Г-convergence of integral functionals with degenerate integrands in periodically perforated domains
We consider a sequence of integral functionals with degenerate integrands in perforated domains of periodic structure. We establish the ¡-convergence of the sequence under consideration to an integral functional defined on a limit weighted Sobolev space. A representation formula for the integrand of...
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Інститут прикладної математики і механіки НАН України
2009
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| Цитувати: | Г-convergence of integral functionals with degenerate integrands in periodically perforated domains / A.A. Kovalevsky, O.A. Rudakova // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2009. — Т. 19. — С. 101-109. — Бібліогр.: 16 назв. — англ. |
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irk-123456789-1239042017-09-14T03:03:06Z Г-convergence of integral functionals with degenerate integrands in periodically perforated domains Kovalevsky, A.A. Rudakova, O.A. We consider a sequence of integral functionals with degenerate integrands in perforated domains of periodic structure. We establish the ¡-convergence of the sequence under consideration to an integral functional defined on a limit weighted Sobolev space. A representation formula for the integrand of the ¡-limit functional is given. 2009 Article Г-convergence of integral functionals with degenerate integrands in periodically perforated domains / A.A. Kovalevsky, O.A. Rudakova // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2009. — Т. 19. — С. 101-109. — Бібліогр.: 16 назв. — англ. 1683-4720 http://dspace.nbuv.gov.ua/handle/123456789/123904 517.9 en Труды Института прикладной математики и механики Інститут прикладної математики і механіки НАН України |
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We consider a sequence of integral functionals with degenerate integrands in perforated domains of periodic structure. We establish the ¡-convergence of the sequence under consideration to an integral functional defined on a limit weighted Sobolev space. A representation formula for the integrand of the ¡-limit functional is given. |
| format |
Article |
| author |
Kovalevsky, A.A. Rudakova, O.A. |
| spellingShingle |
Kovalevsky, A.A. Rudakova, O.A. Г-convergence of integral functionals with degenerate integrands in periodically perforated domains Труды Института прикладной математики и механики |
| author_facet |
Kovalevsky, A.A. Rudakova, O.A. |
| author_sort |
Kovalevsky, A.A. |
| title |
Г-convergence of integral functionals with degenerate integrands in periodically perforated domains |
| title_short |
Г-convergence of integral functionals with degenerate integrands in periodically perforated domains |
| title_full |
Г-convergence of integral functionals with degenerate integrands in periodically perforated domains |
| title_fullStr |
Г-convergence of integral functionals with degenerate integrands in periodically perforated domains |
| title_full_unstemmed |
Г-convergence of integral functionals with degenerate integrands in periodically perforated domains |
| title_sort |
г-convergence of integral functionals with degenerate integrands in periodically perforated domains |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/123904 |
| citation_txt |
Г-convergence of integral functionals with degenerate integrands in periodically perforated domains / A.A. Kovalevsky, O.A. Rudakova // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2009. — Т. 19. — С. 101-109. — Бібліогр.: 16 назв. — англ. |
| series |
Труды Института прикладной математики и механики |
| work_keys_str_mv |
AT kovalevskyaa gconvergenceofintegralfunctionalswithdegenerateintegrandsinperiodicallyperforateddomains AT rudakovaoa gconvergenceofintegralfunctionalswithdegenerateintegrandsinperiodicallyperforateddomains |
| first_indexed |
2025-07-09T00:30:05Z |
| last_indexed |
2025-07-09T00:30:05Z |
| _version_ |
1837127184813129728 |
| fulltext |
ISSN 1683-4720 Труды ИПММ НАН Украины. 2009. Том 19
УДК 517.9
c©2009. A.A. Kovalevsky, O.A. Rudakova
Γ-CONVERGENCE OF INTEGRAL FUNCTIONALSWITH DEGENERATE
INTEGRANDS IN PERIODICALLY PERFORATED DOMAINS
We consider a sequence of integral functionals with degenerate integrands in perforated domains of
periodic structure. We establish the Γ-convergence of the sequence under consideration to an integral
functional defined on a limit weighted Sobolev space. A representation formula for the integrand of the
Γ-limit functional is given.
