second levels have both more dense packing on the cell of the joining.
Thus, except special perturbation of the domain, the boundary conditions are
"-periodically changed in the problem.
1.1. Statement of the Problem
Let B be the �nite union of smooth plane domains which are not crossed
and touched. In addition, the set B is strongly situated in the square f(�1; �2) :
0 < �1 < 1; 0 < �2 < 1g: Let us divide B into two classes: B(1) =
K1S
k=1
B
(1)
k
and
B
(2) =
K2S
k=1
B
(2)
k
(see. Fig. 1).
x
1
2
0 11
x B
B
(2)
(1)
Figure 1.
320 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
Homogenization of the Neumann�Fourier Problem
A model thick two-level junction
" consists of the junction's body
0 = fx 2 R3 : x0 = (x1; x2) 2 Q; 0 < x3 <
(x0)g;
where Q = (0; a) � (0; a);
2 C
1(Q); min
x02Q
(x0) =
0 > 0; and a large
number of the thin cylinders
G
(1)
"
=
N�1[
i;j=0
K1[
k=1
n
x : ("�1x1 � i; "
�1
x2 � j) 2 B(1)
k
; x3 2 (�d1; 0]
o!
;
G
(2)
"
=
N�1[
i;j=0
K2[
k=1
n
x : ("�1x1 � i; "
�1
x2 � j) 2 B(2)
k
; x3 2 (�d2; 0]
o!
:
Here N is a large natural number, " = a=N is a small discrete parameter that
characterizes the distance between nearby thin cylinders and their thicknesses;
0 < d2 � d1: Thus,
" =
0
S
G
(1)
"
S
G
(2)
" : The thin cylinders are divided into
two levels G
(1)
" and G
(2)
" depending on their lengths, and they are "-periodically
alternated along the Ox1-direction and Ox2-direction and they are joined with
0
over the "-homothetic images "(i+j+B
(1)
k
); i; j = 0; 1; : : : ; N�1; k = 1; : : : ;K1;
and "(i + j +B
(2)
k
); i; j = 0; 1; : : : ; N � 1; k = 1; : : : ;K2; of the classes B
(1) and
B
(2) respectively. The cell of alternation is shown on Fig. 2.
x
x
x
1
2
3
�
�
Figure 2.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 321
T.A. Mel'nyk, P.S. Vashchuk
In
" we consider the following problem
��u"(x) = f"(x); x 2
";
@�u"(x) = "g"(x); x 2 S(1)" ;
@�u"(x) = �"k0u"(x); x 2 S(2)" ;
@�u"(x) = 0; x 2 @
"n(S(1)" [ S(2)" );
(1)
where @� = @=@� is the outward normal derivative, S
(i)
" ; i = 1; 2 are the unions
of the lateral surfaces of the thin cylinders from the level G
(i)
" .
Without loss of generality, we can assume that f" 2 L
2(
1) where
1 =
0 [ D1, D1 = Q � (�d1; 0): Analogously we de�ne D2 = Q � (�d2; 0) and
2 =
0 [D2: Assume that
f" �! f0 in L
2(
1) as "! 0: (2)
We also suppose that the function g" and its generalized derivatives with respect
to x1 and x2 belong to L2(D1) and
9 C0 > 0 8 " > 0 k@xmg"kL2(D1) � C0; m = 1; 2;
g" �! g0 in L
2(D1) as "! 0:
(3)
The function u" 2 H1(
") is called a generalized solution to problem (1) if it
satis�es the integral identityZ
"
ru" �r'dx+ "k0
Z
S
(2)
"
u"'d�x =
Z
"
f"'dx+ "
Z
S
(1)
"
g"'d�x 8' 2 H1(
"): (4)
It follows from the fundamental statements of the theory of boundary-value
problems that for every �xed value " > 0 there exists a unique generalized solution
to problem (1).
The aim of the present paper is to study the asymptotic behavior of the
solution to problem (1) as " ! 0; i.e., as the number of thin cylinders increases
in�nitely and their thicknesses tend to zero, and to investigate the in�uence of the
alternation of the boundary Neumann and the Fourier conditions on the asymp-
totic behavior of the solution.
1.2. Features of Investigation and Formulation of the Main Result
For Neumann boundary-value problems in perturbed domains E.Ya. Khruslov
introduced the notion of strongly connected domains D� depending on a small
parameter �: This means that we suppose the existence of an extension operator
322 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
Homogenization of the Neumann�Fourier Problem
from H
1(D�) into H
1(Rn) uniformly bounded with respect to �: Later, D. Cio-
ranescu, J. Saint Jean Paulin, O.A. Oleinik, G.A. Iosif'yan, and A.S. Shamaev
(see, e.g., [4, 10]) proved the existence of the extension operators and proposed a
procedure for their construction in perforated domains of an �-periodic structure.
Uniformly bounded extension operators play a very important role in the investi-
gation of boundary-value problems in the domains with complex dependence on
a small parameter.
