Homogenization of Electrostatic Problems in Nonlinear Medium with Thin Perfectly Conducting Grids
The asymptotic behavior of solutions of the family of nonlinear elliptic equations in domains with thin grids concentrating near a hypersurface when measure of the wires tends to zero and the density tends to infinity is investigated. The homogenized equations and the homogenized boundary conditions...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
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irk-123456789-1066782016-10-03T03:02:14Z Homogenization of Electrostatic Problems in Nonlinear Medium with Thin Perfectly Conducting Grids Goncharenko, M.V. Prytula, V.I. The asymptotic behavior of solutions of the family of nonlinear elliptic equations in domains with thin grids concentrating near a hypersurface when measure of the wires tends to zero and the density tends to infinity is investigated. The homogenized equations and the homogenized boundary conditions are derived. The homogenization technique is based on applying of the abstract theorem on homogenization of the nonlinear variational functionals in the Sobolev-Orlicz spaces. 2006 Article Homogenization of Electrostatic Problems in Nonlinear Medium with Thin Perfectly Conducting Grids / M.V. Goncharenko, V.I. Prytula // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 4. — С. 424-448. — Бібліогр.: 6 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106678 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The asymptotic behavior of solutions of the family of nonlinear elliptic equations in domains with thin grids concentrating near a hypersurface when measure of the wires tends to zero and the density tends to infinity is investigated. The homogenized equations and the homogenized boundary conditions are derived. The homogenization technique is based on applying of the abstract theorem on homogenization of the nonlinear variational functionals in the Sobolev-Orlicz spaces. |
| format |
Article |
| author |
Goncharenko, M.V. Prytula, V.I. |
| spellingShingle |
Goncharenko, M.V. Prytula, V.I. Homogenization of Electrostatic Problems in Nonlinear Medium with Thin Perfectly Conducting Grids Журнал математической физики, анализа, геометрии |
| author_facet |
Goncharenko, M.V. Prytula, V.I. |
| author_sort |
Goncharenko, M.V. |
| title |
Homogenization of Electrostatic Problems in Nonlinear Medium with Thin Perfectly Conducting Grids |
| title_short |
Homogenization of Electrostatic Problems in Nonlinear Medium with Thin Perfectly Conducting Grids |
| title_full |
Homogenization of Electrostatic Problems in Nonlinear Medium with Thin Perfectly Conducting Grids |
| title_fullStr |
Homogenization of Electrostatic Problems in Nonlinear Medium with Thin Perfectly Conducting Grids |
| title_full_unstemmed |
Homogenization of Electrostatic Problems in Nonlinear Medium with Thin Perfectly Conducting Grids |
| title_sort |
homogenization of electrostatic problems in nonlinear medium with thin perfectly conducting grids |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/106678 |
| citation_txt |
Homogenization of Electrostatic Problems in Nonlinear Medium with Thin Perfectly Conducting Grids / M.V. Goncharenko, V.I. Prytula // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 4. — С. 424-448. — Бібліогр.: 6 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT goncharenkomv homogenizationofelectrostaticproblemsinnonlinearmediumwiththinperfectlyconductinggrids AT prytulavi homogenizationofelectrostaticproblemsinnonlinearmediumwiththinperfectlyconductinggrids |
| first_indexed |
2025-07-07T18:51:18Z |
| last_indexed |
2025-07-07T18:51:18Z |
| _version_ |
1837015268005511168 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2006, v. 2, No. 4, p. 424�448
Homogenization of Electrostatic Problems in Nonlinear
Medium with Thin Perfectly Conducting Grids
M.V. Goncharenko
Mathematical Division, B. Verkin Institute for Low Temperature Physics & Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov, 61103, Ukraine
E-mail:goncharenko@ilt.kharkov.ua
V.I. Prytula
Institute for Radiophysics and Electronics, National Academy of Sciences of Ukraine
12 Akad. Proscura Str., Kharkov, 61002, Ukraine
E-mail:vlad-pritula@yandex.ru
Received November 7, 2005
The asymptotic behavior of solutions of the family of nonlinear ellip-
tic equations in domains with thin grids concentrating near a hypersurface
when measure of the wires tends to zero and the density tends to in�nity
is investigated. The homogenized equations and the homogenized boundary
conditions are derived. The homogenization technique is based on apply-
ing of the abstract theorem on homogenization of the nonlinear variational
functionals in the Sobolev-Orlicz spaces.
Key words: homogenization, domains with grids, electrostatic, Sobolev�
Orlicz spaces.
Mathematics Subject Classi�cation 2000: 35B27.
1. Introduction
In the present paper we study the asymptotic behavior of the electrostatic
�eld in the domains
(s) =
n F (s) �
� R
n , s = 1; 2; : : : , where F (s) is
a connected set of the net type with a density that tends to in�nity as s ! 1.
These structures having the form of metal wire nets with di�erent cell shapes are
widely applied in radio engineering, antenna technique and radio-relay links.
Usually a complex structure of the domain in which the initial problem is
considered does not cause additional di�culties in the proof of the existence of
solutions contrary to the way of �nding solution either analytically or numerically.
However, one can expect that when the grid is dense enough, then it acts as
c
M.V. Goncharenko and V.I. Prytula, 2006
Homogenization of Electrostatic Problems in Nonlinear Medium
an e�ective continuous medium (or �lm), and its behavior can be approximately
described by the homogenized di�erential equations (or boundary conditions). To
derive these equations we are to analyze asymptotic behavior of solutions in the
domains with grids.
The homogenized equations describing distribution of the electrostatic poten-
tial in the domains with dense thin grids in the case of linear dependence of the
permeability of the medium on the intensity of electric �eld were derived in [1].
The case of nonlinear medium with a zero potential on the grid was studied in
[2]. In the present work we study the case of nonlinear medium with nonzero
potential on the grid.
The method used is based on the variational principles [3] and the Sobolev�
Orlicz spaces technique [4], [5]. This technique helps us to show that sequence
of solutions of the initial systems converges to the solution of the homogenized
system.
The paper is composed as follows. In Section 2 we consider a number of basic
de�nitions of the Sobolev�Orlicz spaces. In section 3 we formulate the problem
statement and the main result which is proved in Sect. 4. In Section 5 we apply
the main result for studying the asymptotic behavior of the electrostatic potential
in weakly nonlinear medium with thin perfectly conducting grids.
2. Basic De�nitions of the Sobolev�Orlicz Spaces Theory
In this section we present some basic de�nitions and properties of the Orlicz
and Sobolev�Orlicz spaces. More information on the subject can be found, for
example, in [4, 5].
