Integral Conditions for Convergence of Solutions of Non-Linear Robin's Problem in Strongly Perforated Domain
We consider a boundary-value problem for the Poisson equation in a strongly perforated domain Ωε = Ω\Fε ⊂ Rⁿ (n ≥ 2) with non-linear Robin's condition on the boundary of the perforating set Fε. The domain Ωε depends on the small parameter ε > 0 such that the set Fε becomes more and more loos...
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irk-123456789-1405762018-07-11T01:23:36Z Integral Conditions for Convergence of Solutions of Non-Linear Robin's Problem in Strongly Perforated Domain Khruslov, E.Ya. Khilkova, L.O. Goncharenko, M.V. We consider a boundary-value problem for the Poisson equation in a strongly perforated domain Ωε = Ω\Fε ⊂ Rⁿ (n ≥ 2) with non-linear Robin's condition on the boundary of the perforating set Fε. The domain Ωε depends on the small parameter ε > 0 such that the set Fε becomes more and more loosened and distributes more densely in the domain Ω as ε→0. We study the asymptotic behavior of the solution uε(x) of the problem as ε→0. A homogenized equation for the main term u(x) of the asymptotics of uε(x) is constructed and the integral conditions for the convergence of uε(x) to u(x) are formulated. 2017 Article Integral Conditions for Convergence of Solutions of Non-Linear Robin's Problem in Strongly Perforated Domain / E.Ya. Khruslov, L.O. Khilkova, M.V. Goncharenko // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 3. — С. 283-313. — Бібліогр.: 17 назв. — англ. 1812-9471 Mathematics Subject Classification 2000: 35Q70 http://dspace.nbuv.gov.ua/handle/123456789/140576 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We consider a boundary-value problem for the Poisson equation in a strongly perforated domain Ωε = Ω\Fε ⊂ Rⁿ (n ≥ 2) with non-linear Robin's condition on the boundary of the perforating set Fε. The domain Ωε depends on the small parameter ε > 0 such that the set Fε becomes more and more loosened and distributes more densely in the domain Ω as ε→0. We study the asymptotic behavior of the solution uε(x) of the problem as ε→0. A homogenized equation for the main term u(x) of the asymptotics of uε(x) is constructed and the integral conditions for the convergence of uε(x) to u(x) are formulated. |
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Khruslov, E.Ya. Khilkova, L.O. Goncharenko, M.V. |
| spellingShingle |
Khruslov, E.Ya. Khilkova, L.O. Goncharenko, M.V. Integral Conditions for Convergence of Solutions of Non-Linear Robin's Problem in Strongly Perforated Domain Журнал математической физики, анализа, геометрии |
| author_facet |
Khruslov, E.Ya. Khilkova, L.O. Goncharenko, M.V. |
| author_sort |
Khruslov, E.Ya. |
| title |
Integral Conditions for Convergence of Solutions of Non-Linear Robin's Problem in Strongly Perforated Domain |
| title_short |
Integral Conditions for Convergence of Solutions of Non-Linear Robin's Problem in Strongly Perforated Domain |
| title_full |
Integral Conditions for Convergence of Solutions of Non-Linear Robin's Problem in Strongly Perforated Domain |
| title_fullStr |
Integral Conditions for Convergence of Solutions of Non-Linear Robin's Problem in Strongly Perforated Domain |
| title_full_unstemmed |
Integral Conditions for Convergence of Solutions of Non-Linear Robin's Problem in Strongly Perforated Domain |
| title_sort |
integral conditions for convergence of solutions of non-linear robin's problem in strongly perforated domain |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2017 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/140576 |
| citation_txt |
Integral Conditions for Convergence of Solutions of Non-Linear Robin's Problem in Strongly Perforated Domain / E.Ya. Khruslov, L.O. Khilkova, M.V. Goncharenko // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 3. — С. 283-313. — Бібліогр.: 17 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT khrusloveya integralconditionsforconvergenceofsolutionsofnonlinearrobinsprobleminstronglyperforateddomain AT khilkovalo integralconditionsforconvergenceofsolutionsofnonlinearrobinsprobleminstronglyperforateddomain AT goncharenkomv integralconditionsforconvergenceofsolutionsofnonlinearrobinsprobleminstronglyperforateddomain |
| first_indexed |
2025-07-10T10:46:58Z |
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1837256591889399808 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2017, vol. 13, No. 3, pp. 283–313
doi:10.15407/mag13.03.283
Integral Conditions for Convergence of
Solutions of Non-Linear Robin’s Problem in
Strongly Perforated Domain
E.Ya. Khruslov1, L.O. Khilkova2, and M.V. Goncharenko3
1,3B. Verkin Institute for Low Temperature Physics and Engineering
of the National Academy of Sciences of Ukraine
47 Nauky Ave., Kharkiv 61103, Ukraine
E-mail: 1khruslov@ilt.kharkov.ua
3marusya61@yahoo.co.uk
2Institute of Chemical Technologies
of Volodymyr Dahl East Ukrainian National University
31 Volodymyrska Str., Rubizhne 93009, Ukraine
E-mail: 2LarisaHilkova@gmail.com
Received May 27, 2017
We consider a boundary-value problem for the Poisson equation in a
strongly perforated domain Ωε = Ω \ F ε ⊂ Rn (n > 2) with non-linear
Robin’s condition on the boundary of the perforating set F ε. The domain
Ωε depends on the small parameter ε > 0 such that the set F ε becomes more
and more loosened and distributes more densely in the domain Ω as ε→ 0.
We study the asymptotic behavior of the solution uε(x) of the problem as
ε→ 0. A homogenized equation for the main term u(x) of the asymptotics
of uε(x) is constructed and the integral conditions for the convergence of
uε(x) to u(x) are formulated.
Key words: homogenization, stationary diffusion, non-linear Robin’s
boundary condition, homogenized equation.
Mathematical Subject Classification 2010: 35Q70.
1. Introduction
Let Ω be a fixed bounded domain in Rn (n > 2) and F ε be a closed set in Ω,
depending on a small parameter ε so that, as ε → 0, the set F ε becomes more
and more loosened and distributes more densely in Ω. We call F ε the perforating
set. Denote by Ωε = Ω \ F ε a strongly perforated domain. The boundary of the
c© E.Ya. Khruslov, L.O. Khilkova, and M.V. Goncharenko, 2017
E.Ya. Khruslov, L.O. Khilkova, and M.V. Goncharenko
domain Ωε consists of two disjoint parts: ∂Ωε = ∂Ω∪∂F ε, where ∂Ω is the outer
boundary, ∂F ε is the boundary of the perforating set F ε. We assume that these
boundaries are smooth.
In the domain Ωε we consider a boundary-value problem
−∆uε = f ε(x), x ∈ Ωε,
∂uε
∂ν
+ σε(x, uε) = 0, x ∈ ∂F ε,
uε = 0 on ∂Ω,
(1)
where ∆ is the Laplacian, ν is the unit outer normal to the boundary ∂F ε with
respect to the domain Ωε, the functions f ε(x) : Ωε → R1 and σε(x, u) : Ωε ×
R1 → R1 are given.
Problem (1) describes the process of stationary diffusion in a porous medium
Ωε. The function uε(x) is a concentration of a diffusing substance.
The geometry of the domain Ωε is very complicated. It is practically impos-
sible to solve problem (1) either by analytical or numerical methods. In the case
under consideration, the scale ε of the microstructure of the medium is much
smaller than the scale of the physical process, therefore we can proceed to study
the asymptotic behavior of the solution uε(x) as ε → 0. As a result, we ob-
tain the homogenized equation. The corresponding homogenized problem is a
macroscopic model of the process.
The asymptotic analysis of problems similar to problem (1) was carried out
by many authors (see, for example, [1–7,14,16,17]). Note that in all these papers
the domains Ωε = Ω \ F ε are of special type. Namely, the perforating set F ε is
a union of periodically located (with a period ε) identical bodies of a diameter
O(ε).
However, the domains Ωε of an arbitrary form (including the connected per-
forating sets F ε ) are more natural from the physical point view. Earlier, in [9]
we considered problem (1) in the domains Ωε of an arbitrary form satisfying only
one important condition: the condition of strong connectedness (this condition
was introduced in [12]). It was shown that the solution uε(x) of problem (1)
converges as ε→ 0 in the L2(Ωε) metric to the solution u(x) of the homogenized
problem in the domain Ω:
−
n∑
i,k=1
∂
∂xi
(
aik(x)
∂u
∂xk
)
+
1
2
cu(x, u) = f(x), x ∈ Ω, u(x) = 0, x ∈ ∂Ω. (2)
Here {aij}ni,j=1 is the positive-definite tensor characterizing the effective conduc-
tivity of the porous medium, cu(x, u) = ∂
∂uc(x, u), and the function c(x, u) char-
acterizes the effective absorption properties of the medium. They are determined
284 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3
Integral Conditions for Convergence of Solutions
by the so-called mesoscopic characteristics of the domains Ωε ( [12]). These char-
acteristics are introduced in cubes (“mesocubes”), which are small relative to the
whole domain but at the same time are large relative to the microscale. The
structure of the strongly connected domain Ωε and the absorption on the bound-
ary in the cube we characterize with the help of the mesoscopic characteristic
a(x, ε, h). It can be the mean conductivity tensor or the absorption function.
In [9], it is assumed that uniformly with respect to x ∈ Ω there exists a density
of mesoscopic characteristic:
lim
h→0
lim
ε→0
a(x, ε, h)
hn
= lim
h→0
lim
ε→0
a(x, ε, h)
hn
= a(x). (3)
The authors proved that under these conditions the solution of problem (1) con-
verges in L2(Ωε) to the solution u(x) of homogenized problem (2).