However, as it was shown in [20]�[29], thick junctions do not belong to the
class of strongly connected (as well as weakly connected) domains, i.e., for these
domains there are no extension operators that would be bounded uniformly with
respect to the parameter " in the corresponding Sobolev spaces. This is one of the
main speci�c features of investigation of boundary-value problems in thick junc-
tions. In [20]�[29] the procedures were developed for the construction of special
extension operators preserving the class of a space for solutions of boundary-value
problems in thick junctions of di�erent types and with the help of these operators
the asymptotic behavior of solutions was studied and convergence theorems were
proved.
Later, in [36] where the homogeneous Neumann boundary-value problem in
a thick one-level junction was studied, it was shown that if the boundaries of thin
cylinders are rectilinear along the Ox3-axis, then the solution of the boundary-
value problem can be extended by zero to prove the convergence theorem. This is
explained by the fact that due to the rectilinearity of the boundaries of cylinders
this extension preserves the generalized derivative with respect to x3: We use this
fact in the present paper. However, for thick two-level junctions it is necessary
to construct two special operators of zero extension into two di�erent domains.
In the case, when the thin cylinders of a thick two-level junction are of variable
thickness, it is necessary to construct special extension operators (for the thick
plane two-level junctions it was made in [33]).
To formulate the main result we introduce the following operations of extension
by zero for functions from the space H1(
"):
ey"(1)(x) =
(
y"; x 2
0 [G(1)
" ;
0; x 2 D1 nG(1)
" ;
ey"(2)(x) =
(
y"; x 2
0 [G(2)
" ;
0; x 2 D2 nG(2)
" ;
(5)
where D1 = Q� (�d1; 0) and D2 = Q� (�d2; 0) are parallelepipeds �lled up with
thin cylinders of the �rst and the second levels respectively in the limit passage
as "! 0. It is obvious that ey"(1) and ey"(2) belong to the anisotropic Sobolev spaces
W
0;1(Di) =
�
v 2 L2(Di) : there exists a generalized derivative @x3v 2 L2(Di)
,
i = 1; 2.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 323
T.A. Mel'nyk, P.S. Vashchuk
Theorem 1. The solution u" to problem (1) satis�es the following relations
u"
w�! v
+
0 in H
1(
0);eu"(1) w�! jB(1)j v(1;�)0 in W
0;1(D1);eu"(2) w�! jB(2)j v(2;�)0 in W
0;1(D2);
9>=>; as "! 0;
where
v0(x) =
8><>:
v
+
0 (x); x 2
0;
v
(1;�)
0 (x); x 2 D1;
v
(2;�)
0 (x); x 2 D2;
(6)
is a solution of the following problem
��v+0 (x) = f0(x); x 2
0;
@�v
+
0 (x) = 0; x 2 @
0nQ;
�jB(1)j @2
x3
v
(1;�)
0 (x) = jB(1)j f0(x) + l
(1)
g0(x); x 2 D1;
@x3v
(1;�)
0 (x0;�d1) = 0; x
0 2 Q:
�jB(2)j @2
x3
v
(2;�)
0 (x) + k0l
(2)jB(2)jv(2;�)0 (x) = jB(2)j f0(x); x 2 D2;
@x3v
(2;�)
0 (x0;�d2) = 0; x
0 2 Q:
v
(1;�)
0 (x0; 0) = v
(2;�)
0 (x0; 0) = v
+
0 (x
0
; 0); x
0 2 Q:
jB(1)j@x3v
(1;�)
0 (x0; 0) + jB(2)j@x3v
(2;�)
0 (x0; 0) = @x3v
+
0 (x
0
; 0); x
0 2 Q:
(7)
Here jB(i)j =
KiP
k=1
jB(i)
k
j; l(i) =
KiP
k=1
l
(i)
k
; where jB(i)
k
j; l(i)
k
are the area and the perime-
ter of the plane domain B
(i)
k
respectively, i = 1; 2.
2. Auxiliary Asymptotic Estimates
Investigation of the boundary-value problems in thick junctions with inhomo-
geneous Neumann, Fourier, or Steklov boundary conditions on the boundaries of
the attached thin domains encounters special di�culties. In [37, 26, 27, 28] for
the homogenization of these boundary-value problems there was suggested a new
approach with the special integral identities being used.