Let M(u) be a real-valued function of the real variable u satisfying
M(u) =
jujZ
0
M 0(t) dt; (2.1)
where M 0(t) is a positive for t > 0, right�continuous for t � 0, nondecreasing
function such that M 0(0) = 0; M 0(1) = lim
t!1
M 0(t) =1. A function having the
above-mentioned properties is called N function. We will further assume that for
u � 0
uM 0(u) � �M(u) (� > 1): (2.2)
Let us suppose that M(u) satis�es the following condition : there exist such
l > 0; u0 > 0, that
M(u) �
1
2l
M(lu); u � u0: (2.3)
Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4 425
M.V. Goncharenko and V.I. Prytula
Inequality (2.2) guaranties that M(u) satis�es �2-condition: there exists such
a function k(l) � 0 that, for all l � 0 and u � 0
M(lu) � k(l)M(u) (2.4)
where k(l) is a monotone increasing and di�erentiable for l > 0 function such that
k(0) = 0 and for all l > 0
lk0(l) � Ck(
l
2
): (2.5)
For any N function M(u) we introduce the complementary function N(v)
given by
N(v) = max
u>0
[ujvj �M(u)] : (2.6)
N(v) also satis�es �2- condition, because of (2.3).
The following inequalities hold
N [M(u)=u] < M(u); u > 0 (2.7)
uv �M(u) +N(v) (the Young inequality): (2.8)
Let
� R
n be a domain with a piecewise smooth boundary. Then the Orlicz
class LM (
) consists of all the functions u(x) such that
�(u;M;
)
def
=
Z
M (u(x)) dx <1: (2.9)
Let us introduce the Orlicz norm
kuk
M;
= sup
�(v;N;
)�1
������
Z
u(x)v(x) dx
������ ; (2.10)
where N(v) is a complementary function to M(u).
Taking into account the Young inequality, this norm makes sense for all u(x) 2
LM (
), and if M(u); N(u) satisfy �2-condition, then LM(
) becomes a Banach
re�exive space, which is called as Orlicz space [4] denoted by LM (
)).
Let u 2 LM(
) and v 2 LN (
). The following inequalities holdZ
uv dx � kuk
M;
kvkN;
(the Holder inequality) (2.11)
kuk
M;
� �(u;M;
) + 1 (2.12)
and if kuk
M;
� 1, then
�(u;M;
) � kuk
M;
: (2.13)
426 Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4
Homogenization of Electrostatic Problems in Nonlinear Medium
We say that the sequence of functions fuk(x) 2 LM(
); k = 1; 2; : : : g con-
verges on average to u(x) 2 LM(
), if
�(uk � u;M;
) ! 0; k !1: (2.14)
IfM(u) satis�es �2-condition then convergence on average is equivalent to strong
convergence in space LM (
).
Let us now consider the classesW 1
M
(
) consisting of all the functions u(x) from
the Orlicz spaces LM (
) such that distributional derivatives D�u are contained
in LM (
) for all � with j�j � 1. Here we denote by � the multi-index of integers
[�1; : : : ; �n] and by j�j the sum
nP
i=1
�i. These classes of functions can be supplied
with the norm
kuk1
M;
= max
j�j�1
n
kD�uk
M;
o
; (2.15)
where k�k
M;
is a suitable norm in LM (
) (as the norm de�ned above). These
classes are the Banach spaces under this norm. We shall refer to the spaces of
the form W 1
M
(
) as to the Orlicz�Sobolev spaces. They form generalization of
the Sobolev spaces in the same way as the Orlicz spaces form generalization of
Lp spaces. Next we de�ne another Orlicz�Sobolev space, W
1;0
M
(
), as the closure
of C1
0 in W 1
M
(
).
The following imbedding theorems are valid [5]:
Theorem 2.1. Let
be a bounded domain in R
n with a piecewise smooth
boundary, then W 1
M
(
) ,! LM (
), where we use �,!� to indicate the compact
imbedding. Furthermore, the following inequality holds
kuk
M;
� C kDuk
M;
; (2.16)
for any u(x) 2
Æ
W
1
M (
).
Theorem 2.2. Let
be a bounded domain in R
n and let � be a smooth
hypersurface of dimension n � 1 such that � �
. If
1R
1
M
�1(t)
t
1+ 1
n
dt = 1, then
W 1
M
(
) �! L
[M�]
n�1
n
(�), where
(M�)�1(jxj) =
jxjZ
0
M�1(t)
t1+
1
n
dt; (2.17)
and (M�)�1(u) and M�1(u) are the functions inverted to the N -functions of
M�(u) and M(u) respectively.
Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4 427
M.V. Goncharenko and V.I. Prytula
3. Problem Statement and Formulation of the Main Result
Let
be a bounded open set in R
n , n � 2, and F (s) be a closed set in
of the arbitrary shape depending on a parameter s (s = 1; 2; : : : ) such that
lim
s!1
mesF (s) = 0. We will assume that F (s) belongs to the inde�nitely small
neighborhood of some (n � 1)-dimensional smooth surface � �
and that the
distance from each point x 2 � to F (s) tends to zero as s!1.
For every �xed s in domain
(s) =
nF (s) we consider the nonlinear variational
problem
J (s)[u(s)] =
Z
(s)
F (x; u(s);ru(s))dx! inf; (3.1)
u
(s)
j
@F
(s)
= As; x 2 @F (s); (3.2)
u
(s)
j@
= f(x); x 2 @
; (3.3)
where the in�num is taken over the class of functions u(s) 2W 1
M
(
(s)) such that
u(s) = f(x) for x 2 @
, u(s) = As for x 2 @F (s) and parameters As are some
unknown constants. Without loss of generality, we may assume that f(x) 2
C1(
).
Let F (x; u; p) be a function that is de�ned and continuous for f(x; u; p) : x 2
� R
n ; u 2 R1 ; p 2 Rng. Let this function possess continuous partial derivatives
Fu, Fpi , i = 1; : : : ; n and satisfy the following conditions:
F (x; u; p)� F (x; u; q) �
nX
i=1
Fpi(x; u; q)(pi � qi) � 0; (3.4)
A1M(jpj) �A2M(u) � F (x; u; p) � A3[1 +M(u) +M(jpj)]; (3.5)
jF (x; u; p) � F (x; v; q)j
� A4[1 +M 0(juj) +M 0(jvj) +M 0(jpj) +M 0(jqj)](ju � vj+ jp� qj); (3.6)
8x 2
: ju(x)j > k F (x; u; 0) > F (x; k; 0); (3.7)
where Ai > 0, i = 1; 3; 4, A2 � 0, u; v 2 R1 , p; q 2 Rn .
The functional J (s)[u(s)] is assumed to be bounded from below so that for any
function u(s) 2 W 1
M
(
(s)), u(s) = A(s) for x 2 @F (s) and u(s) = f(x) for x 2 @
(f(x) 2 C1(�
)) the following inequality holdsZ
(s)
F (x; u(s);ru(s))dx � �
�
u(s)
(1)
M;
(s)
�
; (3.8)
428 Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4
Homogenization of Electrostatic Problems in Nonlinear Medium
where �(t) is a continuous, increasing for t ! 1, function; k�k
(1)
M;
(s)
is a norm
in the Sobolev�Orlicz space W 1
M
(
(s)).
It is well known [6], that there exists at least one solution u(s)(x) 2W 1
M
(
(s))
of problem (3.1�3.3). We continue u(s) to F (s) by setting u(s)(x) = As, x 2 F (s)
and consider the sequence of the continued functions, still denoted by fu(s)g, as
a sequence in W 1
M
(
). The problem is to describe the asymptotic behavior of
sequence u(s)(x) as s!1.