In the present paper, we obtain a similar result, but under weaker integral
conditions
lim
h→0
lim
ε→0
∫
Ω
∣∣∣∣a(x, ε, h)
hn
− a(x)
∣∣∣∣ dx = 0
imposed on the mesoscopic characteristics a(x, ε, h). We prove that this condition
is sufficient, but it is very probable that it is also necessary for the convergence
of uε(x) to u(x).
The paper is organized as follows. In Section 2, we formulate the problem
and explain the main result. Section 3 contains the proof of the existence and
uniqueness theorem for the weak solution of problem (1) for each fixed ε. In
Section 4 we state auxiliary lemmas. In Section 5 we prove the main theorem.
This section is divided into several subsections correspondingly to the major steps
in our proof.
2. Statement of the Problem and Main Result
Let Ω be a bounded domain in Rn (n > 2), and F ε be a closed set in Ω with
smooth boundary. We assume that the domains Ωε = Ω\F ε satisfy the following
conditions:
1) for any ε > 0 for any subdomain Ω′ ⊂ Ω,
mes{Ωε ∩ Ω′} > C1 ·mes{Ω′} > 0; (4)
2) the domains Ωε satisfy the extension condition, i.e., for any function ϑε(x) ∈
H1(Ωε), there exists a function ϑ̃ε(x) ∈ H1(Ω) such that ϑε(x) = ϑ̃ε(x) for
x ∈ Ωε, and the inequality∥∥∥∇ϑ̃ε∥∥∥
L2(Ω)
6 C2 ‖∇ϑε‖L2(Ωε) (5)
is true. Here the constants C1, C2 are independent of ε.
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3 285
E.Ya. Khruslov, L.O. Khilkova, and M.V. Goncharenko
Remark 1. From conditions (4), (5) it follows that the inequality∥∥∥ϑ̃ε∥∥∥
H1(Ω)
6 C ‖ϑε‖H1(Ωε) (6)
is correct, where the constant C is independent of ε. And, therefore, the domains
Ωε satisfy the condition of strong connectivity (see [12, §4.2]).
In the domain Ωε = Ω \ F ε, we consider boundary-value problem (1) where
the functions f ε(x) ∈ L2(Ωε) and σε(x, u) ∈ C(Ω × R1) are given. Assume that
for any ε the function σε(x, u) satisfies the following conditions:
a1: ∀u1, u2 ∈ R1 : (σε(x, u1)− σε(x, u2)) · (u1 − u2) > 0;
a2: σε(x, 0) = 0;
a3: ∀u ∈ R1 : |σε(x, u)| 6 σ̂ε(x) ·
(
1 + |u|Θ
)
,
(
Θ < n
n−2
)
. Here the function
σ̂ε(x) ∈ C(Ω), σ̂ε(x) > 0, and for any ball B(ρ, z) of radius ρ (0 < ρ < 1)
centered at the point z ∈ Ω,∫
∂F ε∩B(ρ,z)
σ̂ε(x) dΓ < C1ρ
n + C2(ε)ρn−1,
where the constants C1, C2 are independent of z, ρ, C1 is independent of ε,
and C2(ε)→ 0 as ε→ 0.
We denote
H1(Ωε, ∂Ω) = {u(x) ∈ H1(Ωε) : u|∂Ω = 0}.
Definition 1. A generalized solution of problem (1) is a function uε(x) ∈
H1(Ωε, ∂Ω) satisfying the identity∫
Ωε
(∇uε,∇ϕ) dx+
∫
∂F ε
σε(x, uε)ϕdΓ =
∫
Ωε
f εϕdx, ∀ϕ(x) ∈ H1(Ωε, ∂Ω). (7)
Theorem 1. Problem (1) has the unique generalized solution uε(x) ∈
H1(Ωε, ∂Ω) for each fixed ε.
The proof of this theorem is given in Section 3
The main goal of this paper is to study the asymptotic behavior of uε(x) as
ε→ 0. To this end, we first define in what sense we mean the convergence of the
solutions uε(x) and introduce the ”mesoscopic” characteristics of the domain Ωε.
Definition 2. We say that a sequence of functions {uε(x)}ε ∈ Lp(Ωε) con-
verges in Lp(Ωε,Ω) if there exists a function u(x) ∈ Lp(Ω) such that
lim
ε→0
‖uε − χεu‖Lp(Ωε) = 0,
where χε(x) is a characteristic function of the domain Ωε.
286 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3
Integral Conditions for Convergence of Solutions
Let us introduce the “mesoscopic” characteristics of the domain Ωε. These
characteristics are the local characteristics of the microstructure considered in the
cube Kz
h = K(z, h) centered at the point z with the edges of length h oriented
along the coordinate axes.
We define the quantitative characteristic of conductivity by the functional
with respect to an arbitrary vector ` ∈ Rn,
T εh,z(`) = inf
vε
∫
Kz
h∩Ωε
(
|∇vε|2 + h−2−τ |vε − (x− z, `)|2
)
dx, (8)
where the lower bound is taken in a class of functions vε(x) ∈ H1(Kz
h ∩Ωε), τ ∈
(0, 2) is the parameter of penalty. For any ` ∈ Rn, there exists a unique minimizer
of the functional (8) that can be represented in the form (see [12, p. 4, §1])
vε =
n∑
i=1
vεi `
i. (9)
The function vεi minimizes (8) for ` = ei (ei is a basic vector of the axis xi). This
implies that functional (8) is homogeneously quadratic with respect to `, i.e.,
T εh,z(`) =
n∑
i,j=1
aij(z, ε, h)`i`j , (10)
where the coefficients aij(z, ε, h) are defined by
aij(z, ε, h)
=
∫
Kz
h∩Ωε
{(
∇vεi ,∇vεj
)
+ h−2−τ [vεi − (xi − zi)][vεj − (xj − zj)]
}
dx. (11)
The system of numbers {aij(z, ε, h)}ni,j=1 forms a symmetric positive definite ten-
sor in Rn. The tensor characterizes the conductivity of the domains Ωε in the
cube Kz
h.
We define the quantitative characteristic of absorption on the boundary ∂F ε
by the functional for arbitrary s ∈ R1,
c(z, s; ε, h)
= inf
wε
[∫
Kz
h∩Ωε
{
|∇wε|2 + h−2−τ |wε − s|2
}
dx+
∫
Kz
h∩∂F ε
gε(x,wε) dΓ
]
, (12)
where the infimum is taken in a class of functions wε ∈ H1(Kz
h ∩ Ωε), τ ∈ (0, 2)
is the parameter of penalty, and the function gε(x, u) is defined by
gε(x, u) := 2
∫ u
0
σε(x, s) ds. (13)
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3 287
E.Ya. Khruslov, L.O. Khilkova, and M.V. Goncharenko
It can be shown that for any fixed ε, h, s (0 < ε � h � 1, C1 < s < C2), the
functions
aij(X,ε,h)
hn and c(x,s,ε,h)
hn are measurable and bounded functions of x.
The main result of this paper is the following theorem.
Theorem 2. Let the domains Ωε be strongly connected, the function σε(x, u)
satisfy the conditions a1 − a3 and there exist τ ∈ (0, 2) for which the functions
aij(x, ε, h) (i, j = 1, n), c(x, s; ε, h)
1) lim
h→0
lim
ε→0
∫
Ω
∣∣∣∣aij(x, ε, h)
hn
− aij(x)
∣∣∣∣ dx = 0,
where aij(x) are piecewise continuous functions of x, and {aij(x)}ni,j=1 is a
symmetric positive definite tensor in Rn;
2) lim
h→0
lim
ε→0
∫
Ω
∣∣∣∣c(x, s, ε, h)
hn
− c(x, s)
∣∣∣∣ dx = 0, ∀s ∈ R1,
where the function c(x, s) is bounded in x and differentiable with respect to s,
and its derivative cs(x, s) = ∂
∂sc(x, s) satisfies the conditions:
∀ s1, s2 ∈ R1 : (cs(x, s1)− cs(x, s2))(s1 − s2) > 0,
∀ s ∈ R1 : cs(x, s) 6 C(1 + |s|Θ),
(
Θ <
n+ 2
n− 2
)
;
3) the functions f ε(x) extended by zero to F ε converge weakly in L2(Ω) to the
function f(x).
Then the generalized solution uε(x) of problem (1) converges in Lp(Ωε,Ω)(
p < 2n
n−2
)
to the function u(x), which is a generalized solution of homogenization
problem (2).
We recall that a generalized solution of homogenized problem (2) is a function
u(x) from the space H̊(Ω) satisfying the identity∫
Ω
n∑
i,j=1
aij
∂u
∂xi
∂ϕ
∂xj
dx+
1
2
∫
Ω
cu(x, u)ϕdΓ =
∫
Ω
fϕ dx, ∀ϕ(x) ∈ H̊1(Ω). (14)
Remark 2. Since the mesoscopic characteristics aik(x, ε, h) and c(x, s; ε, h)
depend on the parameter of penalty τ , the limit functions aik(x) and c(x, s) must
also depend on τ formally. However, if conditions 1) and 2) of Theorem 2 are
satisfied for some τ ∈ (0, 2), then the solution uε(x) of problem (1) converges
to the solution u(x) of limit problem (2) for any right-hand side f(x). This
solution does not have to depend on τ as a limit of the original solutions uε(x) is
independent of τ . Taking this fact into account, it can be shown that the limit
coefficients aik(x) and c(x, s) are also independent of τ .
288 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3
Integral Conditions for Convergence of Solutions
Let us give the simplest example, where conditions (3) are not satisfied uni-
formly, but the integral conditions are satisfied.