1. Introduction. In this article we consider a sequence of integral functionals with
degenerate integrands in perforated domains of periodic structure. We establish the Γ-
convergence of the sequence under consideration to an integral functional defined on a
limit weighted Sobolev space. At the same time a representation formula for the integrand
of the Γ-limit functional is given.
We note that the Γ-convergence of functionals plays an important part in the study of
convergence of solutions to variational problems (see for instance [1], [3], [5], [8], [15] and
[16]). In particular, the questions related to the investigation of convergence of minimizers
and minimum values of functionals defined on variable weighted Sobolev spaces were
studied in [10]–[13].
The Γ-convergence of quadratic integral functionals having periodic quickly oscillating
coefficients and defined on a fixed weighted Sobolev space was proved in [2].
In the nonweighted case the Γ-convergence of integral functionals associated with
different kinds of periodically perforated domains was established for instance in [6] and
[9]. Moreover, in the nonweighted case representation formulae for coefficients of the
homogenized problem corresponding to the Neumann variational problems for quadratic
integral functionals in periodically perforated domains were given in [4].
Finally, we emphasize that the integral functionals under consideration in the present
article combine the following three features: their domains of definition depend on a
parameter; their integrands, having a quickly oscillating component, depend on the same
parameter; the integrands have a fixed weighted multiplier.
2. Preliminaries. Let Ω be a bounded domain of Rn (n > 2), p ∈ (1, n), and let ν
be a nonnegative function on Ω with the properties: ν > 0 almost everywhere in Ω and
ν ∈ L1
loc(Ω),
(
1
ν
)1/(p−1)
∈ L1
loc(Ω). (2.1)
We denote by Lp(ν, Ω) the set of all measurable functions u : Ω → R such that the
function ν|u|p is summable in Ω. Lp(ν, Ω) is a Banach space with the norm
‖u‖Lp(ν,Ω) =
(∫
Ω
ν|u|p dx
)1/p
.
101
A.A. Kovalevsky, O.A. Rudakova
We note that by virtue of Young’s inequality and the second inclusion of (2.1) we have
Lp(ν,Ω) ⊂ L1
loc(Ω).
We denote by W 1,p(ν, Ω) the set of all functions u ∈ Lp(ν, Ω) such that for every
i ∈ {1, . . . , n} there exists the weak derivative Diu, Diu ∈ Lp(ν, Ω). W 1,p(ν, Ω) is a
reflexive Banach space with the norm
‖u‖1,p,ν =
(∫
Ω
ν|u|p dx +
n∑
i=1
∫
Ω
ν|Diu|p dx
)1/p
.
Due to the first inclusion of (2.1) we have C∞
0 (Ω) ⊂ W 1,p(ν, Ω). We denote by
◦
W 1,p(ν, Ω)
the closure of the set C∞
0 (Ω) in W 1,p(ν,Ω).
Next, let {Ωs} be a sequence of domains of Rn which are contained in Ω.
By analogy with the spaces introduced above we define the functional spaces corres-
ponding to the domains Ωs.
Let s ∈ N. We denote by Lp(ν,Ωs) the set of all measurable functions u : Ωs → R
such that the function ν|u|p is summable in Ωs. Lp(ν, Ωs) is a Banach space with the
norm
‖u‖Lp(ν,Ωs) =
(∫
Ωs
ν|u|p dx
)1/p
.
By virtue of the second inclusion of (2.1) we have Lp(ν,Ωs) ⊂ L1
loc(Ωs). We denote by
W 1,p(ν, Ωs) the set of all functions u ∈ Lp(ν,Ωs) such that for every i ∈ {1, . . . , n} there
exists the weak derivative Diu, Diu ∈ Lp(ν, Ωs). W 1,p(ν, Ωs) is a Banach space with the
norm
‖u‖1,p,ν,s =
(∫
Ωs
ν|u|p dx +
n∑
i=1
∫
Ωs
ν|Diu|p dx
)1/p
.