For problem (1) this will be integral identities (9). Analogously as in [26], for
the 1-periodic extensions with respect to �1 and �2 of solutions Y
(i)
k
; k = 1; : : : ;Ki
324 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
Homogenization of the Neumann�Fourier Problem
of the following problems
��Y
(i)
k
(�) = l
(i)
k
jB(i)
k
j�1; � = (�1; �2) 2 B(i)
k
;
@�(�)Y
(i)
k
(�) = 1; � 2 @B(i)
k
;R
B
(i)
k
Y
(i)
k
(�)d� = 0;
(8)
we prove
"
Z
S
(i)
"
v d�x =
KiX
k=1
l
(i)
k
jB(i)
k
j
0Z
�di
Z
N�1S
i;j=0
"(i+j+B
(i)
k
)
v dx
0
dx3
+ "
KiX
k=1
0Z
�di
Z
N�1S
i;j=0
"(i+j+B
(i)
k
)
r�Y
(i)
k
j
�=x0
"
� rx0 v dx
0
dx3 8 v 2 H1
�
G
(i)
"
�
; i = 1; 2:
(9)
From (9) it follows that for any function v2 2 H1
�
G
(i)
"
�
c
Z
G
(i)
"
v
2
dx � "
Z
S
(i)
"
v
2
d�x + "
KiX
k=1
0Z
�di
Z
N�1S
i;j=0
"(i+j+B
(i)
k
)
jr�Y
(i)
k
j � jrx0 (v
2)j dx0dx3;
where c = minfl(i)
k
=jB(i)
k
jg: Taking into account that sup
�2B
(i)
k
jr�Y
(i)
k
j � ck; we
obtain the following identities
Z
G
(i)
"
v
2
dx � C0"
0B@ Z
S
(i)
"
v
2
d�x +
Z
G
(i)
"
jrx0(v
2)j dx
1CA ; i = 1; 2: (10)
In the Sobolev space H1 along with the norm kukH1(
") =
�R
"
(jruj2 +
u
2)dx
� 1
2 ; we introduce a new norm k � k" generated by the scalar product
(u; v)" =
Z
"
ru � rv dx+ "k0
Z
S
(2)
"
uv d�x 8u; v 2 H1(
"):
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 325
T.A. Mel'nyk, P.S. Vashchuk
Lemma 1. The norms k � k" and k � kH1(
") are uniformly equivalent, i.e.,
there exist constants C1 > 0; C2 > 0 and "0 > 0 such that for any " 2 (0; "0) and
u 2 H1(
") the following relations hold
C1kukH1(
") � kuk" � C2kukH1(
"): (11)
P r o o f. The right inequality in (11) follows from the inequality
"
Z
S
(i)
"
v
2
d�x � C3
� Z
G
(i)
"
v
2
dx+ "
2
Z
G
(i)
"
jrvj2 dx
�
8 v 2 H1
�
G
(i)
"
�
; i = 1; 2; (12)
which was proved in [26]. Let us prove the left inequality in (11). Using (9) and
(10) we get
kuk2
H1(
")
=
Z
0
jruj2 dx+
Z
0[G
(1)
"
u
2
dx+
Z
G
(2)
"
u
2
dx �
Z
0
jruj2 dx+
Z
0[G
(1)
"
u
2
dx
+ "C0
Z
S
(2)
"
u
2
d�x + "C0
Z
G
(i)
"
j2urx0 uj dx0dx3;
whence
kuk2
H1(
")
� c1kuk2" +
Z
0[G
(1)
"
u
2
dx: (13)
Now let us show that there exists a positive constant c2 such that for " small
enough Z
0[G
(1)
"
u
2
dx � c2kuk2" 8u 2 H1(
"): (14)
We argue by contradiction. If not, then there exist sequences f"n : n 2 Ng and
fv"ng 2 H1(
"n) such that lim
n!1
"n = 0;Z
0[G
(1)
"n
v
2
"n
dx = 1; (15)
kv"nk2"n =
Z
"n
jrv"n j2 dx+ "k0
Z
S
(2)
"n
v
2
"n
d�x <
1
n
8n 2 N: (16)
326 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
Homogenization of the Neumann�Fourier Problem
Since the sequence fv"ng is bounded inH1(
0); there exists a subsequence that
is fundamental in L2(
0): Denote this subsequence again by fv"ng: Furthermore,
kv"n � v"mk2H1(
0)
� kv"n � v"mk2L2(
0)
+ 2krv"nk2L2(
0)
+ 2krv"mk2L2(
0)
� kv"n � v"mk2L2(
0)
+
2
n
+
2
m
! 0 as n;m!1:
Hence fv"ng is fundamental in H1(
0) and therefore it converges to an element
v0 2 H
1(
0): Using relation (16) we get
R
0
jrv0j2 dx = 0; which implies that
v0 = const in H1(
0): Taking into account properties of the trace operator, we
conclude that
v"n
��
x3=0
s�! v0 � const in L
2(Q) as n!1: (17)
From (16) and (10) it follows that
Z
B
(2)
"n
v
2
"n
(x0; 0) dx0 � c6
0BB@ Z
G
(2)
"n
jrv"n j2 dx+
Z