Let Ω be a bounded domain in R3, and G be a subdomain compactly em-
bedded in the domain Ω. We assume that the set F ε =
⋃
i F
ε
i is periodically
distributed in subdomain G. The set F ε consists of the sets F εi = εF +xiε (xiε ∈
εZn) that are a contraction of some fixed domain F ∈ K1 with smooth boundary
∂F (K1 is a unit cube). In the domain Ωε = Ω \F ε, we consider boundary-value
problem (1) for σε(x, u) = ε · σ(x, u) where σ(x, u) ∈ C(Ω×R1).
Conditions (3) for this problem do not hold uniformly at the points of Ω, but
the integral conditions are satisfied. And the limit characteristics of the domains
Ωε are calculated by the formulas:
c(x, u) = 2|∂F |χ(x)
∫ u
0
σ(x, r) dr,
aij(x) = δij − χ(x)
[
δij |F |+
∫
K1\F
n∑
k=1
∂Vi(ξ)
∂ξk
∂Vj(ξ)
∂ξk
dξ
]
, i, j = 1, . . . , n.
Here |∂F |, |F | are an area of the surface ∂F and a volume of the domain F , respec-
tively, χ(x) is a characteristic function of the domain G, the function Vj(ξ) (j =
1, n) is a solution of the “cell” problem in the unit cube K1:
n∑
i=1
∂2Vj(ξ)
∂ξ2
i
= 0, ξ ∈ K1 \ F,
∂Vj(ξ)
∂νξ
= cos(ν(ξ), ej), ξ ∈ ∂F,
Vj |Γ+
i
= Vj |Γ−i ,
∂Vj
∂ξi
|Γ+
i
=
∂Vj
∂ξi
|Γ−i , i = 1, n,∫
K1\F
Vj(ξ) dξ = 0,
here Γ±i are opposite sides of the cube K1, ν = ν(ξ) is a unit normal with
respect to the boundary of the domain F .
For more general case, we refer the reader to [10].
3. Proof of Theorem 1
Let us prove the existence and the uniqueness of the generalized solution of
problem (1) for each fixed ε. In the proof we will follow the method developed
in [8].
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3 289
E.Ya. Khruslov, L.O. Khilkova, and M.V. Goncharenko
The solution of problem (1) minimizes the functional
Φε[uε] =
∫
Ωε
|∇uε|2 dx+
∫
∂F ε
gε(x, uε) dΓ− 2
∫
Ωε
f εuε dx (15)
on the class of functions uε(x) ∈ H1(Ωε, ∂Ω). Here gε(x, uε) is defined by (13).
According to [8], the existence of a minimizer of the functional (15) follows
from the coercivity and the weak lower semicontinuity of the functional.
Let us show that the functional Φε[·] is coercive. By the extension condition
(5) and the Friedrichs inequality, we get
‖uε‖L2(Ωε) 6 ‖ũ
ε‖L2(Ω) 6 C1 ‖∇ũε‖L2(Ω) 6 C2 ‖∇uε‖L2(Ωε) ,
where the constants C1, C2 are independent of ε. Here the function ũε(x) is an
extension of the function uε(x) on the whole domain Ω. Further, by the Young
and the Cauchy–Bunyakovski inequalities, taking into account that gε(x, u) > 0,
we obtain
Φε[uε] =
∫
Ωε
|∇uε|2 dx+
∫
∂F ε
gε(x, uε) dΓ− 2
∫
Ωε
f εuε dx
>
∫
Ωε
|∇uε|2 dx− 2
∣∣∣∣∫
Ωε
f εuε dx
∣∣∣∣
>
1
2
∫
Ωε
|∇uε|2 dx+
1
2
∫
Ωε
|∇uε|2 dx− 1
2δ
∫
Ωε
|uε|2 dx− 2δ
∫
Ωε
|f ε|2 dx
>
1
2
∫
Ωε
|∇uε|2 dx+
(
1
2C
− 1
2δ
)∫
Ωε
|uε|2 dx− 2δ‖f ε‖2L2(Ωε).
Setting C1 = min
(
1
2 ,
1
4C
)
and C2 = 4C‖f ε‖2L2(Ωε), we get the inequality
Φε[u] > C1‖u‖2H1(Ωε) − C2
for any function u ∈ H1(Ωε, ∂Ω). It establishes that the functional Φε[·] is coer-
cive in the space H1(Ωε, ∂Ω).
It remains to prove that the functional Φε[·] is weakly lower semicontinuous.
We now proceed analogously to [14].
We write the functional Φε[·] in the form
Φε[uε] =
∫
Ωε
L(∇uε, uε, x) dx,
where L(p, u, x) is the Lagrangian of the functional. To do this, we need the
following lemma.
290 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3
Integral Conditions for Convergence of Solutions
Lemma 1. For any function ϕ ∈ H1(Ωε, ∂Ω), the equality∫
∂F ε
ϕdΓ =
∫
Ωε
(∇ψ0,∇ϕ) dx− |∂F
ε|
|Ωε|
∫
Ωε
ϕdx (16)
holds, where |∂F ε|, |Ωε| are the surface and volume measures of the corresponding
sets, and the function ψ0(x) is a solution of the boundary-value problem
−∆ψ0 =
|∂F ε|
|Ωε|
, x ∈ Rn \ F ε,
∂ψ0
∂ν
= 1, x ∈ ∂F ε,
ψ0(x)→ 0 as |x| → ∞.
(17)
Proof. The existence and the uniqueness of the solution ψ0(x) can be proved
by the standard methods of potential theory ( [15, Chap. 4]), taking into account
the smoothness of the boundary ∂F ε.
We multiply the differential equation of the problem by an arbitrary function
ϕ ∈ H1(Ωε, ∂Ω) and integrate over the domain Ωε. Using the integration by parts
and the boundary conditions, we obtain the required equality (16).
With the help of Lemma 1 we write the functional Φε[·] in the form
Φε[uε] =
∫
Ωε
|∇uε|2 dx+
∫
Ωε
(∇ψ0,∇xgε(x, uε)) dx
− |∂F
ε|
|Ωε|
∫
Ωε
gε(x, uε) dx− 2
∫
Ωε
f εuε dx
=
∫
Ωε
|∇uε|2 dx+ 2
∫
Ωε
∫ uε
0
(∇ψ0,∇xσε(x, s)) ds dx
+
∫
Ωε
(∇ψ0,∇uε) · σε(x, uε) dx
− |∂F
ε|
|Ωε|
∫
Ωε
gε(x, uε) dx− 2
∫
Ωε
f εuε dx =:
∫
Ωε
L(∇uε, uε, x) dx.
Let p = (p1, ..., pn) := ∇uε,
L(p, u, x) =
n∑
i=1
p2
i + 2
∫ u
0
(∇ψ0,∇xσε(x, s)) ds
+ σε(x, u)
n∑
i=1
∂ψ0
∂xi
· pi −
|∂F ε|
|Ωε|
gε(x, u)− 2f εu.
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3 291
E.Ya. Khruslov, L.O. Khilkova, and M.V. Goncharenko
Since
n∑
i,j=1
∂2L
∂pi∂pj
ηiηj = 2
n∑
i=1
η2
i = 2|η|2, ∀ p, η ∈ Rn, x ∈ Ωε,
the function L is uniformly convex in p for each x ∈ Ωε. This means that the
functional Φε[·] is weakly lower semicontinuous on H1(Ωε, ∂Ω).
Thus, the functional Φε[·] is coercive and weakly lower semicontinuous and
there exists at least one minimizer (see [8, Sect. 8.2]). Therefore, there exists at
least one generalized solution of problem (1). Our next goal is to prove that this
solution is unique.
We assume that problem (1) has two generalized solution uε1(x), uε2(x). Then,
by (7) for any function ϕ ∈ H1(Ωε, ∂Ω), the identities∫
Ωε
(∇uε1,∇ϕ) dx+
∫
∂F ε
σε(x, uε1)ϕdΓ =
∫
Ωε
f εϕdx, (18)∫
Ωε
(∇uε2,∇ϕ) dx+
∫
∂F ε
σε(x, uε2)ϕdΓ =
∫
Ωε
f εϕdx (19)
are true. Subtracting (19) from (18) and taking ϕ(x) = uε1(x)− uε2(x) as the test
function, we obtain∫
Ωε
|∇uε1 −∇uε2|2 dx+
∫
∂F ε
(σε(x, uε1)− σε(x, uε2)) · (uε1 − uε2) dΓ = 0.
By the monotonicity of the function σε(x, u) (property a1), we get
uε1 = uε2 almost everywhere in Ωε.
This proves the uniqueness of the generalized solution of boundary-value problem
(1). Theorem 1 is proved.
4. Auxiliary Statements
The proof of the main theorem is based on the variational methods. The main
idea is to construct suitable approximations of the generalized solution uε(x) of
problem (1). These approximations are constructed by using the minimizers of
the functionals (8) and (12). In order to implement this method, as a preliminary,
we prove several statements characterizing the minimizers themselves, as well as
the functions constructed with their help.
Lemma 2. Let conditions 1), 2) of Theorem 2 be satisfied. Then there exist
the sequences {εk}∞k=1, {h′k}∞k=1, {h′′k = h′k+(h′k)
1+τ/2}∞k=1, such that εk, h
′
k, h
′′
k →
0 as k →∞ and, for almost all x ∈ Ω, the following equalities hold
292 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3
Integral Conditions for Convergence of Solutions
1) lim
k→∞
aij(x, εk, h
′
k)
(h′k)
n
= lim
k→∞
aij(x, εk, h
′′
k)
(h′′k)
n
= aij(x), i, j = 1, . . . , n;
2) lim
k→∞
c(x, s, εk, h
′
k)
(h′k)
n
= lim
k→∞
c(x, s, εk, h
′′
k)
(h′′k)
n
= c(x, s), ∀s ∈ R1;
3) for every µ > 0, there exists a closed set Ωµ ⊂ Ω such that mes(Ω \ Ωµ) < µ
and assertions 1), 2) above are satisfied uniformly with respect to x ∈ Ωµ.