We denote by C̃∞
0 (Ωs) the set of all restrictions on Ωs of functions from C∞
0 (Ω). Due to
the first inclusion of (2.1) we have C̃∞
0 (Ωs) ⊂ W 1,p(ν, Ωs). We denote by W̃ 1,p
0 (ν,Ωs) the
closure of the set C̃∞
0 (Ωs) in W 1,p(ν, Ωs).
We observe that if u ∈
◦
W 1,p(ν, Ω) and s ∈ N, then u|Ωs ∈ W̃ 1,p
0 (ν, Ωs).
Definition 2.1. If s ∈ N, qs is the mapping from
◦
W 1,p(ν, Ω) into W̃ 1,p
0 (ν,Ωs) such
that for every function u ∈
◦
W 1,p(ν, Ω), qsu = u|Ωs .
Definition 2.2. Let for every s ∈ N, Is be a functional on W̃ 1,p
0 (ν,Ωs), and let I be
a functional on
◦
W 1,p(ν,Ω). We say that the sequence {Is} Γ-converges to the functional
I if the following conditions are satisfied:
(i) for every function u ∈
◦
W 1,p(ν, Ω) there exists a sequence ws ∈ W̃ 1,p
0 (ν,Ωs) such
that lim
s→∞ ‖ws − qsu‖Lp(ν,Ωs) = 0 and lim
s→∞ Is(ws) = I(u);
(ii) for every function u ∈
◦
W 1,p(ν, Ω) and every sequence us ∈ W̃ 1,p
0 (ν,Ωs) such that
lim
s→∞ ‖us − qsu‖Lp(ν,Ωs) = 0 we have lim inf
s→∞ Is(us) > I(u).
102
Γ-convergence of integral functionals
The given definition was introduced in [10], and the corresponding Γ-compactness
theorem for integral functionals was established in [10] and [13].
Further, we shall use the following notation: for every i ∈ {1, . . . , n}, ei is the unit
vector of the ith axis in Rn; for every y ∈ Rn and ρ > 0, B(y, ρ) = {x ∈ Rn : |x−y| < ρ};
for every y ∈ Rn and t ∈ N,
Qt(y) = {x ∈ Rn : |xi − yi| < 1/(2t), i = 1, . . . , n}.
For every i ∈ {1, . . . , n} we set
Qi
− = {x ∈ ∂Q1(0) : xi = −1/2}, Qi
+ = {x ∈ ∂Q1(0) : xi = 1/2}.
Clearly, if i ∈ {1, . . . , n} and x ∈ Qi−, we have x + ei ∈ Qi
+.
For every function v ∈ C1(Q1(0)) we denote by v̄ the unique continuous extension of
v on Q1(0).
By C1
per(Q1(0)) we denote the set of all functions v ∈ C1(Q1(0)) such that for every
i ∈ {1, . . . , n} and x ∈ Qi−, v̄(x + ei) = v̄(x).
Next, we fix r ∈ (0, 1/2) and set Π = Q1(0) \ B(0, r). By C1
per(Π) we denote all
functions v ∈ C1(Π) such that v = w|Π, where w ∈ C1
per(Q1(0)). Finally, by W 1,p
per(Π) we
denote the closure of the set C1
per(Π) in W 1,p(Π).
Let ĉ1, ĉ2 > 0, ĉ > 0, and let f̂ : Rn×Rn → R be a Carathéodory function such that
the following conditions are satisfied:
for every ξ ∈ Rn the function f̂(·, ξ) is 1-periodic; (2.2)
for almost every x ∈ Rn the function f̂(x, ·) is convex in Rn; (2.3)
for almost every x ∈ Rn and every ξ ∈ Rn,
ĉ1|ξ|p − ĉ 6 f̂(x, ξ) 6 ĉ2|ξ|p + ĉ. (2.4)
Let f̃ : Rn → R be the function such that for every ξ ∈ Rn,
f̃(ξ) = inf
v∈W 1,p
per (Π)
∫
Π
f̂(x, ξ +∇v)dx.