G
(2)
"n
v
2
"n
dx
1CCA �! 0; n!1; (18)
where B
(2)
"n =
N�1S
i;j=0
�
K2S
k=1
"n(i+ j +B
(2)
k
)
�
:
Consider 1-periodic function �2(�); � 2 R2 which is de�ned in the square
[0; 1]2 as follows:
�2(�) =
(
1; � 2 B(2)
;
0; � 2 [0; 1]2 nB(2)
:
It is easy to verify that
�2
�
x
0
"
�
w�! jB(2)j in L
2([0; 1]2) as "! 0; (19)
where jB(2)j denotes the Lebesgue measure of B(2)
: Using relations (17) and (19)
we obtainZ
B
(2)
"n
v
2
"n
(x0; 0) dx0 =
Z
Q
�2
�
x
0
"n
�
v
2
"n
(x0; 0) dx0 �! jB(2)j
Z
Q
v
2
0 dx
0
; n!1:
On the other hand, according to (18) we have
jB(2)j
Z
Q
v
2
0 dx
0 = 0: (20)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 327
T.A. Mel'nyk, P.S. Vashchuk
Since v0 � const in
0; it follows from (20) that v0 � 0 almost everywhere in
0:
Let us �nd the limit of
R
G
(1)
"n
v
2
"n
(x) dx as n ! 1. According to (17) and
(20) we getZ
B
(1)
"n
v
2
"n
(x0; 0) dx0 =
Z
Q
�1
�
x
0
"n
�
v
2
"n
(x0; 0) dx0 �! 0 as n!1; (21)
where B
(1)
"n =
N�1S
i;j=0
�
K1S
k=1
"n(i+ j +B
(1)
k
)
�
and �1(�); � 2 R2 ; is 1-periodic function
which is de�ned in the square [0; 1]2 as follows
�1(�) =
(
1; � 2 B(1)
;
0; � 2 [0; 1]2 n B(1)
:
The inequalityZ
G
(1)
"n
v
2
"n
(x) dx � 2d21
Z
G
(1)
"n
jrv"n j2 dx+ 2d1
Z
B
(1)
"n
v
2
"n
(x0; 0) dx0
and relations (16) , (21) yield that kv"nk2
L2
�
G
(1)
"n
� �! 0 as n!1:
Thus kv"nk2
L2
�
0[G
(1)
"n
� �! 0 as n !1: However, this is at variance with
(15). This contradiction establishes estimate (14). Now, by virtue of (13) and
(14), we obtain the left inequality in (11). The lemma is proved.
R e m a r k 1. Here and in what follows, all constants ci and Ci in inequalities
are independent of ":
Let us prove uniform estimates for the solution of problem (1). Setting ' = u"
in the integral identity (4) and using (12) we obtain
ku"k2" � C4
�
kf"kL2(
") +
p
"kg"k
L2
�
S
(1)
"
�� ku"kH1(
"):
Taking Lemma 1 into account we get
ku"kH1(
") � C5
�
kf"kL2(
") +
p
"kg"k
L2
�
S
(1)
"
�� : (22)
Assuming (3) and using the integral identity (9) we deduce the following inequality
p
"kg"k
L2
�
S
(1)
"
� � C6: (23)
Thus, taking into account this inequality and relation (2) we conclude from (22)
that there exist constant C7 > 0 and "0 > 0 such that for all " 2 (0; "0)
ku"kH1(
") � C7: (24)
328 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
Homogenization of the Neumann�Fourier Problem
3. Proof of the Convergence Theorem
1. We extend the solution u" by zero (see (5)). Since the boundaries of the
thin cylinders are rectilinear, we get eu"(1) 2 W
0;1(D1) and eu"(2) 2 W
0;1(D2).
Furthermore,
@x3
� eu"(1)� = @̂x3u"
(1)
; @x3
� eu"(2)� = @̂x3u"
(2)
: (25)
Let us �nd the limits of the extensions for the solution u": Using relation (24)
we conclude that the quantities ku"kH1(
0); k eu"(1)kW 0;1(D1); k eu"(2)kW 0;1(D2) are
uniformly bounded with respect to ": Hence, there exists a subsequence f"0g �
f"g; again denoted by " such that
u"
w�! v
+
0 in H
1(
0);eu"(1) w�! jB(1)jv(1;�)0 in W
0;1(D1);eu"(2) w�! jB(2)jv(2;�)0 in W
0;1(D2);
@̂xmu"
(1)
w�!
(1)
m in L
2(D1); m = 1; 2; 3;
@̂xmu"
(2)
w�!
(2)
m in L
2(D2); m = 1; 2; 3;
9>>>>>>>=>>>>>>>;
as "! 0; (26)
where v+0 ; v
(i;�)
0 ;
(i)
m ; i = 1; 2, m = 1; 2; 3, are certain functions which will be
determined in what follows.