Proof. From conditions 1), 2) of Theorem 2, it follows that there exists a
monotone function ε̂(h) such that ε̂(h)→ 0 as h→ 0 and, for all εh < ε̂(h),
lim
h→0
∫
Ω
∣∣∣∣aij(x, εh, h)
hn
− aij(x)
∣∣∣∣ dx = 0 = lim
h→0
∫
Ω
∣∣∣∣c(x, s; εh, h)
hn
− c(x, s)
∣∣∣∣ dx. (20)
Hence, taking into account that mean convergence implies convergence almost
everywhere on the subsequence [11, p. 5, §4], we conclude that there exists a
sequence {h′l}∞l=1 such that h′l → 0 as l→∞ and, for almost all x ∈ Ω,
lim
l→∞
aij(x, εl, h
′
l)
(h′l)
n
= aij(x), i, j = 1, . . . , n;
lim
l→∞
c(x, s, εl, h
′
l)
(h′l)
n
= c(x, s), ∀s ∈ R1,
where εl = ε̂(h′l).
Setting in (20) h = h′′l = h′l + (h′l)
1+τ/2 (τ > 0), we have
lim
l→∞
∫
Ω
∣∣∣∣aij(x, εl, h′′l )(h′′l )
n
− aij(x)
∣∣∣∣ dx = 0 = lim
l→∞
∫
Ω
∣∣∣∣c(x, s; εl, h′′l )(h′′l )
n
− c(x, s)
∣∣∣∣ dx.
Then from the sequence {h′l} we can choose a subsequence {h′k} ⊂ {h′l} and,
consequently, the subsequences {h′′k} ⊂ {h′′l } and {εk = ε̂(h′k)}, for which the
assertions 1), 2) of the lemma hold.
Applying Egorov’s theorem [11], we obtain the third assertion of the lemma.
The lemma is proved.
Further, since we will consider the convergence on the sequences {εk}∞k=1,
{h′k}∞k=1 and {h′′k}∞k=1 constructed in Lemma 2, for the for the sake of brevity, we
use the following notation:
Πz
k := Kz
h′′k
\Kz
h′k
, Kz
k := Kz
h′′k
, Ωk := Ωεk , ∂F k := ∂F εk ,
gk(x, u) := gεk(x, u), σk(x, u) := σεk(x, u), fk(x) := f εk(x).
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3 293
E.Ya. Khruslov, L.O. Khilkova, and M.V. Goncharenko
Lemma 3. Let condition 1) of Theorem 2 be satisfied, {εk}∞k=1, {h′k}∞k=1,
{h′′k = h′k + (h′k)
1+τ/2}∞k=1 and Ωµ be the sequences and the set, respectively,
constructed in Lemma 2, the function vzk minimizes functional (8) in the domain
Kz
k ∩ Ωk for ` ∈ Rn.
Then, for every µ > 0,
lim
k→∞
1
(h′′k)
n
∫
Πz
k∩Ωk
|∇vzk(x)|2 dx = 0,
lim
k→∞
1
(h′′k)
n+2+τ
∫
Πz
k∩Ωk
|vzk(x)− (x− z, `)|2 dx = 0,
lim
k→∞
n∑
i,j=1
1
(h′′k)
n
∫
Kz
k∩Ωk
(
∇vzki,∇vzkj
)
`i`j dx 6
n∑
i,j=1
aij(z)`i`j
uniformly with respect to z ∈ Ωµ.
Proof. Let vzki(x) be a minimizer of functional (8) in the domain Kz
k ∩ Ωk as
` ∈ Rn (ei is a basic vector of the axis xi (i = 1, . . . , n)). Using the definition
of functional (8), its representation (10) and the expression for the coefficients
aij(z, ε, h) (11), we obtain
1
(h′′k)
n
∫
Πz
k∩Ωk
(
|∇vzki|2 + (h′′k)
−2−τ |vzki − (xi − zi)|2
)
dx
=
aii(z, εk, h
′′
k)
(h′′k)
n
− 1
(h′′k)
n
∫
Kz
h′
k
∩Ωk
(
|∇vzki|2 + (h′′k)
−2−τ |vzki − (xi − zi)|2
)
dx
6
aii(z, εk, h
′′
k)
(h′′k)
n
−
(h′k)
n
(h′′k)
n
·
aii(z, εk, h
′
k)
(h′k)
n
.
Furthermore,
1
(h′′k)
n
∫
Kz
k∩Ωk
(
∇vzki,∇vzkj
)
`i`j dx 6
aij(z, εk, h
′′
k)
(h′′k)
n
`i`j .
This, Lemma 2 implies that
lim
k→∞
1
(h′′k)
n
∫
Πz
k∩Ωk
|∇vzki|2 dx = 0,
lim
k→∞
1
(h′′k)
n+2+τ
∫
Πz
k∩Ωk
|vzki − (xi − zi)|2 dx = 0,
lim
k→∞
1
(h′′k)
n
∫
Kz
k∩Ωk
(
∇vzki,∇vzkj
)
`i`j dx 6 aij(z)`i`j
uniformly in z ∈ Ωµ. Since vzk =
∑n
i=1 v
z
ki · `i and ` =
∑n
i=1 `ie
i, the assertions
of the lemma are valid. The lemma is proved.
294 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3
Integral Conditions for Convergence of Solutions
Lemma 4. Let condition 2) of Theorem 2 be satisfied, {εk}∞k=1, {h′k}∞k=1,
{h′′k = h′k + (h′k)
1+τ/2}∞k=1 and Ωµ be the sequences and the set, respectively,
constructed in Lemma 2.
Then
1) for every s = ŝ, there exists a unique minimizer wzk of functional (12) in the
domain Kz
k ∩ Ωk, and it satisfies the estimate
|wzk(x)| 6 |ŝ|;
2) for every µ > 0,
lim
k→∞
1
(h′′k)
n
∫
Πz
k∩Ωk
|∇wzk(x)|2 dx = 0,
lim
k→∞
1
(h′′k)
n+2+τ
∫
Πz
k∩Ωk
|wzk(x)− ŝ|2 dx = 0,
lim
k→∞
1
(h′′k)
n
∫
Πz
k∩∂Fk
, gε(x,wzk) dΓ = 0,
lim
k→∞
1
(h′′k)
n+2+τ
∫
Kz
k∩Ωk
|wzk − ŝ|2 dx 6 C,
lim
k→∞
1
(h′′k)
n
{∫
Kz
k∩Ωk
|∇wzk|2 dx+
∫
Kz
k∩∂Fk
gk(x,wzk) dΓ
}
6 c(z, ŝ)
uniformly with respect to z ∈ Ωµ;
3) for every δ > 0, a measure of the set Bzδ
k = {x ∈ Kz
k ∩ Ωk : |wzk − ŝ| > δ}
satisfies the estimate
mes{Bzδ
k } <
C(h′′k)
n+2+τ
δ2
.
Proof. The existence and the uniqueness of wzk ∈ H1(Kz
h ∩Ωk) can be proved
by the standard variational methods (see, for example, the proof of Theorem 1).
Let us prove the estimate for the function wzk. For simplicity, we set ŝ > 0 and
assume that estimate 1) is not true. Then in the domain Kz
k ∩ Ωk there exists a
set E on which wzk > ŝ. We construct a cut-off function ŵzk such that ŵzk = ŝ for
x ∈ E and ŵzk = wzk for x ∈ (Kz
k ∩ Ωk) \ E. The function ŵzk provides a smaller
value for functional (12) than the function wzk. This contradicts our assumption
that wzk is a minimizer of this functional.
Next we prove assertion 2) of the lemma. By (12), we obtain
1
(h′′k)
n
[∫
Πz
k∩Ωk
(
|∇wzk|2 + (h′′k)
−2−τ |wzk − ŝ|2
)
dx+
∫
Πz
k∩∂Fk
gk(x,wzk)dΓ
]
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3 295
E.Ya. Khruslov, L.O. Khilkova, and M.V. Goncharenko
=
c(z, ŝ; εk, h
′′
k)
(h′′k)
n
− 1
(h′′k)
n
∫
Kz
h′
k
∩Ωk
(
|∇wzk|2 + (h′′k)
−2−τ |wzk − ŝ|2
)
dx
+
∫
Kh′
k
∩∂Fk
gk(x,wzk) Γ
6 c(z, ŝ; εk, h
′′
k)
(h′′k)
n
−
(h′k)
n
(h′′k)
n
·
c(z, ŝ; εk, h
′
k)
(h′k)
n
.
We pass to the limit as k →∞. Since all the integrands are positive and asser-
tion 2) of Lemma 2 is true, then the limit equalities of 2) are satisfied uniformly
with respect to z ∈ Ωµ.
In addition, using (12) and assertion 2) of Lemma 2, we have as k →∞,
1
(h′′k)
n
[∫
Kz
k∩Ωk
(
|∇wzk|2 + (h′′k)
−2−τ |wzk − ŝ|2
)
dx+
∫
Kz
k∩∂Fk
gk(x,wzk)dΓ
]
= c(z, ŝ) + o(1) 6 C.
Therefore,
1
(h′′k)
n
[∫
Kz
k∩Ωk
|∇wzk|2 dx+
∫
Kz
k∩∂Fk
gk(x,wzk)dΓ
]
6 c(z, ŝ) + o(1), k →∞
and ∫
Kz
k∩Ωk
|wzk − ŝ|2 dx 6 C(h′′k)
n+2+τ .