Let us introduce quantitative characteristics of the sets F (s) [3]. Consider an
arbitrary piece S of the hypersurface �. Let us draw normals of the length h > 0
from every point of S to both sides of �. For h small enough these normals are
not intersected and their ends form two smooth surfaces �+
h
and ��
h
. We denote
by Th(S) the subdomain of
which is formed by the segments of the mentioned
above normals (the layer of 2h thickness with the central surface S).
For every �xed h > 0 F (s) belongs to Th(S) for su�ciently large s (s > s(h)).
Consider the functional
C(S; s; h; b) = inf
v(s)
Z
Th(S)
fF (x; 0;rv(s)) + �(h)M(v(s) � b)gdx; (3.9)
where the in�num is taken over the class of functions v(s) 2W 1
M
(
(s)) vanishing on
F (s) \ Th(S); �(h) = k( 1
h1+
),
> 0, the function k(l) is de�ned by �2-condition
(2.4) and b 2 R1 .
It is seen clearly that C(S; s1; h; b) � C(S; s2; h; b) for F
(s1)
T
Th(S) � F (s2)
T
Th(S). Hence, function C(S; s; h; b) is a local characteristic of massiveness of the
set F (s) generated by F (y; u; p).
Theorem 3.1. Let the following condition hold: for any arbitrary piece S of
the surface �; 8 b 2 Rn and
> 0 there exist the following limits:
lim
h!0
lim sup
s!1
C(S; s; h; b) = lim
h!0
lim inf
s!1
C(S; s; h; b) =
Z
S
c(x; b)d�; (3.10)
where c(x; b) is a nonnegative continuous on � function.
Then, for any sequence fu(s)(x)g of the solutions of problem (3.1�3.3) (contin-
ued on F (s) by setting u(s) = A(s)) there exists a subsequence fusj (x)g that weakly
converges in the space W 1
M
(
) to function u(x) such that the pair fu(x); Ag is
a solution of the following problem:Z
F (x; u;ru)dx+
Z
�
c(x; u �A)d�! inf; (3.11)
uj@
= f(x); (3.12)
where A = lim
s!1
As.
Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4 429
M.V. Goncharenko and V.I. Prytula
4. Proof of Theorem 3.1
Since u(s) is the minimum of variational problem (3.1)�(3.3), we haveZ
(s)
F (x; u(s);ru(s))dx �
Z
(s)
F (x; ~fB(x);r ~fB(x) dx
�
Z
jF (x; ~fB(x);r ~fB(x))jdx; (4.1)
where
~fB(x) =
�
f(x); x 2 @
;
B; x 2 F (s) ;
(4.2)
it follows from (3.8) that
�
�
u(s)
(1)
M;
(s)
�
� Const (does not depend on s): (4.3)
This means that the sequence of the continued functions fu(s)(x)g is weakly
compact in W 1
M
(
), hence, one can extract a subsequence u(s); s = sj !1 that
weakly converges to function u(x) 2 W 1
M
(
). Let us prove that fu(x); Ag is
a solution of problem (4.94�4.95).
Let us cover � with a �nite number of segments S0
i
, i = 1; : : : ; N with su�-
ciently small diameter Æ = di(N). Every set S0
i
is immersed into a more extensive
open on � set ~Si, moreover, we choose a subset Si = S0
i
n (
S
i6=j
~Sj) to satisfy the
following conditions:
1. Si � S0
i
� S0
i
� ~Si; diam ~Si < Cdi(N)! 0; N !1; (4.4)
2. ~Si are bounded with the �nite set of smooth (n � 2)-dimension manifolds.
The number of intersections between ~Si do not exceed a �xed number M ,
that doesn't depend on N ;
3.
NP
i=1
�
�
~Si n Si
�
� �(N); �(N) ! 0; N !1: (4.5)
It is obvious that the unit of sets f ~Sig covers �. With this coverage we
associate a partition of unity f'i(x); x = ft1; : : : ; tn�1; 0g 2 �g satisfying the
conditions: 0 � 'i(x) � 1; 'i(x) = 0, x =2 ~Si, 'i(x) = 1, x 2 Si,
P
i
'i(x) � 1,
jD�'i(x)j � C�(N).
Consider an arbitrary function w(x) 2 C2(
) such that w(x) = f(x), x 2 @
.
Let U� be a subset of layers Th( ~Si) covering � so that jw(x)� ~fB(x)j > � > 0 for
430 Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4
Homogenization of Electrostatic Problems in Nonlinear Medium
any point x 2 Th( ~Si). We denote by bi = w(xi) �B, xi 2 � for Th( ~Si) 2 U� and
bi = 1 for Th( ~Si) =2 U�. Let v
(s)
i
(x) be a function that minimizes C( ~Si; s; h; b).
For any Th( ~Si) we de�ne
Bi("; s; h) = fx 2 Th( ~Si) : v
(s)
i
(x) sign bi � jbij � "g (4.6)
and
v̂si (x) =
8<
:
vs
i
(x); x 2 Bi("; s; h);
b"
i
= (jbij � ") sign bi; x 2 Th(Si) n Bi("; s; h);
(4.7)
where 0 < " < �
2 .
In the domain
(s) we de�ne the function
w
(s)
h
(x)
=
w(x) +
NÆX
i=1
[w(x) � ~fB(x)] (v̂
s
i (x)� b"i )(b
"
i )
�1 'i(x)
!
(x) + w(x) [1� (x)] ;
(4.8)
where (x) 2 C1
�
�
, such that 0 � (x) � 1, (x) = 1, x 2 Th0(�), (x) = 0,
x =2 Th(�) and jr (x)j � Cr�1; Th0(�) is the layer of 2h0 = 2(h � r) thickness,
r = h1+
. It follows from the properties of functions v̂s
i
(x), 'i(x), (x) that w
s
h
belongs to W 1
M
(
(s)), ws
h
is equal to f(x) on @
(s) and to some constant B on
F (s). Since u(s)(x) is the solution of problem (3.1�3.3) we get
J (s)
�
u(s)
�
� J (s)
�
w
(s)
h
�
: (4.9)
It is clear thatZ
(s)
F (x;ws
h
;rws
h
)dx �
Z
nTh(�)
F (x;w
(s)
h
;rw
(s)
h
) dx
+
Z
Th(�)nTh0 (�)
T
(s)
F (x;w
(s)
h
;rw
(s)
h
) dx+
X
i
Z
T
h0
(Si)
T
(s)
F (x;w
(s)
h
;rw
(s)
h
) dx
+
X
i;j
i6=j
Z
T
h0
( ~Si)
T
T
h0
( ~Sj)
T
(s)
���F (x;w(s)
h
;rw
(s)
h
)
��� dx: (4.10)
The second term in the right-hand side of (4.10) can be estimated with the use
of (3.5) Z
Th(�)nTh0 (�)
F (x;ws
h;rw
s
h) dx
Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4 431
M.V. Goncharenko and V.I. Prytula
� C
Z
Th(�)nTh0 (�)
[M(jws
hj)] + [M(jrws
hj)] dx+ C �mes(Th(�) n Th0(�)): (4.11)
M(u) is a convex function that satis�es �2�condition, so M(u+ v) � C(M(u) +
M(v)). Further we will use this inequality to estimate
R
Th(�)
M(w
(s)
h
) dx and
R
T 0
h
(�)
M(rw
(s)
h
) dx.