From (2.4) it follows that for every ξ ∈ Rn,
−ĉmeasΠ 6 f̃(ξ) 6 (ĉ2|ξ|p + ĉ)measΠ. (2.5)
We also observe that owing to (2.3) the function f̃ is convex in Rn.
3. Statement of the main result. We shall assume that ν ∈ L1(Ω). We define
b = ĉν and for every s ∈ N we set ψs = b|Ωs . Moreover, we set
ē =
1
2
n∑
i=1
ei.
103
A.A. Kovalevsky, O.A. Rudakova
Now let for every s ∈ N, fs : Ωs × Rn → R be the function such that for every
(x, ξ) ∈ Ωs × Rn,
fs(x, ξ) = ν(x)f̂(sx− ē, ξ).
Clearly, for every s ∈ N and ξ ∈ Rn the function fs(·, ξ) is measurable in Ωs. Moreover,
owing to conditions (2.3) and (2.4) the following assertions hold true: for every s ∈ N
and almost every x ∈ Ωs the function fs(x, ·) is convex in Rn; for every s ∈ N, almost
every x ∈ Ωs and every ξ ∈ Rn we have
ĉ1ν(x)|ξ|p − ψs(x) 6 fs(x, ξ) 6 ĉ2ν(x)|ξ|p + ψs(x). (3.1)
Let for every s ∈ N, Js : W̃ 1,p
0 (ν, Ωs) → R be the functional such that for every
u ∈ W̃ 1,p
0 (ν,Ωs),
Js(u) =
∫
Ωs
fs(x,∇u)dx.
We denote by F the set of all functions f : Ω×Rn → R satisfying the conditions: for
every ξ ∈ Rn the function f(·, ξ) is measurable in Ω; for almost every x ∈ Ω the function
f(x, ·) is convex in Rn; for almost every x ∈ Ω and every ξ ∈ Rn we have
−b(x) 6 f(x, ξ) 6 ĉ2ν(x)|ξ|p + b(x).
Definition 3.1. If f ∈ F , Jf :
◦
W 1,p(ν, Ω) → R is the functional such that for every
u ∈
◦
W 1,p(ν,Ω),
Jf (u) =
∫
Ω
f(x,∇u)dx.
Let f̄ : Ω× Rn → R be the function such that for every (y, ξ) ∈ Ω× Rn,
f̄(y, ξ) = ν(y)f̃(ξ).
Observe that due to (2.5) and the convexity of the function f̃ we have f̄ ∈ F .
In what follows we shall suppose that
Ω = {x ∈ Rn : |xi| < 1, i = 1, . . . , n}.
For every s ∈ N we set Z̃s = {z ∈ Ω : szi − 1/2 ∈ Z, i = 1, . . . , n}. We have
∀ s ∈ N,
⋃
z∈Z̃s
Qs(z) = Ω, (3.2)
∀ s ∈ N, ∀ z, z′ ∈ Z̃s, z 6= z′, Qs(z) ∩Qs(z′) = ∅. (3.3)
We shall assume that the domains Ωs have the following structure: for every s ∈ N,
Ωs = Ω \
⋃
z∈Z̃s
B(z, r/s).
104
Γ-convergence of integral functionals
Theorem 3.2. Suppose that the function ν is positive and continuous in Ω \ {0}.
Then the sequence {Js} Γ-converges to the functional J f̄ .
4. Scheme of the proof of Theorem 3.2. Step 1. For every k ∈ N we set
Ω(k) = {x ∈ Rn : |xi| < 1− 1/(2k), i = 1, . . . , n} \Q2k(0).
Evidently, {Ω(k)} is a sequence of nonempty open sets of Rn, and the following assertions
hold true: for every k ∈ N, Ω(k) ⊂ Ω(k+1) ⊂ Ω; meas(Ω \ Ω(k)) → 0; for every k ∈ N the
functions ν and b are bounded in Ω(k).
These assertions along with the properties of the functions b, ψs and fs provide the
fulfilment of all the conditions under which in [13] Theorem 2 on the Γ-compactness of
a sequence of integral functionals was proved. Thus, some necessary constructions given
in the proof of this theorem may be utilized. These ones are as follows.