This establishes the limit inequalities of 2) and the third assertion of the lemma.
The lemma is proved.
Lemma 5. Let condition 2) of Theorem 2 be satisfied, {εk}∞k=1, {h′k}∞k=1,
{h′′k = h′k + (h′k)
1+τ/2}∞k=1 and Ωµ be the sequences and the set, respectively,
constructed in Lemma 2. Let the function wzk minimizes functional (12) in the
domain Kz
k ∩ Ωk for s = ŝ.
Then the set
Bz
k = {x ∈ Kz
k ∩ Ωk : |wzk − ŝ| > (h′′k)
1+τ/3} (21)
and the function
ŵzk =
wzk, x ∈ Bz
k ,
ŝ− (h′′k)
1+τ/3, ŝ > 0, x ∈ (Kz
k ∩ Ωk) \Bz
k ,
ŝ+ (h′′k)
1+τ/3, ŝ 6 0, x ∈ (Kz
k ∩ Ωk) \Bz
k ,
(22)
satisfy the properties:
296 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3
Integral Conditions for Convergence of Solutions
1) a measure of Bz
k,
mes{Bz
k} < C(h′′k)
n+τ/3;
2) for every µ > 0,
lim
k→∞
1
(h′′k)
n
∫
Πz
k∩Ωk
|∇ŵzk(x)|2 dx = 0,
lim
k→∞
1
(h′′k)
n+2+τ
∫
Πz
k∩Ωk
|ŵzk(x)− ŝ|2 dx = 0,
lim
k→∞
1
(h′′k)
n
∫
Πz
k∩∂Fk
gk(x, ŵzk) dΓ = 0,
lim
k→∞
1
(h′′k)
n+2+τ
∫
Kz
k∩Ωk
|ŵzk − ŝ|2 dx 6 C,
lim
k→∞
1
(h′′k)
n
{∫
Kz
k∩Ωk
|∇ŵzk|2 dx+
∫
Kz
k∩∂Fk
gk(x, ŵzk) dΓ
}
6 c(z, ŝ)
uniformly with respect to z ∈ Ωµ.
Proof. The first assertion of the lemma follows from conclusion 3) of Lemma 4.
We proceed to prove the second assertion of the lemma. Without loss of
generality, with regard for smallness of h′′k, we assume that ŝ > (h′′k)
1+τ/3 > 0.
By (13), (22) and the properties a1, a2 of the function σk(x, u), the functions ŵzk,
gk(x, ŵzk) satisfy the estimates
|∇ŵzk(x)|2 6 |∇wzk(x)|2,
|ŵzk(x)− ŝ|2 6 |wzk(x)− ŝ|2 + (h′′k)
2+2τ/3,
gk(x, ŵzk) = gk(x,wzk) +O
(
(h′′k)
1+τ/3
)
for every x ∈ Kz
k ∩ Ωk.
From these estimates and the conclusions of part 2) of Lemma 4, it follows
that the second assertion of the lemma is valid for ŝ > 0. Similarly, we consider
the case ŝ < 0. The lemma is proved.
Lemma 6. Let the domain Ωµ be a subdomain of Ω. Then we can construct
a covering of the domain Ω with disjoint cubes Kα
h of the size h centered at xα,
α = 1, . . . , Nk >
mes{Ω}
hn , such that the number of cubes with centers not belonging
to the domain Ωµ satisfy the inequality
N ′k 6
mes{Ω \ Ωµ}
mes{Ω}
Nk. (23)
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3 297
E.Ya. Khruslov, L.O. Khilkova, and M.V. Goncharenko
Proof. Let us cover Ω with disjoint (having no common interior points) cubes
K̃α
h = K(x̃α, h) of the size h centered at x̃α. Let χα be a characteristic function
of the domain (Ω \ Ωµ) ∩ K̃α
h . We have
mes{Ω \ Ωµ} =
∑
α
∫
K̃α
h
χα(x) dx. (24)
With the help of shift, we combine all the cubes K̃α
h in one K0
h centered at 0.
Thus, we get∑
α
∫
K̃α
h
χα(x) dx =
∑
α
∫
K0
h
χα(x− x̃α) dx =
∫
K0
h
∑
α
χα(x− x̃α) dx. (25)
We denote
χ(x) =
∑
α
χα(x− x̃α). (26)
It is clear that χ(x) is an integer-valued function in K0
h, which determines the
multiplicity of coverage of the point x ∈ K0
h by the image of the set Ω \ Ωµ. We
choose a point x̂ ∈ K0
h with the smallest coverage multiplicity N ′k. Hence,∫
K0
h
χ(x) dx > N ′kh
n. (27)
We take the points xα as unknown centers of the cubes Kα
h , which are the images
of the point x̂ under reverse shift. The number of the centers of the cubes in the
set Ω \ Ωµ is equal to N ′k. Suppose that inequality (23) does not hold, then by
(24)–(27), we get
mes{Ω \ Ωµ} > N ′khn >
mes{Ω \ Ωµ}
mes{Ω}
Nkh
n > mes{Ω \ Ωµ}.
We have obtained a contradiction
mes{Ω \ Ωµ} > mes{Ω \ Ωµ}.
Thus inequality (23) is true. The lemma is proved
Lemma 7. Let the sets Bk ⊂ Ωk ⊂ Ω be given and
lim
k→∞
mes{Bk} = 0.
If condition 1) of Theorem 2 is fulfilled, then there exists a set B̂k ⊂ Ω and the
functions v̂ki ∈ H1(Ω) (i = 1, n) satisfying the following conditions:
298 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3
Integral Conditions for Convergence of Solutions
1) Bk ⊂ B̂k, lim
k→∞
mes{B̂k} = 0;
2) lim
k→∞
max
x∈Ω
|v̂ki(x)− xi| = 0;
3) lim
k→∞
∫
B̂k
|∇v̂ki|2 dx = 0;
4) for any vector function `(x) = (`1(x), ..., `n(x)) ∈ (C(Ω))n,
lim
k→∞
∫
Ωk
n∑
i,j=1
(∇v̂ki,∇v̂kj) `i(x)`j(x) dx 6
∫
Ω
n∑
i,j=1
aij(x)`i(x)`j(x) dx.
Proof. Let {εk}∞k=1, {h′k}∞k=1, {h′′k = h′k + (h′k)
1+τ/2}∞k=1 be the sequences
constructed in Lemma 2. Let {µk}k be the sequence of numbers and µk → 0 as
k → ∞. Due to assertion 3) of Lemma 2, we can construct a sequence of sets
Ωµ1 ⊂ Ωµ2 ⊂ · · · ⊂ Ωµk ⊂ · · · ⊂ Ω such that mes{Ω \ Ωµk} < µk.
Let us cover the domain Ω by disjoint cubes Kα
hk
(
Ω ∈ ∪αKα
hk
)
of the size
hk =
h′′k+h′k
2 centered at the points xα as in Lemma 6. Besides the cubes Kα
hk
, we
also consider the cubes Kα
h′k
and Kα
h′′k
. With this covering, we associate a partition
of the unity {ϕαk (x)}α of twice continuously differentiable functions satisfying the
following conditions:
ϕαk (x) = 0, if x /∈ Kα
h′′k
; ϕαk (x) = 1, if x ∈ Kα
h′k
;
∀x ∈ Ω : 0 6 ϕαk (x) 6 1,
∑
α
ϕαk (x) = 1, |Dϕαk (x)| < C(h′k)
−1−τ/2. (28)
In each cube Kα
k := Kα
h′′k
, we construct the set Bα
k by Lemma 5 and the set
Bk = ∪αBα
k , whose measure, by virtue of assertion 1) of Lemma 5, satisfies the
inequality
mes{Bk} 6
∑
α
mes{Bα
k } < C(h′′k)
τ/3.
The set Bk satisfies the condition of the lemma.
Let vαki (i = 1, n) be a minimizer of (8) in the cube Kα
k with xα ∈ Ωµk for ` =
ei. Then, by the maximum principle,
max
x∈Kα
k ∩Ωk
|vαki(x)| 6 1
2
h′′k, (29)
and the functions vαki (i = 1, n) satisfy the assertions of Lemma 3.
In Ωk, we consider the functions
vki(x) = xi +
∑
α
[vαki(x)− (xi − xαi )]ϕαk (x) (i = 1, n), (30)
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3 299
E.Ya. Khruslov, L.O. Khilkova, and M.V. Goncharenko
where {ϕαk (x)}α is a partition of unity (28). Notice that vαki(x) = xi − xαi if xα ∈
Ω \ Ωµk or the cube Kα
k does not lie entirely in Ω.
It is clear that
∂vki(x)
∂xj
=
∑
α
∂vαki(x)
∂xj
ϕαk (x) +
∑
α
[vαki(x)− (xi − xαi )]
∂ϕαk (x)
∂xj
.
Let `(x) = {`1(x), `2(x), ..., `n(x)} be an arbitrary vector function continuous in
Ω. By Lemma 3 and the properties of the function ϕαk (x), we obtain∫
Ωk
n∑
i,j=1
(∇vki,∇vkj)`i(x)`k(x) dx 6
∫
Ω
n∑
i,j=1
aij(x)`i(x)`j(x) + o(1), k →∞.
(31)
Furthermore, by (29) and (30), we have
max
x∈Ωk
|vki − xi| 6
1
2
h′′k. (32)
Set uki = vki − xi. Then from (31) and (32), we obtain the estimates
max
x∈Ωk
|uki| 6
1
2
h′′k and ‖uki‖H1(Ωk) 6 C, (33)
where C is independent of k.