Let us denote ~w(x) = w(x) � ~fB(x). It follows from (4.8) that
jw
(s)
h
(x)j � jw(x)j +
NÆX
i
j ~w(x)j
jb"
i
j
jv̂s
i
� b"i jj'i(x)j: (4.12)
Using the properties of 'i(x) and the fact that M(u) satis�es �2-condition, we
get Z
Th(�)nTh0 (�)
M(jw
(s)
h
j) dx
� C
X
i
Z
Th(S
0
i
)nT
h0
(S0
i
)
M(jv̂s
i
� b"�j) dx+ const �mes(Th(�) n Th0(�)): (4.13)
To estimate
X
i
Z
Th(S
0
i
)nT
h0
(S0
i
)
M(jv̂s
i
� b"�j) dx we divide Th(S
0
i
) n Th0(S
0
i
) into
two subsets:
Bi \
�
Th(S
0
i) n Th0(S
0
i)
�
:= U1
i ; (4.14)�
Th(S
0
i) n Th0(S
0
i) n Bi
�
:= U2
i : (4.15)
Since v̂s
i
= b"
i
when x =2 Bi, thenZ
Th(S
0
i
)nT
h0
(S0
i
)
M(jv̂s
i
� b"i j) dx =
Z
U2
i
+
Z
U1
i
=
Z
U1
i
M(jv̂s
i
� b"i j) dx: (4.16)
From the last equality, the condition of Th. 3.1 and the fact that jv̂s
i
�b"
i
j � jvs
i
�bij
when x 2 Bi, we haveZ
Th(S
0
i
)nT
h0
(S0
i
)
M(jv̂s
i
� b"�j) dx �
Z
U1
i
M(jvsi � bij) dx � O(�(S0i))�
�1(h): (4.17)
432 Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4
Homogenization of Electrostatic Problems in Nonlinear Medium
The estimation for Th(S
0
i
) =2 U� is obtained in the same way as for Th(S
0
i
) 2 U�.
So, Z
Th(S
0
i
)nT
h0
(S0
i
)
M(jv̂si � 1j) �
Z
U1
i
M(jvsi � 1j)dx � O(�(S0i))�
�1(h): (4.18)
Then from (4.13), (4.17) and (4.18) we deduce
lim
h!0
lim sup
s!1
Z
Th(�)nTh0 (�)
M(w
(s)
h
) dx = 0: (4.19)
Now we derive the estimation forZ
Th(S
0
i
)nT
h0
(S0
i
)
M(jrws
hj) dx: (4.20)
It follows from de�nition (4.8) of the function w
(s)
h
that
jrws
hj � jrwj+
X
i
�
jr ~wj
jv̂s
i
� b"
i
j
jb"
i
j
+
j ~wj
jb"
i
j
jrv̂si j+
j ~wjjv̂s
i
� b"
i
j
jb"
i
j
C(NÆ)
�
+jr (x)j
X
i
j ~wjjv̂s
i
� b"
i
j
jb"
i
j
: (4.21)
The terms containing
j ~wj
jb"
i
j
jv̂si � b"i j and
jr ~wj
jb"
i
j
jv̂si � b"i j are estimated similarly to
the previous ones. To get the estimation for the terms containing jrv̂s
i
j we need
the following lemma.
Lemma 4.1. Under the assumptions of Th. 3.1 we haveZ
Th( ~Si)nTh0 (
~Si)
h
F (x; 0;rv
(s)
i
) + �(h)M(v
(s)
i
� b"i )
i
dx = o(1); h! 0: (4.22)
From Lemma 4.1 it immediately follows that
lim
h!0
lim sup
s!1
Z
Th(S
0
i
)nT
h0
(S0
i
)
M(jrv̂si j) dx = 0: (4.23)
Now let us get the estimation forZ
Th(S
0
i
)nT
h0
(S0
i
)
M
��� ~w (v̂si � b"i )(b
"
i )
�1
�� jr (x)j� dx:
(4.24)
Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4 433
M.V. Goncharenko and V.I. Prytula
Since Lemma 4.1 implies that
�(h)
Z
Th(S
0
i
)nT
h0
(S0
i
)
M(jv̂si � b"i j) dx = o(1); h! 0; (4.25)
then, taking into account the properties of the function (x) and that M(u)
satis�es �2-condition, we haveZ
Th(S
0
i
)nT
h0
(S0
i
)
M
�
jr j
j ~wjjv̂s
i
� b"
i
j
jb"
i
j
�
dx � k(
1
h1+
)��1(h)o(1); h! 0; (4.26)
and so
lim
h!0
lim sup
s!1
Z
Th(S
0
i
)nT
h0
(S0
i
)
M
�
jr j
j ~wjjv̂s
i
� b"
i
j
jb"
i
j
�
dx = 0: (4.27)
Correspondingly
lim
h!0
lim sup
s!1
Z
Th(�)nTh0 (�)
M
�
jrw
(s)
h
j
�
dx = 0: (4.28)
It follows from (4.19), (4.11) and (4.28) that
lim
h!0
lim sup
s!1
Z
Th(�)nTh0 (�)
F (x;w
(s)
h
;rw
(s)
h
) dx = 0: (4.29)
Moreover, it can be shown that
lim
Æ!0
lim
h!0
lim sup
s!1
X
i;j
i6=j
Z
T
h0
( ~Si)
T
T
h0
( ~Sj)
jF (x;w
(s)
h
;rw
(s)
h
)j dx = 0: (4.30)
Further a useful estimation for mesBi("; s; h) is to be obtained asZ
Th( ~Si)
T
(s)
M(v
(s)
i
� bi) dx = O(�( ~Si))�
�1(h); (4.31)
then Z
Th( ~Si)
T
(s)
M(v
(s)
i
� bi) dx =
Z
Bi
M(v
(s)
i
� bi) dx+
Z
Th( ~Si)nBi
M(v
(s)
i
� bi) dx
�
Z
Bi
M(v
(s)
i
� bi) dx: (4.32)
434 Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4
Homogenization of Electrostatic Problems in Nonlinear Medium
Since jbi � v
(s)
i
j � " when x 2 Bi then, with the use of (4.32), we get
mesBi � O(�( ~Si))�
�1(h)
1
M(")
: (4.33)
From now on we assume that " = M�1(��1+Æ(h)), where M�1(v) is the
function inverse to M(u), 0 < Æ < 1. It is clear that if h is small enough and
s � ŝ(h), then
mesBi("; s; h) � o(�( ~Si))�
�Æ(h): (4.34)
Denote Bi
1(s; h) = Th0(Si) \ B
i("; s; h) \
(s) and Bi
2(s; h) =
(s) \ Th0(Si) n
Bi
1(s; h). So long as w(x) is smooth in
and w
(s)
h
(x) = w(x) for x 2 Bi
2(s; h), it
follows from (4.34) thatZ
Bi
1(s;h)
F (x;ws
h;rw
s
h)dx =
Z
Bi
2(s;h)
F (x;w;rw)dx
=
Z
T
h0(Si)
F (x;w;rw) dx+O(�(Si)) +
1
M(b)
��1(h)O(�( ~Si)): (4.35)
Let us consider the case Th( ~Si) � U�.