Let us determine
(i)
3 ; i = 1; 2: Consider an arbitrary function 2 C
1
0 (Di):
Using (25) we getZ
Di
@̂x3u"
(i)
dx =
Z
Di
@x3 eu"(i) dx = �
Z
Di
eu"(i) @x3 dx 8 2 C10 (Di); i = 1; 2:
Passing to the limit as "! 0 in this equality we obtainZ
Di
(i)
3 dx = �jB(i)j
Z
Di
v
(i;�)
0 @x3 dx 8 2 C10 (Di); (27)
which implies that
(i)
3 = jB(i)j@x3v
(i;�)
0 almost everywhere in Di; i = 1; 2:
Let us determine
(i)
m ; i = 1; 2; m = 1; 2: Let (b
(i)
1 (k); b
(i)
2 (k)) be the geometric
center of gravity of the domain B
(i)
k
(k = 1; : : : Ki): Consider the functions
Z
(i)
m;k
(�m) = ��m + b
(i)
m
(k) + [�m]; k = 1; : : : Ki; i = 1; 2; m = 1; 2;
where [t] is the integer part of t: With the help of this functions we determine the
following test function
�(1)
m (x) =
(
0; x 2
0
S
G
(2)
" ;
"Z
(1)
m;k
�
xm
"
�
(x); x 2 G(1)
" (k); k = 1; : : : ;K1; m = 1; 2;
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 329
T.A. Mel'nyk, P.S. Vashchuk
8 2 C10 (D1);
�(2)
m (x) =
(
0; x 2
0
S
G
(1)
" ;
"Z
(2)
m;k
�
xm
"
�
(x); x 2 G(2)
" (k); k = 1; : : : ;K2; m = 1; 2;
8 2 C10 (D2);
where G
(i)
" (k) =
N�1S
i;j=0
�
fx : ("�1x1 � i; "
�1
x1 � j) 2 B(i)
k
; x3 2 (�di; 0]g
�
: It is
easy to see that �
(i)
m (x) 2 H1(
") and
r�
(i)
1 =
�
� + "Z
(i)
1;k
�
x1
"
�
@x1 ; "Z
(i)
1;k
�
x1
"
�
@x2 ; "Z
(i)
1;k
�
x1
"
�
@x3
�
;
r�
(i)
2 =
�
"Z
(i)
2;k
�
x2
"
�
@x1 ; � + "Z
(i)
2;k
�
x2
"
�
@x2 ; "Z
(i)
2;k
�
x2
"
�
@x3
�
;
x 2 G(i)
" (k); k = 1; : : : Ki; i = 1; 2:
Substituting the functions �
(1)
1 and �
(1)
2 into the integral identity (4) we get
K1X
k=1
Z
G
(1)
" (k)
�
� @u"
@xm
+ "Z
(1)
m;k
@u"
@x1
@
@x1
+ "Z
(1)
m;k
@u"
@x2
@
@x2
+ "Z
(1)
m;k
@u"
@x3
@
@x3
�
dx
=
K1X
k=1
0BB@ Z
G
(1)
" (k)
"f"Z
(1)
m;k
dx+ "
2
Z
S
(1)
" (k)
Z
(1)
m;k
g" d�x
1CCA ; m = 1; 2:
Then, using relations (2), (3),(23) and (24) we have�������
Z
G
(1)
"
@u"
@xm
dx
�������
� "
K1X
k=1
0BB@ Z
G
(1)
" (k)
��Z(1)
m;k
(ru" � r � f" )
��dx+ "
Z
S
(1)
" (k)
��Z(1)
m;k
��jg" j d�x
1CCA
� "c1
�
kru"k
L2
�
G
(1)
"
�kr k
L2
�
G
(1)
"
� + kf"k
L2
�
G
(1)
"
�k k
L2
�
G
(1)
"
�
+
p
"kg"k
L2
�
S
(1)
"
�p"k k
L2
�
S
(1)
"
�
�
� "c1
�
c2k k
H1
�
G
(1)
"
�
+ c3k k
H1
�
G
(1)
"
� + c4k k
H1
�
G
(1)
"
�
�
� "c5k kH1(D1); m = 1; 2:
330 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
Homogenization of the Neumann�Fourier Problem
Passing to the limit as "! 0 in these inequalities we obtainZ
D1
(1)
m
dx = 0 8 2 C10 (D1); m = 1; 2; (28)
i.e.,
(1)
1 =
(1)
2 = 0 almost everywhere in D1:
Substituting the functions �
(2)
1 and �
(2)
2 into the integral identity (4) we get
K2X
k=1
Z
G
(2)
" (k)
�
� @u"
@xm
+ "Z
(2)
m;k
@u"
@x1
@
@x1
+ "Z
(2)
m;k
@u"
@x2
@
@x2
+ "Z
(2)
m;k
@u"
@x3
@
@x3
�
dx
+ "
2
k0
K2X
k=1
Z
S
(2)
" (k)
Z
(2)
m;k
u" d�x =
K2X
k=1
Z
G
(2)
" (k)
"f"Z
(2)
m;k
dx; m = 1; 2:
Analogously we obtain that
(2)
1 =
(2)
2 = 0 almost everywhere in D2:
2. It remains to determine the functions v+0 ; v
(1;�)
0 and v
(2;�)
0 : First, we �nd the
traces of these functions on Q: By virtue of the compactness of the trace operator
in the anisotropic spaces W 0;1 and the �rst three relations in (26) we have
v"(x
0
; 0)
s�! v
+
0 (x
0
; 0) in L
2(Q) as "! 0;ev"(1)(x0; 0) s�! jB(1)jv(1;�)0 (x0; 0) in L
2(Q) as "! 0;ev"(2)(x0; 0) s�! jB(2)jv(2;�)0 (x0; 0) in L
2(Q) as "! 0:
(29)
Since
ev"(1)(x0; 0) = �1
�x0
"
�
v"(x
0
; 0); ev"(2)(x0; 0) = �2
�x0
"
�
v"(x
0
; 0); x
0 2 Q; (30)
then, passing to the limit as "! 0 in(30) and using relation (29), we obtain
v
+
0 (x
0
; 0) = v
(1;�)
0 (x0; 0) = v
