On the convergence of solutions of the variational problems with integral constraints and degeneration in variable domains
In this article we deal with a sequence of integral functionals defined on weighted Sobolev spaces associated with a sequence of n-dimensional domains. For the given functionals we consider variational problems with sets of constraints of an integral kind. We establish sufficient conditions of conve...
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| Cite this: | On the convergence of solutions of the variational problems with integral constraints and degeneration in variable domains / O.A. Rudakova // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2013. — Т. 26. — С. 172-180. — Бібліогр.: 14 назв. — англ. |
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Rudakova, O.A. 2017-09-21T16:11:23Z 2017-09-21T16:11:23Z 2013 On the convergence of solutions of the variational problems with integral constraints and degeneration in variable domains / O.A. Rudakova // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2013. — Т. 26. — С. 172-180. — Бібліогр.: 14 назв. — англ. 1683-4720 https://nasplib.isofts.kiev.ua/handle/123456789/124166 517.9 In this article we deal with a sequence of integral functionals defined on weighted Sobolev spaces associated with a sequence of n-dimensional domains. For the given functionals we consider variational problems with sets of constraints of an integral kind. We establish sufficient conditions of convergence of minimizers and minimum values of the variational problems under consideration. В настоящей статье для последовательности интегральных функционалов, определенных на весовых пространствах Соболева, связанных с последовательностью n-мерных областей, рассмотрены вариационные задачи с множествами ограничений интегрального вида. Установлены достаточные условия сходимости минимизантов и минимальных значений рассматриваемых вариационных задач. Для послiдовностi iнтегральних функцiоналiв, визначених на вагових просторах Соболєва, пов’язаних з послiдовнiстю n-вимiрних областей, розглянуто варiацiйнi задачi з множинами обмежень iнтегрального вигляду. Встановлено достатнi умови збiжностi мiнiмiзантiв i мiнiмальних значень розглянутих варiацiйних задач. en Інститут прикладної математики і механіки НАН України Труды Института прикладной математики и механики On the convergence of solutions of the variational problems with integral constraints and degeneration in variable domains О сходимости решений вариационных задач с интегральными ограничениями и вырождением в переменных областях Про збiжнiсть розв’язкiв варiацiйних задач з iнтегральними обмеженнями i виродженням в змiнних областях published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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On the convergence of solutions of the variational problems with integral constraints and degeneration in variable domains |
| spellingShingle |
On the convergence of solutions of the variational problems with integral constraints and degeneration in variable domains Rudakova, O.A. |
| title_short |
On the convergence of solutions of the variational problems with integral constraints and degeneration in variable domains |
| title_full |
On the convergence of solutions of the variational problems with integral constraints and degeneration in variable domains |
| title_fullStr |
On the convergence of solutions of the variational problems with integral constraints and degeneration in variable domains |
| title_full_unstemmed |
On the convergence of solutions of the variational problems with integral constraints and degeneration in variable domains |
| title_sort |
on the convergence of solutions of the variational problems with integral constraints and degeneration in variable domains |
| author |
Rudakova, O.A. |
| author_facet |
Rudakova, O.A. |
| publishDate |
2013 |
| language |
English |
| container_title |
Труды Института прикладной математики и механики |
| publisher |
Інститут прикладної математики і механіки НАН України |
| title_alt |
О сходимости решений вариационных задач с интегральными ограничениями и вырождением в переменных областях Про збiжнiсть розв’язкiв варiацiйних задач з iнтегральними обмеженнями i виродженням в змiнних областях |
| description |
In this article we deal with a sequence of integral functionals defined on weighted Sobolev spaces associated with a sequence of n-dimensional domains. For the given functionals we consider variational problems with sets of constraints of an integral kind. We establish sufficient conditions of convergence of minimizers and minimum values of the variational problems under consideration.
В настоящей статье для последовательности интегральных функционалов, определенных на весовых пространствах Соболева, связанных с последовательностью n-мерных областей, рассмотрены вариационные задачи с множествами ограничений интегрального вида. Установлены достаточные условия сходимости минимизантов и минимальных значений рассматриваемых вариационных задач.