If Th( ~Si) � U�, we can rewrite the integral over the set B
i
1(s; h) in the following
way Z
Bi
1(s;h)
F (x;ws
h;rw
s
h)dx =
Z
Bi
1(s;h)
F (x; 0;rv
(s)
i
)dx
+
Z
Bi
1(s;h)
[F (x;ws
h;rw
s
h)� F (x; 0;rv
(s)
i
)]dx: (4.36)
Since w(x), rw(x), v
(s)
i
(x) are bounded in Bi
1(s; h) and (x) � 1, 'i(x) � 1 for
all x 2 Th0(Si), then with the use of (4.8), (3.6) we getZ
Bi
1(s;h)
h
F (x;ws
h;rw
s
h)� F (x; 0;rv
(s)
i
)
i
dx � C
Z
B�
1 (s;h)
M 0(C1 + C2jrv
(s)
i
j) dx
+�(Æ; h)
Z
Bi
1(s;h)
M(jrv
(s)
i
j) dx; (4.37)
where constants C1, C2, C do not depend on s, h, ".
Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4 435
M.V. Goncharenko and V.I. Prytula
To get the estimation for the �rst term of the right-hand side of (4.37) we
divide set Bi
1(s; h) into two subsets Bi
11(s; h) and B
i
12(s; h)
Bi
11(s; h) =
n
x 2 Bi
1(s; h) : a
(s)
i
(x) � m(h)
o
;
Bi
12(s; h) =
n
x 2 Bi
1(s; h) : a
(s)
i
(x) > m(h)
o
;
(4.38)
where a
(s)
i
(x) = C8 + C9jrv
(s)
i
(x)j and m(h) = sup
�
s : M 0(s) < �Æ=2(h)
. It
follows from the properties of functions �(h) and M 0(s) that m(h) ! 1 as
h! 0. The convexity and monotony of the N -function imply that M 0(a
(s)
i
(x)) �
M 0(m(h)) when x 2 Bi
11(s; h). ThereforeZ
Bi
11(s;h)
M 0(C8 + C9jrv
(s)
i
(x)j) dx � �Æ=2mesBi
11(s; h): (4.39)
From (2.2) for x 2 Bi
12(s; h) we have
m(h)M 0(a
(s)
i
(x)) < a
(s)
i
(x)M 0(a
(s)
i
(x)) � CM(a
(s)
i
(x)): (4.40)
This means that M 0(a
(s)
i
(x)) < CM(a
(s)
i
(x))m�1(h). And using the convexity of
M(u) and (4.34), we getZ
Bi
12(s;h)
M 0(C8 + C9jrv
(s)
i
(x)j) dx � Cmes[Bi
12(s; h)]m
�1(h) +m�1(h)O(�( ~Si)):
(4.41)
It follows from (4.39), (4.41) and (3.9) that for all Th( ~Si) � U� the following
inequality holds
lim sup
s!1
Z
T
h0
(Si)
T
(s)
F (x;w
(s)
h
;rw
(s)
h
) dx �
Z
T
h0
(Si)
F (x;w;rw) dx
+ lim sup
s!1
C(S0i; s; h; bi) + �(Æ; h)O(�( ~Si)) + o(1); h! 0; (4.42)
where bi = w(xi)� ~fB(x
i).
Now we shall study the case when Th0( ~Si) 6� U�.
Using the same technique we haveZ
Bi
2(s;h)
F (x;w
(s)
h
;rw
(s)
h
) dx �
Z
T
h0
(Si)
F (x;w;rw)+O(�( ~Si))�
�Æ(h)+o(1); s!1:
(4.43)
436 Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4
Homogenization of Electrostatic Problems in Nonlinear Medium
The integral over the set Bi
1(s; h) can be written in the following form:Z
Bi
1(s;h)
F (x;w
(s)
h
;rw
(s)
h
) dx =
Z
Bi
1(s;h)
F (x;w;rw)
+
Z
Bi
1(s;h)
h
F (x;w
(s)
h
;rw
(s)
h
)� F (x;w;rw)
i
dx: (4.44)
Since w(x), rw(x), vs�(x) are bounded on the set B�
1 (s; h) and jw(x) �
~fB(x)j < � it follows from (3.6) thatZ
Bi
1(s;h)
h
F (x;w
(s)
h
;rw
(s)
h
)� F (x;w;rw)
i
dx
� A � C1
Z
Bi
1(s;h)
M 0(C2 + C3jrv
(s)
i
j) dx+ k(�)
Z
Bi
1(s;h)
M(jrv
(s)
i
j) dx; (4.45)
where k(�) is de�ned in (2.4). Then
lim sup
s!1
Z
T
h0
T
(s)
F (x;w
(s)
h
;rw
(s)
h
) �
Z
T
h0
F (x;w;rw)
+k(�)O(�( ~Si)) + o(1); h! 0: (4.46)
Now let us summarize (4.42) and (4.46) over corresponding elements Th( ~Si)
covering layer T (�; h) and pass to the limit �rst where h ! 0, then Æ ! 0 and
�nally �! 0. Then from (4.10), (4.29), (4.30) and the condition of Th. 3.1, we get
lim
Æ
lim
h!0
Z
(s) lim sup
s!1
F (x;w
(s)
h
;rw
(s)
h
)
�
Z
F (x;w;rw) dx+
Z
�
c (x;w(x) �B) d�; (4.47)
and from (4.10)
lim
h!0
lim sup
s!1
J (s)
h
w
(s)
h
i
�
Z
F (x;w;rw) dx+
Z
�
c (x;w(x) �B) d� = Jc [w;B] ;
(4.48)
as well as using (4.9), we get
lim
s!1
J (s)
h
u(s)
i
� Jc [w;B] : (4.49)
Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4 437
M.V. Goncharenko and V.I. Prytula
This inequality was obtained under the assumption that w(x) 2 C1(
). Hence
this holds true for any w(x) 2 W 1
M
(
) such that w(x) = f(x) when x 2 @
.
This statement goes out from the density of C1(
) in the space W 1
M
(
) and the
following lemma.
Lemma 4.2. The functional Jc(w) is continuous on the Sobolev�Orlicz space
W 1
M
(
).
Let the sequence fu(s)(x); s = 1; 2; 3; : : : g of the solutions of problem (3.1)�
(3.3) (continued on F (s) by setting u(s) = A(s)) weakly converge in the space
W 1
M
(
) to some function u(x). Let us show that
lim
s=sk!1
J (s)
h
u(s)
i
� Jc [u;A] ; (4.50)
where A is the limit of A(s) as s ! 1. It is clear that jAj < 1 because of the
principle of the maximum [6] and (3.7).