A. For every t ∈ N we set Yt = {y ∈ Rn : tyi ∈ Z, i = 1, . . . , n}. Observe that
∀ t ∈ N,
⋃
y∈Yt
Qt(y) = Rn;
∀ t ∈ N, ∀ y, y′ ∈ Yt, y 6= y′, Qt(y) ∩Qt(y′) = ∅.
For every t ∈ N we define Y ′
t = {y ∈ Yt : Qt(y) ⊂ Ω}. Obviously, there exists t0 ∈ N
such that for every t ∈ N, t > t0, the set Y ′
t is nonempty.
Let for every t ∈ N, t > t0, s ∈ N and y ∈ Y ′
t ,
Vt,s(y) =
{
u ∈ W̃ 1,p
0 (ν, Ωs) :
∫
Qt(y)∩Ωs
ν|u|p dx 6 t−n−3p
}
.
Now for every t ∈ N, t > t0, s ∈ N, y ∈ Y ′
t and ξ ∈ Rn we set
Ft,s(y, ξ) = tn inf
u∈Vt,s(y)
∫
Qt(y)∩Ωs
fs(x, ξ +∇u)dx.
B. Let {s̄k} ⊂ N be an arbitrary increasing sequence. From (3.1) and the convexity
of the functions fs(x, ·) for almost every x ∈ Ωs it follows that there exist an increasing
sequence {sj} ⊂ {s̄k} and a sequence of functions Φt : Rn ×Rn → R such that for every
t ∈ N, t > t0, y ∈ Y ′
t and ξ ∈ Rn we have
lim
j→∞
Ft,sj (y, ξ) = Φt(y, ξ). (3.4)
C. Let for every t ∈ N and y ∈ Ω such that Qt(y) ⊂ Ω, χt,y : Ω → R be the
characteristic function of the set Qt(y).
For every k, t ∈ N we set Yk,t = {y ∈ Yt : Qt(y) ⊂ Ω(k)}.
Let us give the following definition: if k, t ∈ N and Yk,t 6= ∅, H
(k)
t is the function on
Ω× Rn such that for every pair (x, ξ) ∈ Ω× Rn,
H
(k)
t (x, ξ) =
∑
y∈Yk,t
χt,y(x)Φt(y, ξ);
105
A.A. Kovalevsky, O.A. Rudakova
if k, t ∈ N and Yk,t = ∅, H
(k)
t is the function on Ω × Rn such that for every pair
(x, ξ) ∈ Ω× Rn, H
(k)
t (x, ξ) = 0.
D. In accordance with the considerations given within steps 4–11 of the proof of
Theorem 2 in [13] there exist an increasing sequence {ti} ⊂ N and a Carathéodory
function f : Ω× Rn → R such that f ∈ F and the following assertions hold:
k ∈ N, ξ ∈Rn, ϕ ∈ L∞(Ω) ⇒ lim
i→∞
∫
Ω(k)
H
(k)
ti
(·, ξ)ϕ dx =
∫
Ω(k)
f(·, ξ)ϕdx; (3.5)
the sequence {Jsj} Γ-converges to the functional Jf . (3.6)
Now the aim is to prove that for almost every x ∈ Ω and every ξ ∈ Rn, f(x, ξ) = f̄(x, ξ).
Step 2. Taking into account the inclusions ν ∈ L1(Ω) and f ∈ F , we establish that
there exists a set E ⊂ Ω with measure zero such that
for every z ∈ Ω \E and ξ ∈ Rn, τn
∫
Qτ (z)
f(·, ξ)dx → f(z, ξ). (3.7)
Step 3. We fix z0 ∈ Ω \ (E ∪ {0}) and ξ ∈ Rn. Clearly, there exists τ0 ∈ N such that
0 /∈ Qτ0(z0) and Qτ0(z0) ⊂ Ω. Then there exists k ∈ N such that Qτ0(z0) ⊂ Ω(k). Since
the function ν is positive and continuous in Ω \ {0}, there exists Mk > 0 such that
∀x ∈ Ω(k), 1/Mk 6 ν(x) 6 Mk. (3.8)
We fix ε > 0. Due to the continuity of ν in Ω \ {0} there exists δ > 0 such that
for every x′, x′′ ∈ Ω(k), |x′ − x′′| 6 δ, we have |ν(x′)− ν(x′′)| 6 ε. (3.9)
Let τ ∈ N, τ > τ0 + 1 + n/δ. We fix t ∈ N such that t > max{t0, 2τ(τ − 1)} and
define Xt = {y ∈ Yt : Qt(y) ∩Qτ (z0) 6= ∅}. It is easy to see that Xt 6= ∅.