Для послiдовностi iнтегральних функцiоналiв, визначених на вагових просторах Соболєва, пов’язаних з послiдовнiстю n-вимiрних областей, розглянуто варiацiйнi задачi з множинами обмежень iнтегрального вигляду. Встановлено достатнi умови збiжностi мiнiмiзантiв i мiнiмальних значень розглянутих варiацiйних задач.
|
| issn |
1683-4720 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/124166 |
| citation_txt |
On the convergence of solutions of the variational problems with integral constraints and degeneration in variable domains / O.A. Rudakova // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2013. — Т. 26. — С. 172-180. — Бібліогр.: 14 назв. — англ. |
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| first_indexed |
2025-11-24T16:07:11Z |
| last_indexed |
2025-11-24T16:07:11Z |
| _version_ |
1850482485826158592 |
| fulltext |
ISSN 1683-4720 Труды ИПММ НАН Украины. 2013. Том 26
UDK 517.9
c©2013. O.A. Rudakova
ON THE CONVERGENCE OF SOLUTIONS OF THE VARIATIONAL
PROBLEMS WITH INTEGRAL CONSTRAINTS AND DEGENERATION
IN VARIABLE DOMAINS
In this article we deal with a sequence of integral functionals defined on weighted Sobolev spaces
associated with a sequence of n-dimensional domains. For the given functionals we consider variational
problems with sets of constraints of an integral kind. We establish sufficient conditions of convergence
of minimizers and minimum values of the variational problems under consideration.
Keywords: varying weighted Sobolev spaces, variational problem, integral functional, degeneration,
integral constraint, Γ-convergence.
1. Introduction. In this article for a sequence of integral functionals defined on
varying weighted Sobolev spaces we consider variational problems with sets of constraints
of an integral kind. The strong connectedness of the given weighted Sobolev spaces with a
"limit" weighted Sobolev space and the Γ-convergence of the involved integral functionals
are the main conditions under which we establish the convergence of the minimizers and
minimum values of the given variational problems.
The role of the strong connectedness of the corresponding spaces in the study of
the homogenization of boundary value problems and variational problems in variable
domains (particularly, in strongly perforated domains) is well known (see for instance
[7-12]). The notion of the strong connectedness used in the present work was introduced
and studied in [11].
The Γ-convergence is a special kind of the convergence introduced by E. De Giorgi in
the seventies of last century to propose a framework for the study of the asymptotic
behaviour of families of minimum problems. Its first definition as well as the main
properties were presented in [5]. There are many works devoted to the Γ-convergence
of functionals including integral functionals with degenerate integrands and variable
domains of definitions (see for instance [1-6], [8], [9] and [11-13] and the bibliography
in [1], [2]). We note that the effectiveness of the Γ-convergence in the study of the
homogenization of variational problems is connected with the possibility of obtaining
converging subsequences from sequences of minimizers of minimum problems.
2. Preliminaries. Let n ∈ N, n > 2, and let Ω be a bounded domain of Rn. Let
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). (1)
We denote by Lp(ν,Ω) the set of all measurable functions u : Ω → R such that the
172
On the convergence of solutions of the variational problems with integral constraints
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
.
Note that by virtue of Young’s inequality and the second inclusion of (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 (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 Young’s inequality and the second inclusion of (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 (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 1. If s ∈ N, then qs :
◦
W 1,p(ν,Ω) → W̃ 1,p
0 (ν,Ωs) is the mapping such that
for every function u ∈
◦
W 1,p(ν, Ω), qsu = u|Ωs .
Definition 2.We say that the sequence of the spaces W̃ 1,p
0 (ν, Ωs) is strongly connected
with the space
◦
W 1,p(ν, Ω) if there exists a sequence of linear continuous operators ls :
W̃ 1,p
0 (ν, Ωs) →
◦
W 1,p(ν, Ω) such that:
173
O.A. Rudakova
(i) the sequence of the norms ‖ls‖ is bounded;
(ii) for every s ∈ N and for every u ∈ W̃ 1,p
0 (ν, Ωs) we have qs(lsu) = u a. e. in Ωs.
Proposition 1. Suppose that the embedding of
◦
W 1,p(ν,Ω) into Lp(ν,Ω) is compact,
and the sequence of the spaces W̃ 1,p
0 (ν, Ωs) is strongly connected with the space
◦
W 1,p(ν,Ω).