We denote by
Æ
W
1
M (
; F (s)) a class of functions u(x) 2 W 1
M
(
) that are
vanishing on F (s). The following lemma is essential.
Lemma 4.3. Let w(x) be an arbitrary function fromW 1
M
(
) such that kwk1
M;
< 1 and let conditions of Th. 3.1 be satis�ed. Then there exists a sequence
fŵ(x) 2
Æ
W
1
M (
; F (s)), s = 1; 2; : : : g that weakly converges in W 1
M
(
) to w(x)
and
kŵ(x)k1
M;
� �
�
kwk1
M;
�
; (4.51)
for su�ciently large s (s � s(w)), where �(t) is a nonnegative function de�ned
on [0;1) and �(t)! 0 as t! 0.
Let u(x) 2W 1
M
(
) be a weak limit in W 1
M
(
) of the subsequence of solutions
fu(s)(x), s = sk ! 1g of problem (3.1)�(3.3), continued to F (s) by setting
u(s)(x) = A(s), A(s) ! A <1. Since space C1(
) is dense in W 1
M
(
), it follows
that for any " > 0 there exists a function u"(x) 2 C
1(
) such that
ku" � uk1
M;
� ": (4.52)
Furthermore, according to Lem. 4.3, there exists a sequence of functions
fw
(s)
" (x) 2
Æ
W
1
M (
; F (s))g that weakly converges in W 1
M
(
) to u" � u. We set
u
(s)
" = u(s) � w
(s)
" . Then u
(s)
" = A(s) in F (s), and u
(s)
" weakly converges to u" as
s = sk !1. Therefore, by Lem. 4.3 we have
lim
s!1
w(s)
"
1
M;
� �
�
ku" � uk1
M;
�
: (4.53)
438 Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4
Homogenization of Electrostatic Problems in Nonlinear Medium
By this equality and using (4.52) we get
lim
"!0
lim sup
s=sk!1
u(s)" � u(s)
1
M;
= 0; (4.54)
from which one can easily obtain
lim
"!0
lim sup
s=sk!1
jJ (s)[u(s)" ]� J (s)[u(s)]j = 0: (4.55)
Lemma 4.2 and (4.52) imply that
lim
"!0
Jc[u"] = Jc[u]: (4.56)
One can see now that to obtain (4.50) it is su�cient to prove the
following inequality:
lim
s=sk!1
J (s)[u(s)" ] � Jc[u"]: (4.57)
Now let us divide the layer T (�; h) into the elements T (S0
i
; h), that were de�ned
above, and introduce the notation:
T�
�
= fx 2 T (�; h); �(u" �A) > �g ; ~T�
�
=
([
i
T (S0i; h); T (S
0
i; h) 2 T
�
�
)
;
T� = T+
�
[
T�
�
; ~T� = ~T+
�
[
~T�
�
; G� = T (�; h) n T�;
T
(s)
�
= T�
\
(s); ~T
(s)
�
= ~T�
\
(s); G
(s)
�
= G�
\
(s): (4.58)
Since u"(x) and f(x) are smooth in
, we have
lim
Æ!0
lim
h!0
h
T
(s)
�
n ~T
(s)
�
i
= 0; (4.59)
uniformly with respect to s. It is clear thatZ
T (�; h)
T
F (s)
F (x; u(s)" ;ru(s)" ) dx =
Z
~T
(s)
�
F (x; u(s)" ;ru(s)" ) dx
+
Z
T�n ~T
(s)
�
F (x; u(s)" ;ru(s)" ) dx+
Z
G
(s)
�
F (x; u(s)" ;ru(s)" ) dx: (4.60)
Now we get the following estimation
F (x; u(s)" ;ru(s)" ) � F (x; u";ru") +
nX
i=1
Fp(x; u";r")
@u
(s)
"
@xi
�
@u"
@xi
!
Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4 439
M.V. Goncharenko and V.I. Prytula
�
�
1 + 2M 0(jru(s)" j) +M 0(ju(s)" j) +M 0(ju"j)
�
(ju(s)" � u"j); (4.61)
which follows from (3.4) and (3.6). Applying the Holder inequality, we obtainZ
T�n ~T
(s)
�
F (x; u(s)" ;ru(s)" ) dx
�
Z
T�n ~T
(s)
�
F (x; u";ru") dx+
nX
i=1
Z
T�n ~T
(s)
�
Fp(x; u";r")
@u
(s)
"
@xi
�
@u"
@xi
!
dx
�
�1 + 2M 0(jru(s)" j) +M 0(ju(s)" j) +M 0(ju"j)
�
N;
u(s)" � u"
M;
: (4.62)
Since ru
(s)
" weakly converges to ru" in the space LM (
), u
(s)
" converges to u" in
LM (
), u
(s)
" is bounded in W 1
M
(
) and Fp(x; u; p) belongs to the space LN (
),
with the help of (4.61), (4.59)and (2.7) we get
lim
Æ!0
lim
h!0
lim inf
s=sk!1
Z
T�n ~T
(s)
�
F (x; u(s)" ;ru(s)" ) dx � 0: (4.63)
The fact that u
(s)
" = As when x 2 F (s) implies thatZ
G
(s)
�
F (x; u(s)" ;ru(s)" ) dx =
Z
G�
F (x; u(s)" ;ru(s)" ) dx�
Z
G�
T
F (s)
F (x;A(s); 0) dx:
(4.64)
In the same way we obtain that
lim sup
s=sk!1
Z
G
(s)
�
F (x; u(s)" ;ru(s)" ) dx �
Z
G�
F (x; u";ru") dx: (4.65)
Consider the �rst term in the right -hand side of (4.60). Let T (S0
i
; h) be
an arbitrary element from ~
+
�
. We set
b^i = A+ Æ; b_i = min
Th(S
0
i
)
u"(x)� Æ; bi = b_i � b^i ; Æ > 0; (4.66)
where Æ is a su�ciently small parameter that will be de�ned later.