Step 4. We fix y ∈ Xt and take a function w ∈ W 1,p
per(Π) such that
f̃(ξ) =
∫
Π
f̂(x, ξ +∇w)dx. (3.10)
The existence of such a function follows from (2.3), (2.4) and the known results on the
existence of minimizers of functionals (see for instance [14]).
Finally, we fix s0 ∈ N such that
2ns−p
0 Mk
∫
Π
|w|p dx 6 t−n−3p,
and after that fix s ∈ N, s > max{s0, 2t}.
Let ws : Ωs → R be a function such that
ws(x) = s−1w(s(x− z)) if z ∈ Z̃s and x ∈ Qs(z) \B(z, r/s).
106
Γ-convergence of integral functionals
Taking into account assertions (3.2) and (3.3) and the inclusion w ∈ W 1,p
per(Π), we establish
that ws ∈ W 1,p(Ωs). Then involving into consideration the function wsϕt, where ϕt is a
function in C∞
0 (Ω) such that ϕt = 1 in Qt(y), we obtain the inequality
Ft,s(y, ξ) 6 tn
∫
Qt(y)∩Ωs
fs(x, ξ +∇ws)dx. (3.11)
Using the definitions of the functions ws, fs and f̄ along with (2.2), (2.4) and (3.8)–(3.11),
we get
Ft,s(y, ξ)− f̄(y, ξ) 6 2nn(ĉ2/ĉ1 + 1)(|f̃(ξ)|+ ĉ)Mk(ε + ts−1). (3.12)
Step 5. With the use of the definition of Ft,s(y, ξ), properties of the function fs and
(3.8) we establish that there exists a function vs ∈ C̃∞
0 (Ωs) such that vs = 0 in Ωs \Qt(y)
and ∫
Qt(y)∩Ωs
fs(x, ξ +∇vs)dx 6 t−nFt,s(y, ξ) + ĉ3(1 + |ξ|p)Mkt
−n−1, (3.13)
where ĉ3 is a positive constant depending only on n, p, ĉ1, ĉ2 and ĉ.
Step 6. By means of the function vs we construct a function belonging to C1
per(Π).
We need this in order to obtain a suitable estimate from below for the left-hand side of
inequality (3.13).
First, we observe that due to the inclusion vs ∈ C̃∞
0 (Ωs) there exists a function
v ∈ C1(Rn) such that supp v ⊂ Ω and vs = v|Ωs .
We define Z̃ ′s = {z ∈ Z̃s : Qs(z)∩Qt(y) 6= ∅}. Owing to (3.2) the set Z̃ ′s is nonempty.
We denote by n′s the number of elements of the set Z̃ ′s.
For every z ∈ Z̃ ′s we define the function gs,z : Π → R by gs,z(x) = sv(s−1x + z),
x ∈ Π, and after that we set
gs =
1
n′s
∑
z∈Z̃′s
gs,z.