Since the domains Ωk satisfy the extension condition (6), there exists a func-
tion uki(x) ∈ H1(Ω) equal to uki(x) for x ∈ Ωk which satisfies inequalities (33) in
Ω. We extend uki(x), maintaining these inequalities, on the parallelepiped Π ⊃
Ω and then approximate a twice continuously differentiable in Π function ũki(x)
such that
‖ũki − uki‖H1(Π) 6 εk, max
x∈Π
|ũki| 6 C1h
′′
k, ‖ũki‖H1(Π) 6 C2, (34)
where C1, C2 are independent of k.
Now we apply Lemma 1.3 [12, Chap. 3] to the function ũki(x) and the set
Bk. By the lemma, there exists the function ûki(x) ∈ H1(Π) and the set B̂k ⊂ Π
satisfying the conditions:
Bk ⊂ B̂k, mes(B̂k) 6 C1Am
2| lnm|−1/3
(
m| lnm|
−2/3
+ | lnm|−1/6
)
, (35)
ûki(x) = ũki(x), x ∈ Π \ B̂k,
max
x∈Π
|ûki(x)| 6 C2h
′′
k, ‖ûki‖H1(B̂k)
6 C3A
(
m| lnm|
−2/3
+ | lnm|−1/6
)
, (36)
where m = mes(Bk), A = ‖ũki‖H1(Π) and the constants C1, C2, C3 are indepen-
dent of k.
300 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3
Integral Conditions for Convergence of Solutions
We set v̂ki = ûki + xi and consider the restrictions of the set B̂k ∈ Π and
the function v̂ki(x) ∈ H1(Π) on the domain Ω (preserving the original notation).
We obtain the set and the function which by (34)–(36) satisfy assertions 1)–3)
of the lemma. We now verify estimate 4) of the lemma. Set ṽki(x) = ũki(x) +
xi. According to (31), (34), and (36) taking into account the positivity of the
integrand and estimate 3) of the lemma, the inequality∫
Ωk
n∑
i,j=1
(∇v̂ki,∇v̂kj)`i(x)`j(x) dx
6
∫
Ω
n∑
i,j=1
(∇v̂ki,∇v̂kj)`i(x)`j(x) dx
=
∫
Ω\B̂k
n∑
i,j=1
(∇ṽki,∇ṽkj)`i(x)`j(x) dx
+
∫
B̂k
n∑
i,j=1
(∇v̂ki,∇v̂kj)`i(x)`j(x) dx
=
∫
Ωk
n∑
i,j=1
(∇vki,∇vkj)`i(x)`j(x) dx+ o(1)
6
∫
Ω
n∑
i,j=1
aij(x)`i(x)`j(x) + o(1), k →∞
holds for all `(x) ∈ (C(Ω))n. This proves estimate 4). The lemma is proved.
5. Proof of Main Theorem 2
We briefly outline the scheme of the proof. It is similar to the scheme devel-
oped in [9], but it should be taken into account that the conditions of the uniform
convergence does not hold in the entire region Ω.
Earlier, in the proof of Theorem 1, we determined the energy functional Φε[·]
(15) of the original problem (1). The generalized solution uε(x) of problem (1)
minimizes this functional in a class of functions uε(x) ∈ H1(Ωε,Ω).
We also determine an energy functional of homogenized problem (2),
Φ[u] =
∫
Ω
n∑
i,j=1
aij
∂u
∂xi
∂u
∂xj
dx+
∫
Ω
c(x, u) dx− 2
∫
Ω
fu dx. (37)
The generalized solution of homogenized problem (2) minimizes Φ[u] in the class
of functions u(x) ∈ H̊1(Ω).
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3 301
E.Ya. Khruslov, L.O. Khilkova, and M.V. Goncharenko
In Subsection 5.1, we show that the generalized solutions uε(x) of prob-
lem (1) can be extended on the set F ε such that the extended solutions ũε(x)
are uniformly bounded with respect to ε in H̊1(Ω). The resulting sequence
{ũε(x)}ε is weakly compact in H̊1(Ω). Hence, we can select the subsequence
{ũεm(x), m = 1, . . . ,∞} that is weakly convergent in H̊1(Ω) and, by the embed-
ding theorem, strongly convergent in Lp(Ω)
(
p < 2n
n−2
)
to some function u(x) ∈
H̊1(Ω).
In Subsections 5.2–5.3, we prove that the function u(x) is the minimizer of
functional (37) and, consequently, the generalized solution of homogenized prob-
lem (2). This is done as follows. In Subsection 5.2, we define the special test
function wε(x) approximating the minimizer of functional (15). The function is
constructed on an arbitrary function w(x) ∈ C2
0 (Ω). Since the solution uε(x) of
problem (1) minimizes functional (15) in H1(Ωε, ∂Ω), the inequality below holds,
Φε[uε] 6 Φε[wε]. (38)
Further we show that under the conditions of Theorem 2,
lim
ε→0
Φε[wε] = Φ[w], (39)
where Φ[·] is defined by (37). By virtue (38), (39) and the density of C2
0 (Ω) in
the space H̊1(Ω), it follows that
lim
ε→0
Φε [uε] 6 Φ[w], ∀w ∈ H̊1(Ω). (40)
In Subsection 5.3, we show that if uε(x) converges weakly in H̊1(Ω) to a
function u(x) on some subsequence ε = εm → 0, the reverse inequality holds,
lim
ε=εm→0
Φε[uε] > Φ[u]. (41)
From (40), (41), it follows that the limit function u(x) satisfies the inequality
Φ[u] 6 Φ[w] for an arbitrary function w(x) ∈ H̊1(Ω). Therefore, u(x) minimizes
the functional Φ[w] in the class H̊1(Ω).
In Subsection 5.4, we prove that homogenized problem (2) has the unique
generalized solution u(x) and, therefore, the whole sequence of the extended
solutions {ũε(x)}ε converges in Lp(Ω) to the function u(x). Thus the sequence
of solutions {uε(x)}ε of initial problem (1) converges in Lp(Ωε,Ω) to the solution
u(x) of homogenized problem (2).
302 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3
Integral Conditions for Convergence of Solutions
5.1. Compactness of the generalized solutions of problem (1). Since
Φε[uε] 6 Φε[0] = 0, the inequality
0 6
∫
Ωε
|∇uε|2 dx+
∫
∂F ε
gε(x, uε) dΓ 6 2
∫
Ωε
f εuε dx
is true. By virtue of the properties a1, a2 of the function σε(x, u), we have
gε(x, uε) > 0.
By the Cauchy–Buniakovski inequality and nonnegativity of the function
gε(x, uε), we get
‖∇uε‖2L2(Ωε) 6 2 ‖f ε‖L2(Ωε) ‖u
ε‖L2(Ωε) . (42)
On account of the extension condition (5) and the Friedrichs inequality, there
exists a function ũε(x) ∈ H̊1(Ω) such that ũε(x) = uε(x) for x ∈ Ωε and
‖uε‖L2(Ω) 6 ‖ũ
ε‖L2(Ω) 6 C1 ‖∇ũε‖L2(Ω) 6 C2 ‖∇uε‖L2(Ωε) , (43)
where the constants C1, C2 are independent of ε.
From (42), (43) and the uniform boundedness of the norm ‖f ε‖L2(Ωε) with
respect to ε, it follows that the sequence of the functions {ũε(x)}ε is uniformly
bounded and weakly compact in H̊1(Ω) with respect to ε. Hence, we can extract
a subsequence {ũεm(x), m = 1, . . . ,∞} that is weakly convergent in H̊1(Ω) and
strongly convergent in Lp(Ω) to some function u(x) ∈ H̊1(Ω).
Now we will show that the function u(x) is a minimizer of (15).
5.2. Proof of inequality (40). Let {εk}∞k=1, {h′k}∞k=1, {h′′k = h′k +
(h′k)
1+τ/2}∞k=1 be the sequences constructed in Lemma 2. By assertion 3) of
this lemma, we construct a sequence of sets Ωµ1 ⊂ Ωµ2 ⊂ · · · ⊂ Ωµk ⊂ · · · ⊂ Ω
such that mes{Ω \ Ωµk} < µk. Setting µk = h′′k, we get mes{Ω \ Ωµk} < h′′k.
As in Lemma 6, we cover the domain Ω by disjoint cubes Kα
hk
(
Ω ∈ ∪αKα
hk
)
of the size hk =
h′′k+h′k
2 centered at the points xα. Taking into account that
mes s{Ω \ Ωµk} < h′′k and the total number of cubes Nk ∼ mes s{Ω}
hn , we get the
estimate for the number of cubes with centers xα /∈ Ωµk :
N ′k = O
(
(hk)
1−n) . (44)
We also consider the cubes Kα
h′′k
and Kα
h′k
centered at xα.
We denote
Λk := {α ∈ {1, ..., Nk} : xα ∈ Ωµk ,K
α
k ∈ Ω},
Ωk := Ωεk , F k := F εk , Kα
k := Kα
h′′k
, Πα
k := Kα
h′′k
\Kα
h′k
,
Φk[u] := Φεk [u], gk(x, u) := gεk(x, u), σk(x, u) := σεk(x, u), fk(x) := f εk(x).