440 Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4
Homogenization of Electrostatic Problems in Nonlinear Medium
Note that b_
i
� b^
i
= minu"(x)�A� 2Æ > �
2 for Æ < �
4 . Let us decompose the
set T (S0
i
; h)
T
(s) into three nonintersecting subsets:
(s)
1i =
n
x 2 T (S0
i
; h)
T
(s) : u
(s)
" < b^
i
o
;
(s)
2i =
n
x 2 T (S0
i
; h)
T
(s) : b^
i
� u
(s)
" � b_
i
o
;
(s)
3i =
n
x 2 T (S0
i
; h)
T
(s) : u
(s)
" > b_
i
o
:
(4.67)
From the convergence of u
(s)
" to u" in space LM (
) one can show that for su�-
ciently large s = s(h) the following equality holdsZ
T (S0
i
; h)
T
(s)
M(u(s)" � u") dx = O(�(S0i))�
�2�
(h); 0 <
< 1: (4.68)
Therefore, Z
(s)
1i
S
(s)
2i
M(ju(s)" � u"j) dxu"j) dx
�
Z
(s)
1i
S
(s)
2i
M(jmin
ThS
0
i
u(s)" � u"j) dx �M(Æ)mes
h
(s)
1i
[
(s)
2i
i
; (4.69)
this yields
M(Æ)mes
h
(s)
1i
[
(s)
2i
i
� O(�(S0i))�
�2�
(h): (4.70)
Let us choose Æ =M�1(��1(h)), where M�1 is a inverse function to M(u). Then
for s > s(h) we have
mes
h
(s)
1i
[
(s)
2i
i
= O(�(S0i))�
�1�
(h): (4.71)
It is obvious thatZ
T (S0
i
; h)
T
(s)
F (x; u(s)" ;ru(s)" ) =
Z
(s)
2i
F (x; u(s)" ;ru(s)" ) dx
+
Z
(s)
1i
S
(s)
3i
F (x; u(s)" ;ru(s)" ) dx: (4.72)
Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4 441
M.V. Goncharenko and V.I. Prytula
To estimate the integral over the set
(s)
2i we introduce the following function
~vs� =
8>>>>><
>>>>>:
0; x 2
(s)
1i
S�
F (s)
T
T (S0
i
; h)
�
;
u
(s)
" � b^
i
; x 2
(s)
2i ;
bi = b_
i
� b^
i
; x 2
(s)
3i :
(4.73)
Keeping in mind the de�nition of ~vs�(x), we �nd that ru
(s)
" (x) = r~vs� when
x 2
(s)
2i . So, Z
(s)
2i
F (x; u(s)" ;ru(s)" ) dx
=
Z
(s)
2i
h
F (x; u(s)" ;ru(s)" )� F (x; 0;ru(s)" )
i
dx+
Z
(s)
2i
F (x; 0;rvs�) dx: (4.74)
Using (3.6) and (4.71), and according to boundness of u
(s)
" in W 1
M
(
) we obtainZ
(s)
2i
h
F (x; u(s)" ;ru(s)" )� F (x; 0;ru(s)" )
i
dx
� A
Z
(s)
2i
h
1 +M 0(u(s)" ) + 2M 0(jru(s)" j)
i
ju(s)" j dx
� Cmes
h
(s)
1i
\
(s)
1i
i
+ C
Z
(s)
2i
M 0(jru(s)" j) dx: (4.75)
Analogously to (4.39) and (4.41) we �nd that for su�ciently large s, s > s(h),
the following equality holdsZ
(s)
2i
h
F (x; u(s)" ;ru(s)" )� F (x; 0;ru(s)" )
i
dx = O(�(S0i))o(1); h! 0: (4.76)
Then from (4.74) and the de�nition of ~vs� we deduce thatZ
(s)
2i
F (x; u(s)" ;ru(s)" ) dx =
Z
Th(S
0
i
)
F (x; 0;rvs�) dx+ �(h)
Z
Th(S
0
i
)
M(vs� � bi) dx
442 Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4
Homogenization of Electrostatic Problems in Nonlinear Medium
+O(�(S0i))o(1); h! 0; (4.77)
therefore, from the de�nition of C(S0
i
; s; h; b)Z
(s)
2i
F (x; u(s)" ;ru(s)" ) dx � C(S0i; s; h; bi) +O(�(S0i))o(1); h! 0: (4.78)
It is clear that Z
(s)
3i
F (x; u(s)" ;ru(s)" ) dx�
Z
Th(S
0
i
)
F (x; u";ru") dx
=
Z
Th(S
0
i
)
F (x; u";ru") dx�
Z
Th(S
0
i
)
F (x; u";ru") dx�
Z
Th(S
0
i
)n
(s)
3i
F (x; ai; 0) dx;
(4.79)
where
u" =
8><
>:
u
(s)
" ; x 2
(s)
3i
ai; x 2 Th(S
0
i
) n
(s)
3i :
(4.80)
Then from (3.4) and (4.61) we obtainZ
Th(S
0
i
)
F (x; u";ru") dx�
Z
Th(S
0
i
)
F (x; u";ru") dx�
Z
Th(S
0
i
)n
(s)
3i
F (x; ai; 0) dx
�
nX
j=1
Z
Th(S
0
i
)
Fu"xj
(x; u";ru")(u" � u")xj dx
+const � ku" � u"kM;Th(S
0
i
) � Cmes
�
Th(S
0
i) n
(s)
3i
�
; (4.81)
thus weak convergence of u
(s)
" to u" in W
1
M
(
) and de�nition of u" give us
lim
s=sk!1
Z
(s)
3i
F (x; u(s)" ;ru(s)" ) dx �
Z
Th(S
0
i
)
F (x; u";ru") dx+O(�(S0i))�
�1�
(h):
(4.82)
Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4 443
M.V. Goncharenko and V.I. Prytula
Now we obtain the estimation for
Z
(s)
1i
F (x; u(s)" ;ru(s)" ) dx. It follows from (3.5)
that F (x; u; p) � F (x; u; 0) � A1M(jpj) � 0, henceZ
(s)
1i
F (x; u(s)" ;ru(s)" ) dx �
Z
(s)
1i
h
F (x; u(s)" ; 0)� F (x; u"; 0)
i
dx+
Z
(s)
1i
F (x; u"; 0) dx:
(4.83)
With the help of (4.61) the last inequality leads toZ
(s)
1i
F (x; u(s)" ;ru(s)" ) dx
�
Z
(s)
1i
�
1 + 2M 0(jru(s)" j) +M 0(ju(s)" j) +M 0(ju"j)
�
(ju(s)" � u"j) dx
+
Z
(s)
1i
F (x; u"; 0) dx: (4.84)
Taking into account the weak convergence of u
(s)
" to u" in space W 1
M
(
) and
smothness of u", we deduce from (4.71) and (4.84) that
lim
s=sk!1
Z
(s)
1i
F (x; u(s)" ;ru(s)" ) dx � O(�(S0i))�
�1�
(h): (4.85)
Combining of (4.76), (4.82), (4.85), (4.78) in the case when Th(S
0
i
) �
+
�
yields
lim
s=sk!1
Z
Th(S
0
i
)
T
(s)
F (x; u(s)" ;ru(s)" )
�
Z
Th(S
0
i
)
F (x; u";ru") dx+ lim
s=sk!1
C(S0i; s; h; b) + o(1); h! 0: (4.86)
Besides, it follows from (4.61), (2.7) and from weak convergence of u
(s)
" to u" in
W 1
M
(
) that
lim
s=sk!1
Z
nT (�; h)
F (x; u(s)" ;ru(s)" ) dx �
Z
nT (�; h)
F (x; u";ru") dx: (4.87)
444 Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4
Homogenization of Electrostatic Problems in Nonlinear Medium
Similarly to (4.86) inequalities hold for the elements Th(S
0
i
) � T�
�
. We sum-
marize them all over the elements Th(S
0
i
) � ~T+
�
S
~T�
�
and pass to the limit �rst
as h! 0 and then as Æ ! 0.