(2;�)
0 (x0; 0); x
0 2 Q:
Using the extension operators (5) and the integral identities (9) we rewrite
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 331
T.A. Mel'nyk, P.S. Vashchuk
the integral identity (4) in the following wayZ
0
ru" � r' dx+
Z
D1
�
@̂x1u"
(1)
@x1'+ @̂x2u"
(1)
@x2'+ @̂x3u"
(1)
@x3'
�
dx
+
Z
D2
�
@̂x1u"
(2)
@x1'+ @̂x2u"
(2)
@x2'+ @̂x3u"
(2)
@x3'
�
dx
+ k0
K2X
k=1
l
(2)
k
jB(2)
k
j
0Z
�d2
Z
N�1S
i;j=0
"(i+j+B
(2)
k
)
u"' dx
0
dx3
+ "k0
K2X
k=1
0Z
�d2
Z
N�1S
i;j=0
"(i+j+B
(2)
k
)
r�Y
(2)
k
� rx0 (u"') dx
0
dx3
=
Z
0
f"'dx+
Z
D1
�1
�
x
0
"
�
f"'dx+
Z
D2
�2
�
x
0
"
�
f"'dx
+
K1X
k=1
l
(1)
k
jB(1)
k
j
0Z
�d1
Z
N�1S
i;j=0
"(i+j+B
(1)
k
)
g"' dx
0
dx3
+ "
K1X
k=1
0Z
�d1
Z
N�1S
i;j=0
"(i+j+B
(1)
k
)
r�Y
(1)
k
� rx0 (g"') dx
0
dx3 8' 2 H1(
1): (31)
Then, passing to the limit as "! 0 in (31) and taking into account relations (2),
(3), (19), (26)�(28) and the fact that the last terms both in the left-hand side and
in the right-hand side tend to zero, we obtainZ
0
rv+0 � r' dx+ jB(1)j
Z
D1
@v0
@x3
(1;�)
@'
@x3
dx+ jB(2)j
Z
D2
@v0
@x3
(2;�)
@'
@x3
dx
+ k0jB(2)j l(2)
Z
D2
v
(2;�)
0 ' dx =
Z
0
f0' dx+ jB(1)j
Z
D1
f0' dx
+ jB(2)j
Z
D2
f0' dx+ l
(1)
Z
D1
g0' dx 8' 2 H1(
1):
(32)
Identity (32) is the corresponding integral identity for problem (7) in the
332 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
Homogenization of the Neumann�Fourier Problem
following anisotropic Sobolev vector-space
H = fu = (u0; u1; u2) 2 V := L
2(
0)� L
2(D1)� L
2(D2)
�� u0 2 H1(
0);
9 @x3u1 2 L2(D1); 9 @x3u2 2 L2(D2); u0(x
0
; 0) = u1(x
0
; 0) = u2(x
0
; 0); x0 2 Qg
with the scalar product
(u;v)H =
Z
0
ru0 � rv0 dx+
2X
i=1
jB(i)j
Z
Di
@x3ui @x3vi dx+ k0jB(2)j l(2)
Z
D2
u v dx:
Obviously, the space H continuously embeds in V: By using standard Hilbert
space methods, we can state that there exists a unique weak solution v0 2 H to
problem (7), which is called the limit problem for problem (1).
Due to the uniqueness of the solution to problem (7), the above reasoning
holds for any subsequence of f"g chosen at the beginning of the proof. Therefore,
the theorem is proved.
The fact that extensions of solutions to boundary-value problems in perforated
domains converge weakly in the spaces H1 enables to prove the convergence of
the energy integrals (see, i.g., [5, 10]). Theorem 1 gives this possibility as well.
We introduce the following notation
E"(u") :=
Z
"
jru"j2 dx+ "
Z
S
(2)
"
u
2
" d�x = (u"; u")";
E0(v0) = (v0;v0)H :=
Z
0
jrv+0 j2 dx+ jB(1)j
Z
D1
j@x3v
(1;�)
0 j2 dx
+ jB(2)j
Z
D2
j@x3v
(2;�)
0 j2 dx+ k0jB(2)j l(2)
Z
D2
(v
(2;�)
0 )2 dx:
The quantities E"(u") and E0(v0) determine the energy of the systems simulated
by problems (1) and (7) respectively. It is easy to see that
E"(u") =
Z
"
f"u" dx+ "
Z
S
(1)
"
g"u" d�x:
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 333
T.A. Mel'nyk, P.S. Vashchuk
Passing to the limit as " ! 0 in this equality and taking into account relations
(2), (3) and (26), as in Theorem 1 we obtain
lim
"!0
E"(u") =
Z
0
f0v
+
0 dx+ jB(1)j
Z
D1
f0v
(1;�)
0 dx
+ jB(2)j
Z
D2
f0v
(2;�)
0 dx+ l
(1)
Z
D1
g0v
(1;�)
0 dx = (v0;v0)H:
Corollary 1. lim"!0E"(u") = E0(v0):
Conclusions
In the present paper we have studied the in�uence of boundary conditions on
the asymptotic behavior of the solution to problem (1). We have shown that the
limit boundary-value problem (7) consists of three boundary value problems joined
together into one limit problem by certain conjugation conditions in the joint zone.