Let for every s ∈ N, us ∈ W̃ 1,p
0 (ν,Ωs), and let the sequence of the norms ‖us‖1,p,ν,s be
bounded. Then there exist an increasing sequence {sj} ⊂ N and a function u ∈
◦
W 1,p(ν, Ω)
such that lim
j→∞
‖usj − qsju‖Lp(ν,Ωsj ) = 0.
The proof of the proposition is simple (see [11]).
Definition 3. 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 for 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).
3. Main result. Let c1, c2 > 0, and let for every s ∈ N, ψs ∈ L1(Ωs) and ψs > 0 in
Ωs. We shall assume that
the sequence of the norms ‖ψs‖L1(Ωs) is bounded. (2)
Let fs : Ωs × Rn → R, s ∈ N, be a sequence of functions such that:
for every s ∈ N and for every ξ ∈ Rn the function fs(·, ξ) is measurable in Ωs; (3)
for every s ∈ N and for almost every x ∈ Ωs the function fs(x, ·) is convex in Rn;
(4){
for every s ∈ N, for almost every x ∈ Ωs and for every ξ ∈ Rn,
c1ν(x)|ξ|p − ψs(x) 6 fs(x, ξ) 6 c2ν(x)|ξ|p + ψs(x).
(5)
From (3)-(5) it follows that for every s ∈ N, fs is a Carathéodory function and if
s ∈ N and u ∈ W̃ 1,p
0 (ν,Ωs), the function fs(x,∇u) is summable in Ωs.
For every s ∈ N we define the functional Js : W̃ 1,p
0 (ν,Ωs) → R by
Js(u) =
∫
Ωs
fs(x,∇u) dx, u ∈ W̃ 1,p
0 (ν, Ωs).
Next, let c3, c4 > 0, and let ψ be a function in L1(Ω) such that ψ > 0 in Ω.
Let g : Ω× R→ R be a function such that:
for every η ∈ R the function g(·, η) is measurable in Ω; (6)
174
On the convergence of solutions of the variational problems with integral constraints
for almost every x ∈ Ω the function g(x, ·) is strictly convex in R; (7)
{
for almost every x ∈ Ω and for every η ∈ R,
c3ν(x)|η|p − ψ(x) 6 g(x, η) 6 c4ν(x)|η|p + ψ(x).
(8)
From (6)-(8) it follows that g is a Carathéodory function and if s ∈ N and u ∈
W̃ 1,p
0 (ν, Ωs), the function g(x, u) is summable in Ωs. Moreover, for every u ∈
◦
W 1,p(ν,Ω),
the function g(x, u) is summable in Ω.
For every s ∈ N we define the functional Gs : W̃ 1,p
0 (ν, Ωs) → R by
Gs(u) =
∫
Ωs
g(x, u) dx, u ∈ W̃ 1,p
0 (ν, Ωs).
We observe that by virtue of (2), (4), (5) and (7), (8) for every s ∈ N the functional
Js + Gs is weakly lower semicontinuous on W̃ 1,p
0 (ν, Ωs), strictly convex and there exist
c′, c′′, c′′′ > 0 such that for every s ∈ N and u ∈ W̃ 1,p
0 (ν, Ωs),
c′‖u‖p
1,p,ν,s − c′′′ 6 (Js + Gs)(u) 6 c′′‖u‖p
1,p,ν,s + c′′′. (9)
In view of known results on the existence of the minimizers of functionals (see for instance
[14]), these properties of the functionals Js +Gs imply that the next assertion holds true:
{
if s ∈ N and U is a nonempty convex and closed set in W̃ 1,p
0 (ν, Ωs),
there exists a unique function u ∈ U minimizing the functional Js + Gs on U.
(10)
Further, let c > 0, b ∈ L1(Ω), b > 0 in Ω, and let ϕ : Ω× R→ R be a Carathéodory
function such that:
for almost every x ∈ Ω the function ϕ(x, ·) is convex in R; (11)
for almost every x ∈ Ω ϕ(x, 0) = 0; (12)
{
for almost every x ∈ Ω and for every η ∈ R,
|ϕ(x, η)| 6 cν(x)|η|p + b(x).