Since Z
(s)
F (x; u(s)" ;ru(s)" ) dx
=
Z
n T (�; h)
F (x; u(s)" ;ru(s)" ) dx+
Z
T (�; h)
T
F (s)
F (x; u(s)" ;ru(s)" ) dx; (4.88)
it follows from (4.63), (4.60), (4.65), (4.87), (4.86), (4.88) and the condition of
Th. 3.1 that
lim
s=sk!1
Z
(s)
F (x; u(s)" ;ru(s)" ) dx �
Z
F (x; u";ru") dx+
Z
��
c(x; u" �A) d�;
(4.89)
where �� = fx 2 � : �(u"(x)�A) > �g.
It is clear that
S
�>0
T� = fx 2 T (�; h) : ju"(x)�Aj > 0g. Now we pass to the
limit in (4.89) as �! 0 and obtain (4.57) and, therefore, (4.50).
It follows from (4.50) and (4.49) that
Jc[u] � Jc[w]
holds for any w(x) 2 W 1
M
(
). This means that any weak limit in W 1
M
(
) of the
solutions of problem (3.1)�(3.3) (continued to set F (s) by setting u(s) = A(s)) is
a solution of problem (4.94)�(4.95). Theorem 3.1 is proved.
Remark 1. The theorem proved above corresponds to the generalization of
the case of distribution of the electrostatic �eld in weakly nonlinear medium with
a nonzero potential and a zero charge on the net. With the same methods being
used one can show that Th. 3.1 can be modi�ed in the following way.
Theorem 4.1. Let the following condition hold: for any arbitrary piece S of
surface �; 8 b 2 Rn and
> 0 there exist the following limits:
lim
h!0
lim sup
s!1
C(S; s; h; b) = lim
h!0
lim inf
s!1
C(S; s; h; b) =
Z
S
c(x; b)d�; (4.90)
where c(x; b) is a continuous on � nonnegative function such as if b is large enough,
then c(x; b) = O(b2).
Then for any sequence fu(s)(x)g of the problem solutions
Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4 445
M.V. Goncharenko and V.I. Prytula
J (s)[u(s)] =
Z
(s)
F (x;ru(s))dx� 2Asq ! inf; (4.91)
u
(s)
j
@F
(s)
= As; x 2 @F (s); (4.92)
u
(s)
j@
= f(x); x 2 @
(4.93)
(continued on F (s) by setting u(s) = A(s) ) there exists a subsequence fusj (x)g
that weakly converges in space W 1
M
(
) to the function u(x) such that the pair
fu(x); Ag is a solution of the following problem:Z
F (x;ru)dx +
Z
�
c(x; u�A)d�� 2Aq ! inf (4.94)
uj@
= f(x); x 2 @
; (4.95)
where A = lim
s!1
As.
5. Asymptotic Behavior of the Electrostatic Potential
in a Weakly Nonlinear Medium with Thin Perfectly
Conducting Grids
Let
� R
3 be a dielectric. We suppose that a certain part of this dielectric
is penetrated by thin perfectly conducting wires forming a periodic grid F (s) that
concentrates in the neighborhood of plane � b
. We also suppose that the
dielectric permeability "(E) depends on the electric �eld strength E as follows
"(E) = "0 + � ln�(1 + jEj2) ("0 > 0; � > 0; 0 � � � 1): (5.1)
Thus the dielectric is a weakly nonlinear medium.
Assume that F (s) depends on a parameter s and has the following structure:
Q(s)(�)
T
F
(s)Q , where Q(s)(�) is the neighborhood of � such as 8x 2 Q(s)(�) :
x ! �, s ! 1, and F
(s)Q is a periodic set in R3 . We suppose that F
(s)Q consists
of the circular cylinders with radius r(s) = C
s
. The axes of the cylinders form
a periodic net in R2 of the period Æ(s) and
Æ(s) �
8><
>:
Æ
�
ln 1
r(s)
���1
; 0 � � < 1;
Æ
�
ln ln 1
r(s)
��1
; � = 1;
(5.2)
as s!1, where Æ > 0.
446 Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4
Homogenization of Electrostatic Problems in Nonlinear Medium
Let f(x) be a potential de�ned on @
and let total charge on the grid be
equal to q. Then potential u(s)(x) in domain
(s) =
n F (s) is described by the
following boundary value problem
3X
i=1
@
@xi
"(ru(s))
@u(s)
@xi
!
= 0; x 2
(s); (5.3)
u(s) = A(s); x 2 @F (s); (5.4)
u(s) = f(x); x 2 @
; (5.5)Z
@F (s)
"(ru(s))
@u(s)
@�
d� = q; (5.6)
where " : R3 ! R
1 is de�ned by (5.1).
It is clear that equations (5.4) and (5.1) are the Euler equation for functional
(3.1) with the following integrant
F (x; u; �) = "0j�j
2 + �
j�j2Z
0
ln�(1 + t) dt = F (�): (5.7)
It is easy to prove that F (�) satis�es (3.4)�(3.6) with the function
M(u) = u2 ln�(1 + u2): (5.8)
We continue u(s)(x) to F (s) by setting u(s)(x) = A(s), still denoting them as u(s).
From Th. 4.1 it follows that u(s)(x) strongly converges, as s ! 1, in L2(
)
to the solution u(x) of the following problem
3X
i=1
@
@xi
�
"(ru)
@u
@xi
�
= 0; x 2
n �; (5.9)
�
"(ru)
@u
@�
��
= u� C�
1
j�j
8<
:
Z
�
u d� + q
9=
; ; x 2 �;
[u]� = 0; x 2 �;
(5.10)
u = f(x); x 2 @
; (5.11)
Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4 447
M.V. Goncharenko and V.I. Prytula
where
C� =
8>>>>><
>>>>>:
4(�+ "0)�Æ; � = 0;
2�+2��Æ
(1��) ; 0 < � < 1;
8��Æ; � = 1:
(5.12)
References
[1] M. Goncharenko and E. Khruslov, Homogenization of Electrostatic Problems in
Domains with Nets. � Math. Sci. Appl., Gakuto 9 (1995), 215�223.
[2] E. Khruslov and L. Pankratov, Homogenization of the Dirichlet Variational Prob-
lems in Orlich�Sobolev Spaces. � Fields Inst. Comm. 25 (2000), 345�366.
[3] V.A. Marchenko and E.Ya. Khruslov, Homogenized Models of the Micro Inhomo-
geneous Media. Naukova Dumka, Kyiv, 2005. (Russian)
[4] M.A. Krasnoselki and Ya.B Ruticki, Convex Functions and Sobolev�Orlicz Spaces.
Fizmatgiz, Moscow, 1958. (Russian)
[5] Th.K. Donaldson and N.S. Trudinger, Sobolev�Orlicz Spaces and Imbedding Theo-
rems. � J. Funct. Anal. 8, (1971), 52�75.
[6] O.A. Ladyzhenskaya and N.N. Ural'tseva, Linear and Quasilinear Elliptic Equations.
Acad. Press, London, 1968.
448 Journal of Mathematical Physics, Analysis, Geometry, 2006, v. 2, No. 4
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