The inhomogeneity in the Neumann boundary conditions on the lateral sides of
the cylinders from the �rst level results in the appearance of a new term in the
right-hand side of the homogenized boundary-value problem in the parallelepiped
D1. This fact was noted in [37] where the homogenization of elliptic equations
that describe processes in strongly inhomogeneous thin perforated domains with
rapidly varying thickness was made. Furthermore,in the di�erential equations of
problem (7) there appear the coe�cients jB(i)j=l(i), i = 1; 2 which characterize
"density of the packing" of the thin cylinders from the �rst and second levels.
It was noted in [5] that for functionals, that are de�ned on re�exive spaces
and grow faster than the norm, there is, in fact, only one natural de�nition of
homogenization of such functionals, namely, the de�nition in terms of the con-
vergence of energies. For this reason, Corollary 1 is a very important result that
enables to investigate variational problems in thick multi-level junctions.
References
[1] N.S. Bakhvalov and G.P. Panasenko, Homogenization: Averaging Processes in Pe-
riodic Media. Kluwer Acad. Publ., Ser. Math. and its Appl., 1989. (Russian Ed.:
Nauka, Moscow, 1984.)
[2] A. Bensoussan, J. Lions, and G. Papanicolau, Asymptotic Analysis for Periodic
Structure. North Holland, Amsterdam, 1978.
[3] P.G. Ciarlet, Plates and Junctions in Elastic Multi-Structures. Masson, Paris,
1990.
334 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
Homogenization of the Neumann�Fourier Problem
[4] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures.
Springer�Verlag, Ser. Appl. Math. Sci. 136 (1999).
[5] V.V. Zhikov, S.M. Kozlov, and O.A. Oleinik, Homogenization of Di�erential Oper-
ators and Integral Functionals. Springer, Berlin�Heidelberg, 1994.
[6] A.A. Kovalevskij, Averaging of the Neumann Problems for Nonlinear Elliptic Equa-
tions in Domains with Accumulators. � Ukr. Mat. Zh. 47 (1995), No. 2, 194�212.
(Russian)
[7] V.A. Kozlov, V.G. Maz'ya, and A.B. Movchan, Asymptotic Representation of an
Elastic Field in a Multi-Structure. � Asymp. Anal. 11 (1995), 343�415.
[8] V.A. Marchenko and E.Ya. Khruslov, Boundary-Value Problems in Domains with
Finegrained Boundary. Naukova Dumka, Kiev, 1974. (Russian)
[9] S.A. Nazarov, Asymptotic Analysis of the Thin Plates and Rods. Vol. 1. Novosi-
birsk, 2002. (Russian)
[10] O.A. Oleinik, G.A. Yosifyan and A.S. Shamaev, Mathematical Problems to Theories
Powerfully Lumpy Springy Ambiences. Izd-vo MGU, Moscow, 1990. (Russian)
[11] O.A. Oleinik and T.A. Shaposhnikova, On the One Method of Approach in the
Problem of the Averaging in Partly Perforated Domains. � Di�. Eq. 30 (1994),
No. 11, 1994�2000. (Russian)
[12] O.A. Oleinik and T.A. Shaposhnikova, On the Aaveraging of the Poisson Equation
in the Partly Perforated Domains with the Mixed Type of Conditions on Boundary
of the Cavities for all their Concentration. � Rendiconti Lincei: Matematica e
Applicazioni 7 (1996), 129�146.
[13] J. Sanchez-Hubert and E. Sanchez-Palencia, Vibration and Coupling of Continuous
Systems. Springer�Verlag, Berlin, Heidelberg, 1989.
[14] I.V. Skrypnyk, The Methods of the Investigation of the Nonlinear Elliptical
Boundary-Value Problems. Nauka, Moscow, 1990. (Russian)
[15] F. Fleury and E. Sanchez�Palencia, Asymptotic and Spectral Properties of the
Acoustic Vibrations of Body Perforated by Narrow Channels. � Bull. Sci. Math. 2
(1986), No. 110, 149�176.
[16] G.A. Chechkin, A. Friedman, and A. Piatnitski, The Boundary-Value Problem in
Domains with Very Rapidly Oscillating Boundary. � J. Math. Anal. Appl. 231
(1999), 213�234.
[17] V.P. Kotliarov and E.Ya. Khruslov, On a Limit Boundary Condition of Some Neu-
mann Problem. � Teor. Funkts., Funkts. Anal. i Prilozh. (1970), No. 10, 83�96.