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3 303
E.Ya. Khruslov, L.O. Khilkova, and M.V. Goncharenko
Let w(x) be an arbitrary function from C2
0 (Ω), ŵαk and Bα
k be the functions
and sets constructed in Lemma 5 with ŝ = w(xα) for the cubes Kα
k ; v̂ki (i =
1, . . . , n), and B̂k be the functions and sets constructed in Lemma 7 for the set
Bk = ∪
α
Bα
k ; {ϕαk (x)}α be a partition of unity (28) associated with the covering of
Ω by the cubes Kα
h′k
and Kα
h′′k
. Consider the function
wk(x) = w(x) +
n∑
i=1
∂w(x)
∂xi
[v̂ki(x)− xi] +
∑
α
[ŵαk (x)− w(xα)]ϕαk (x). (45)
Here we set ŵαk (x) = w(xα) if α /∈ Λk. It is obvious that wk(x) ∈ H1(Ωk, ∂Ω).
We substitute the function wk(x) into functional (15) and estimate each term.
We estimate the first term. By (45), we can write the derivatives of wk(x) in
the form
∂wk
∂xj
=
n∑
i=1
∂w
∂xi
∂v̂ki
∂xj
+
∑
α∈Λk
∂ŵαk
∂xj
ϕαk (x) +
2∑
s=1
As(x), (46)
where
A1(x) =
n∑
i=1
∂2w
∂xi∂xj
[v̂ki(x)− xi], A2(x) =
∑
α∈Λk
[ŵαk (x)− w(xα)]
∂ϕαk (x)
∂xj
.
The selected terms in (46) give a finite contribution to the functional Φk[wk], the
contribution of the terms A1(x), A2(x) vanishes. Indeed,∫
Ωk
|∇wk|2 dx 6
n∑
i,j=1
∫
Ωk
(∇v̂ki,∇v̂kj)
∂w
∂xi
∂w
∂xj
dx
+
∑
α∈Λk
∫
Kα
k ∩Ωk
|∇ŵαk |2 dx+
∑
α∈Λk
Eαk . (47)
Here, Eαk denotes the sum (over the sets Πα
k ∩ Ωk and Kα
k ∩Bα
k ) of integrals of
quadratic and linear combinations of the function (v̂ki − xi) and [ŵαk (x)− w(xα)]·
∂ϕαk
∂xj
with bounded coefficients that are equal to 1 or depend on w ∈ C2
0 (Ω) for
the quadratic terms, and on
∂ŵαk
∂xj
tϕαk or ∂v̂ki
∂xj
for the linear terms. We use the
properties of the functions ϕαk and Lemmas 5 and 7 to estimate
∑
Eαk . As a
result, we get
lim
k→∞
∑
α∈Λk
Eαk = 0. (48)
We can now proceed to estimate the surface integral in the functional Φk[wk].
To this end, we write wk(x) in the form
wk(x) =
∑
α∈Λk
[ŵαk (x) + (w(x)− w(xα))]ϕαk (x)
304 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3
Integral Conditions for Convergence of Solutions
+
∑
α/∈Λk
w(x)ϕαk (x) +
n∑
i=1
∂w(x)
∂xi
[v̂ki(x)− xi].
For the second term, by taking into account (13), the properties a1–a3 of the
function σk(x, u), estimate 2) of Lemma 7, and (44), we obtain∫
∂Fk
gk(x,wk) dΓ =
∑
α∈Λk
∫
Kα
k ∩∂Fk
gk(x,wk) dΓ +
∑
α/∈Λk
∫
Kα
k ∩∂Fk
gk(x,wk) dΓ
=
∑
α∈Λk
∫
Kα
k ∩∂Fk
gk(x, ŵαk ) dΓ
+
∑
α/∈Λk
∫
Kα
k ∩∂Fk
gk(x,w) dΓ +O(h′′k)
=
∑
α∈Λk
∫
Kα
k ∩∂Fk
gk(x, ŵαk ) dΓ +O(h′′k) + C(εk),
where C(εk)→ 0 as k →∞. Thus,∫
∂Fk
gk(x,wk) dΓ =
∑
α∈Λk
∫
Kα
k ∩∂Fk
gk(x, ŵαk ) dΓ + o(1), k →∞. (49)
Consider now the third term in Φk[wk]. By (45), condition 3) of Theorem 2,
and the estimates from Lemmas 5 and 7, we obtain∫
Ωk
fk(x)wk(x) dx =
∫
Ωk
fk(x)w(x) dx+ o(1), k →∞. (50)
Thus, by virtue of (47)–(50), the functional Φk[wk] can be estimated as fol-
lows:
Φk[wk] =
n∑
i,j=1
∫
Ωk
(∇v̂ki,∇v̂kj)
∂w
∂xi
∂w
∂xj
dx− 2
∫
Ωk
fkw dx
+
∑
α∈Λk
[∫
Kα
k ∩Ωk
|∇ŵαk |2 dx+
∫
Kα
k ∩∂Fk
gk(x, ŵαk ) dΓ
]
+ o(1), k →∞.
Passing to the limit as k →∞ and taking into account the conditions of Theorem
2 and the estimates from Lemmas 5 and 7, we obtain
lim
k→∞
Φk[wk] 6
∫
Ω
n∑
i,j=1
aij(x)
∂w
∂xi
∂w
∂xk
dx
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3 305
E.Ya. Khruslov, L.O. Khilkova, and M.V. Goncharenko
+
∫
Ω
c(x,w) dx− 2
∫
Ω
f(x)w(x) dx. (51)
Since the function uε minimizes the functional Φε[·] and from any sequence ε→ 0
we can choose the subsequence {ε = εk → 0, k = 1, . . . ,∞}, for which inequality
(51) is true, then
lim
ε→0
Φε[uε] 6 lim
k→∞
Φk[wk] 6 Φ[w],
where Φ[w] is the energy functional of homogenized problem (2) defined by for-
mula (37).
Hence, ∀w(x) ∈ C2
0 (Ω), inequality (40) is true. Since C2
0 (Ω) is dense in
H̊1(Ω), this inequality is true ∀w(x) ∈ H̊1(Ω).
5.3. Proof of inequality (41). Let u(x) be the weak limit in H1(Ω) of
the extended solutions ũε(x) of problem (1) on some subsequence {ε = εm, m =
1, . . . ,∞}. From the embedding theorems it follows that the traces of ũεm(x) and
u(x) on ∂Ω are preserved and equal to 0. Thus the function u(x) ∈ H̊1(Ω).
As before, for the sake of brevity, we introduce the following notation:
um(x) := uεm(x), ũm(x) := ũεm(x), Ωm := Ωεm ,
Fm := F εm , Φm[u] := Φεm [u], gm(x, u) := gεm(x, u),
σm(x, u) := σεm(x, u), σ̂m(x) := σ̂εm(x), fm(x) := f εm(x).
Let uδ(x) ∈ C2
0 (Ω) be an approximation of u(x) such that
‖uδ − u‖H1(Ω) < δ. (52)
In the domains Ωm, we consider the functions
ũδm(x) = ũm(x) + uδ(x)− u(x) ∈ H̊1(Ω),
uδm(x) = ũδm(x)|Ωm = um(x) + uδ(x)− u(x) ∈ H1(Ωm, ∂Ω).
By (52), we have
‖ũδm − ũm‖H1(Ω) < δ, ‖uδm − um‖H1(Ωm) < δ, (53)
in addition, as m→∞,
‖ũδm − uδ‖L2(Ω) → 0, ‖uδm − uδ‖L2(Ωm,Ω) → 0. (54)
We define the functions
vδm(x) = ũδm(x)− uδ(x) ∈ H̊1(Ω).
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Since, the functions vδm converge strongly to 0 in L2(Ω) and weakly to 0 in H1(Ω)
as m → ∞, we conclude that ‖vδm‖H1(Ω) 6 C and the functions vδm(x) converge
in measure to 0. Namely, there exist the sets Gm := Gεm ⊂ Ω and the numbers
βm := β(εm) such that
∀x ∈ Ω \Gm : |vδm(x)| < βm, lim
m→∞
βm = 0, lim
m→∞
mes{Gm} = 0.
By virtue of Lemma 1.4 [12, Chap. 3], on the bases of the functions vδm and the
sets Gm, we can construct the functions v̂δm ∈ H̊1(Ω) and the sets Ĝm ⊂ Ω such
that Ĝm ⊃ Gm, v̂δm = vδm for x ∈ Ω \ Ĝm, and
max
Ω
|v̂δm(x)| 6 Cβm, lim
m→∞
∥∥∥v̂δm∥∥∥
H1(Ĝm)
= 0, lim
m→∞
mes s{Ĝm} = 0.
Using these functions, we define the functions
ûδm(x) = uδ(x) + v̂δm(x) ∈ H̊1(Ω). (55)
Let us cover Ω with disjoint cubes Kα
m = K(xα, hm) of the size hm centered
at xα. The centers xα and the sizes hm are chosen as follows: hm is defined as a
minimum value for which the following inequalities are true:
max
Ω
|v̂δm(x)| < hm, ‖ûδm − uδ‖L2(Ω) < h2
m,
mes{Ĝm} < h2+τ1
m , ‖v̂δm‖H1(Ĝm)
< h2+τ1
m , τ1 > τ > 0. (56)
By Lemma 2, we construct a sequence of sets Ωµ1 ⊂ Ωµ2 ⊂ . . . ⊂ Ωµm ⊂ . . . ⊂ Ω
such that mes{Ω \ Ωµm} < hm. We choose the centers xα of the cubes Kα
m as in
Lemma 6. According to the assertions of Lemma 2, the equalities
lim
m→∞
1
hnm
|c(xα, s; εm, hm)− c(xα, s) · hnm| = 0,
lim
m→∞
1
hnm
|aij(xα, εm, hm)− aij(xα) · hnm| = 0, i, j = 1, . . . , n (57)
hold uniformly with respect to xα ∈ Ωµm .
We denote
Λm := {α ∈ {1, . . . , Nm} : xα ∈ Ωµm , K
α
m ∈ Ω}.