With the use of considerations analogous to those given in the proof of Lemma 2.2.1 of
[7] we establish that gs ∈ C1
per(Π). Therefore, gs ∈ W 1,p
per(Π). Then, taking into account
the definitions of the functions f̃ and gs and (2.3), we get
f̃(ξ) 6
∫
Π
f̂(x, ξ +∇gs)dx 6 1
n′s
∑
z∈Z̃′s
∫
Π
f̂(x, ξ +∇gs,z)dx. (3.14)
Moreover, using (2.2), (2.4), (3.8) and (3.9), we obtain that for every z ∈ Z̃ ′s,
ν(y)
∫
Π
f̂(x, ξ +∇gs,z)dx 6 sn
∫
Qs(z)\B(z,r/s)
fs(x, ξ +∇vs)dx
+ εĉ2Mks
n
∫
Qs(z)\B(z,r/s)
ν|ξ +∇vs|p dx + εĉM2
k . (3.15)
107
A.A. Kovalevsky, O.A. Rudakova
From (3.13)–(3.15), taking into account (3.1), (3.8) and the definitions of f̄ and
Ft,s(y, ξ), we get the inequality
f̄(y, ξ)− Ft,s(y, ξ) 6 ĉ4(1 + |ξ|p)M2
k (ε + t−1 + ts−1), (3.16)
where ĉ4 is a positive constant depending only on n, p, ĉ1, ĉ2 and ĉ.
Step 7. From (2.5), (3.12) and (3.16) we deduce that
|Ft,s(y, ξ)− f̄(y, ξ)| 6 ĉ5(1 + |ξ|p)M2
k (ε + t−1 + ts−1),
where ĉ5 is a positive constant depending only on n, p, ĉ1, ĉ2 and ĉ.
Hence, taking into account that s is an arbitrary natural number greater than or
equal to max{s0, 2t} and using (3.4), we infer that for every y ∈ Xt,
|Φt(y, ξ)− f̄(y, ξ)| 6 ĉ5(1 + |ξ|p)M2
k (ε + t−1). (3.17)
Step 8. Taking into account the definition of the function H
(k)
t and the equality
∑
y∈Xt
meas[Qτ (z0) ∩Qt(y)] = τ−n, (3.18)
we obtain that
∣∣∣∣
∫
Qτ (z0)
H
(k)
t (·, ξ)dx− f̄(z0, ξ)τ−n
∣∣∣∣
6
∑
y∈Xt
|Φt(y, ξ)− f̄(y, ξ)|meas[Qτ (z0) ∩Qt(y)]
+
∑
y∈Xt
|f̄(y, ξ)− f̄(z0, ξ)|meas[Qτ (z0) ∩Qt(y)].
Hence, taking into account (3.17), (3.18), (3.9), the inequality τ > 1 + n/δ and the
definition of the function f̄ , we derive that for every t ∈ N, t > max{t0, 2τ(τ − 1)},
∣∣∣∣
∫
Qτ (z0)
H
(k)
t (·, ξ)dx− f̄(z0, ξ)τ−n
∣∣∣∣ 6 (ĉ5 + 1)(1 + |ξ|p + |f̃(ξ)|)M2
k (ε + t−1)τ−n.
This and (3.5) imply that for every τ ∈ N, τ > τ0 + 1 + n/δ,
∣∣∣∣τn
∫
Qτ (z0)
f(·, ξ)dx− f̄(z0, ξ)
∣∣∣∣ 6 (ĉ5 + 1)(1 + |ξ|p + |f̃(ξ)|)M2
kε.
Hence, using (3.7) and after that taking into account the arbitrariness of ε > 0, we
obtain f(z0, ξ) = f̄(z0, ξ).
Thus, for almost every x ∈ Ω and every ξ ∈ Rn we have f(x, ξ) = f̄(x, ξ). Therefore,
Jf = J f̄ . From this and (3.6) it follows that the sequence {Jsj} Γ-converges to the
functional J f̄ .
108
Γ-convergence of integral functionals
Step 9. The result obtained allows us to affirm that the following assertion holds true:
for every increasing sequence {s̄k} ⊂ N there exists an increasing sequence {sj} ⊂ {s̄k}
such that the sequence {Jsj} Γ-converges to the functional J f̄ . Hence we deduce the
conclusion of the theorem.
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Ин-т прикл. математики и механики НАН Украины, Донецк
alexkvl@iamm.ac.donetsk.ua, rudakova@iamm.ac.donetsk.ua
Получено 01.12.09
109
титул
Научное издание
Том 19
содержание
Том 19
Донецк, 2009
Основан в 1997г.
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