(Russian)
[18] G.V. Suzikov and E.Ya. Khruslov, On Advancing Sound Waves Through Narrow
Channels in a Re�acting Layer. � Teor. Funkts., Funkts. Anal. i Prilozh. (1976),
No. 5, 35�49. (Russian)
[19] E.Ya. Khruslov, On the Resonance Phenomenas in One Problem of Di�raction. �
Teor. Funkts., Funkts. Anal. i Prilozh. 10 (1968), 113�120. (Russian)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 335
T.A. Mel'nyk, P.S. Vashchuk
[20] T.A. Mel'nyk and S.A. Nazarov, Asymptotic Structure of the Spectrum of the
Neumann Problem in a Thin Comb-Like Domain. � C.R. Acad. Sci. Paris, Ser. 1.
319 (1994), 1343-1348.
[21] T.A. Mel'nyk and S.A. Nazarov, On the Asymptotics of Eigenvalues of the Neumann
Problem in a Domain of "Thick Comb". � Dokl. Russian Akad. Nauk 342 (1995),
No. 1, 23�25. (Russian)
[22] T.A. Mel'nyk and S.A. Nazarov, Asymptotics of the Neumann Spectral Problem
Solution in a Domain of "Thick Comb". � Tr. Sem. im. I.G. Petrovskogo 19 (1996),
138�174 (Russian); Engl. transl. � In: J. Math. Sci. 85 (1997), No. 6, 2326�2346.
[23] T.A. Mel'nyk, Homogenization of the Poisson Equation in a Thick Periodic Junction.
� Zeitschrift f�ur Analysis und ihre Anwendungen 18 (1999), No. 4, 953�975.
[24] T.A. Mel'nyk, Asymptotic Analysis of a Spectral Problem in a Periodic Thick
Junction of Type 3:2:1. � Math. Meth. Appl. Sci. 23 (2000), No. 4, 321�346.
[25] T.A. Mel'nyk and S.A. Nazarov, Asymptotic Analysis of the Neumann Problem of
the Junction of a Body and Thin Heavy Rods. � Algebra i Analiz 12 (2000), No. 2,
188�238. (Russian); Engl. transl. � In: St.-Petersburg Math. J. 12 (2001), No. 2,
317�351.
[26] T.A. Mel'nyk, Homogenization of a Singularly Perturbed Parabolic Problem in a
Thick Periodic Junction of Type 3:2:1. � Ukr. Mat. Zh. 52 (2000), No. 11, 1524�
1534. (Russian); Engl. transl. � In: Ukr. Math. J. 52 (2000), No. 11, 1737�1749.
[27] T.A. Mel'nyk, Asymptotic Behavior of Eigenvalues and Eigenfunctions of the
Steklov Problem in a Thick Periodic Junction. � Nonlinear Oscillations 4 (2001),
No. 1, 91�105.
[28] T.A. Mel'nyk, Asymptotic Behavior of Eigenvalues and Eigenfunctions of the Fourier
Problem in a Thick Periodic Junction of Type 3:2:1. � Grouped and Anal. Meth.
Math. Physics, Akad. Nauk Ukr., Kiev 36 (2001), 187�196.
[29] T.A. Mel'nyk, Vibrations of a Thick Periodic Junction with Concentrated Masses.
� Math. Models, Methods Appl. Sci. 11 (2001), No. 6, 1001�1029.
[30] S.A. Nazarov, Junctions of Singularly Degenerating Domains with Di�erent Limit
Demension. � Tr. Sem. im. I.G. Petrovskogo 18 (1995), 1�78 (P. I); Tr. Sem. im.
I.G. Petrovskogo 20 (2000), 155�196 (P. II). (Russian)
[31] T.A. Mel'nyk, Eigenmodes and Pseudo-Eigenmodes of Thick Multi-Level Junctions.
� Proc. Int. Conf. "Days on Di�raction-2004" 11 (2004), 51�52.
[32] T.A. Mel'nyk, Asymptotic Behavior of Eigenvalues and Eigenfunctions of the Fourier
Problem in a Thick Multi-Level Junction. � Ukr. Math. J. 58 (2006), No. 2, 195�
216.
[33] U. De Maio, T.A. Mel'nyk, and C. Perugia, Homogenization of the Robin Problem
in a Thick Multi-Level Junction. � Nonlinear Oscillations 7 (2004), No. 3, 336�356.
336 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
Homogenization of the Neumann�Fourier Problem
[34] T.A. Mel'nyk and P.S. Vashchuk, Homogenization of the Boundary-Value Problem
with Varying Type of Boundary Conditions in a Thick Multi-Level Junction. �
Nonlinear Oscillations 8 (2005), No. 2, 241�257.
[35] U. De Maio, T. Durante, and T.A. Mel'nyk, Asymptotic Approximation for the
Solution to the Robin Problem in a Thick Multi-Level Junction. � Math. Models,
Methods Appl. Sci. 15 (2005), No. 12, 1897�1921.
[36] R. Brizzi and J.P. Chalot, Boundary Homogenization and Neumann Boundary
Value Problem. � Ricerche di Matimatica 46 (1997), 341�387.
[37] T.A. Mel'nyk, Averaging of Elliptic Equations Describing the Processes in Strongly
Non-Uniform Thin Perforated Domains with Rapidly Varying Thickness. � Dokl.
Akad. Nauk UkrSSR (1991), No. 10, 15�19. (Russian)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 337
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