In the intersections Kα
m ∩ Ωm, we consider the functions
wδαm1(x) = ûδm(x)− uδ(xα). (58)
Since uδ(x) ∈ C2
0 (Ω), the inequality
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3 307
E.Ya. Khruslov, L.O. Khilkova, and M.V. Goncharenko
∫
Kα
m∩Ωm
|wδαm1(x)− (x− xα, `)|2 dx 6 3
∫
Kα
m∩Ωm
|ûδm(x)− uδ(x)|2 dx
+ 3
∫
Kα
m∩Ωm
[(∇uδ(xα), x− xα)− (x− xα, `)]2 dx+O(hn+4
m )
holds for all ` ∈ Rn and any fixed number δ > 0. Setting ` = ∇uδ(xα) and taking
into account (56), we obtain∫
Kα
m∩Ωm
|wδαm1(x)− (∇uδ(xα), x− xα)|2 dx = O(hn+4
m ). (59)
By virtue of definition (8) of the functional T εh,z(`) and its presentation (10) for
` = ∇uδ(xα), we have∫
Kα
m∩Ωm
|∇wδαm1(x)|2 + h−2−τ |wδαm1(x)− (∇uδ(xα), x− xα)|2 dx
>
n∑
i,j=1
aij(x
α, εm, hm)
∂uδ
∂xi
(xα)
∂uδ
∂xj
(xα). (60)
From (55), (56), (58)–(60), taking into account that α = 1, . . . , Nm ∼ mes{Ω}
hnm
,
uδm(x) = ûδm(x) for x ∈ Ωm \ Ĝm and ∇wδαm1(x) = ∇ûδm(x) for x ∈ Ωm, we obtain
∑
α
∫
Kα
m∩Ωm\Ĝm
|∇uδm(x)|2 dx =
∑
α
∫
Kα
m∩Ωm\Ĝm
|∇ûδm(x)|2 dx
>
∑
α
n∑
i,j=1
aij(x
α, εm, hm)
∂uδ
∂xi
(xα)
∂uδ
∂xj
(xα)
−
∫
Ωm∩Ĝm
|∇ûδm(x)|2 dx−O(h2−τ
m )
=
∑
α
n∑
i,j=1
aij(x
α, εm, hm)
∂uδ
∂xi
(xα)
∂uδ
∂xj
(xα)−O(h2−τ
m ).
Thus,
∑
α
∫
Kα
m∩Ωm\Ĝm
|∇uδm(x)|2 dx
>
∑
α
n∑
i,j=1
aij(x
α, εm, hm)
∂uδ
∂xi
(xα)
∂uδ
∂xj
(xα)−O(h2−τ
m ). (61)
308 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3
Integral Conditions for Convergence of Solutions
Further, in the intersections Kα
m ∩ Ωm, we consider the functions
wδαm2(x) = uδ(xα) + uδm(x)− ûδm(x). (62)
By virtue of definition (12) of the functional c(x, s; ε, h), we have∑
α
c(xα, uδ(xα); εm, hm) 6
∑
α
∫
Kα
m∩Ωm
|∇wδαm2|2 dx
+ h−τ−2
m
∑
α
∫
Kα
m∩Ωm
|wδαm2(x)− uδ(xα)|2 dx
+
∑
α
∫
Kα
m∩∂Fm
gm(x,wδαm2) dΓ. (63)
We now consider each term of the right-hand side. By the Minkowski inequality,
the definitions of wδαm2 (62) and of ûδm(x) (55) and (56) for the first and second
terms, we obtain∑
α
∫
Kα
m∩Ωm
|∇wδαm2|2 dx =
∫
Ωm∩Ĝm
|∇wδαm2|2 dx
=
∫
Ωm∩Ĝm
|∇(uδm − uδ − v̂δm)|2 dx
6
(√∫
Ωm∩Ĝm
|∇uδm|2 dx+O
(
h1+τ1/2
m
))2
=
∫
Ωm∩Ĝm
|∇uδm|2 dx+O
(
h1+τ1/2
m
)
, (64)∑
α
∫
Kα
m∩Ωm
|wδαm2(x)− uδ(xα)|2 dx
=
∫
Ωm∩Ĝm
|uδm(x)− ûδm(x)|2 dx = O(h2+τ1
m ). (65)
Let us estimate the surface integral. By the smoothness of the function uδ(x)
and (56), we write wδαm2 = uδm(x) − (uδ(x) − uδ(xα)) − v̂δm(x) = uδm(x) + O(hm)
for x ∈ Ωm ∩Kα
m. By the properties a1 − a3 of the function σm(x, u),
∑
α
∫
Kα
m∩∂Fm
gm(x,wδαm2) dΓ =
∫
∂Fm
gm(x, uδm) dΓ
+O(hm)
∫
∂Fm
|uδm|Θσ̂m(x) dΓ +O(hm)
=
∫
∂Fm
gm(x, uδm) dΓ +O(hm)‖uδm‖ΘLΘ(Ω,µm) +O(hm).
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3 309
E.Ya. Khruslov, L.O. Khilkova, and M.V. Goncharenko
Here LΘ(Ω, µm) is a space with measure dµm = σ̂m(x) dΓ, Θ < n
n−2 . By the
property a3 of the function σm(x, u), we have the inequality∫
∂Fm∩B(ρ,z)
dµm < C1ρ
n + C2(εm)ρn−1,
where C2(εm) → 0 as m → ∞. The embedding of the space H1(Ω) into
LΘ(Ω, µm) follows from Sobolev’s generalized theorem ( [13, p. 58]). Under the
condition H1(Ω) ⊂ LΘ(Ω, µm), (53) and the fact that the sequence of extended
solutions of problem (1) is uniformly bounded in H1(Ω), the functions uδm and
ũδm satisfy the inequalities
‖uδm‖LΘ(Ω,µm) = ‖ũδm‖LΘ(Ω,µm) 6 C1‖ũδm‖H1(Ω)
6 C1
(
‖ũm‖H1(Ω) + δ
)
6 C2. (66)
Thus, for the surface integral, the estimate∑
α
∫
Kα
m∩∂Fm
gm(x,wδαm2) dΓ =
∫
∂Fm
gm(x, uδm) dΓ +O(hm) (67)
holds.
By (64)–(67), from (63) it follows∑
α
c(xα, uδ(xα); εm, hm) 6
∫
Ωm∩Ĝm
|∇uδm|2 dx
+
∫
∂Fm
gm(x, uδm) dΓ +O(hτ1−τm ). (68)
Hence, by virtue of the obtained estimates (61), (68) and the positivity of
summable functions, we have
Φm[uδm] =
∑
α
∫
Kα
m∩Ωm\Ĝm
|∇uδm|2 dx+
∫
Ωm∩Ĝm
|∇uδm|2 dx
+
∫
∂Fm
gm(x, uδm) dΓ− 2
∫
Ωm
fmuδm dx
>
∑
α∈Λm
n∑
i,j=1
aij(x
α, εm, hm)
∂uδ
∂xi
(xα)
∂uδ
∂xj
(xα)
+
∑
α∈Λm
c(xα, uδ(xα); εm, hm)− 2
∫
Ωm
fmuδm dx−O(hτ1−τm ).
By (57), taking into account mes{Ωµm} → mes{Ω} as m→∞, the equalities
lim
m→∞
∑
α∈Λm
n∑
i,j=1
aij(x
α, εm, hm)
∂uδ
∂xi
(xα)
∂uδ
∂xj
(xα)
310 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3
Integral Conditions for Convergence of Solutions
= lim
m→∞
∫
Ωµm
n∑
i,j=1
aij(x)
∂uδ
∂xi
∂uδ
∂xj
dx
=
∫
Ω
n∑
i,j=1
aij(x)
∂uδ
∂xi
∂uδ
∂xj
dx,
lim
m→∞
∑
α∈Λm
c(xα, uδ(xα); εm, hm) = lim
m→∞
∫
Ωµm
c(x, uδ) dx =
∫
Ω
c(x, uδ) dx.
are true.
When passing to the limit as m → ∞ for fixed δ, we take into account the
above equalities, condition 3) of Theorem 2 and (54), to get
lim
m→∞
Φm[uδm] >
∫
Ω
n∑
i,j=1
aij(x)
∂uδ
∂xi
∂uδ
∂xj
dx+
∫
Ω
c(x, uδ) dx− 2
∫
Ω
fuδ dx
= Φ[uδ].
We now pass to the limit as δ → 0. In view of the smoothness of the function
uδ(x) and inequality (52), we get limδ→0 Φ[uδ] = Φ[u] on the right-hand side. On
the left-hand side, passing to the limit in the surface integral, we use Sobolev’s
generalized theorem and inequality (52). As a result, we obtain the required
inequality (41).
5.4. Uniqueness of generalized solution of homogenized prob-
lem (2). In the proof we follow all steps as in the proof of Theorem 1. Assume
that problem (2) has two generalized solutions u1, u2. Applying (14) for the
functions u1, u2, subtracting one equality from another, and substituting ϕ(x) =
u1(x)− u2(x) as a test function, we obtain
∫
Ω
n∑
i,k=1
aik(x)
∂ (u1 − u2)
∂xk
∂ (u1 − u2)
∂xi
dx
+
1
2
∫
Ω
(cu(x, u1)− cu(x, u2)) (u1 − u2) dΓ = 0.
From the above, by the positive definiteness of the tensor {aik(x)}ni,k=1 and mono-
tonicity of the function cu(x, u), we get
u1 = u2 almost everywhere in Ω.
Thus, the uniqueness of the generalized solution of homogenized problem (2) is
proved. Theorem 2 is proved.
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 3 311
E.Ya. Khruslov, L.O. Khilkova, and M.V. Goncharenko
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