(13)
For every s ∈ N we define the functional Φs : W̃ 1,p
0 (ν,Ωs) → R by
Φs(u) =
∫
Ωs
ϕ(x, u) dx, u ∈ W̃ 1,p
0 (ν, Ωs).
Using (8) and (13) along with Egoroff’s theorem, we establish the following fact:
{
for every v ∈
◦
W 1,p(ν, Ω) and for every sequence vs ∈ W̃ 1,p
0 (ν, Ωs) such that
‖vs − qsv‖Lp(ν,Ωs) → 0 we have Gs(vs)−Gs(qsv) → 0 and Φs(vs)− Φs(qsv) → 0.
(14)
175
O.A. Rudakova
For every s ∈ N we define
Vs = {u ∈ W̃ 1,p
0 (ν, Ωs) : Φs(u) 6 1}.
Due to (12) for every s ∈ N the set Vs is nonempty.
For every function σ ∈ L∞(Ω) we define the functionals Gσ, Φσ :
◦
W 1,p(ν, Ω) → R by
Gσ(u) =
∫
Ω
σg(x, u) dx, Φσ(u) =
∫
Ω
σϕ(x, u) dx, u ∈
◦
W 1,p(ν,Ω),
and we set
V σ = {u ∈
◦
W 1,p(ν,Ω) : Φσ(u) 6 1}.
In view of (11) for every s ∈ N the set Vs is convex. This fact and (10) imply that for
every s ∈ N there exists a unique function us ∈ Vs minimizing the functional Js + Gs on
Vs.
Theorem 1. Suppose that the following conditions are satisfied:
(∗1) the embedding of
◦
W 1,p(ν, Ω) into Lp(ν,Ω) is compact;
(∗2) the sequence of the spaces W̃ 1,p
0 (ν,Ωs) is strongly connected with
◦
W 1,p(ν, Ω);
(∗3) there exists a positive bounded measurable function σ on Ω such that for every
open cube Q ⊂ Ω,
lim
s→∞meas(Q ∩ Ωs) =
∫
Q
σ dx;
(∗4) the sequence {Js} Γ-converges to a functional J :
◦
W 1,p(ν, Ω) → R.
Assume that for every s ∈ N, us is the function in Vs minimizing the functional Js+Gs
on Vs.
Then there exists a function u ∈ V σ such that the following assertions hold true:
the function u is a unique minimizer of the functional J + Gσ on V σ; (15)
lim
s→∞ ‖us − qsu‖Lp(ν,Ωs) = 0; (16)
lim
s→∞(Js + Gs)(us) = (J + Gσ)(u). (17)
Proof. We observe that in view of condition (∗3) for every function v ∈
◦
W 1,p(ν, Ω)
Gs(qsv) → Gσ(v) and Φs(qsv) → Φσ(v). This and (14) imply that
{
for every v ∈
◦
W 1,p(ν, Ω) and for every sequence vs ∈ W̃ 1,p
0 (ν, Ωs) such that
‖vs − qsv‖Lp(ν,Ωs) → 0 we have Gs(vs) → Gσ(v) and Φs(vs) → Φσ(v).
(18)
For every s ∈ N we set Is = Js + Gs and I = J + Gσ. From condition (∗4) and
assertion of (18) it follows that
the sequence {Is} Г-converges to the functional I. (19)
176
On the convergence of solutions of the variational problems with integral constraints
Moreover, taking into account that for every s ∈ N the function us minimizes the
functional Js + Gs on Vs and using (9) we obtain that
the sequence of the norms ‖us‖1,p,ν,s is bounded. (20)
This fact along with conditions (∗1), (∗2) and Proposition 1 implies that there exist an
increasing sequence {sj} ⊂ N and a function u ∈
◦
W 1,p(ν, Ω) such that
lim
j→∞
‖usj − qsju‖Lp(ν,Ωsj ) = 0. (21)
Now we define the sequence {ūs} by
ūs =
{
us if s = sj for some j ∈ N,
qsu if s 6= sj for every j ∈ N.
It is evident that for every s ∈ N, ūs ∈ W̃ 1,p
0 (ν, Ωs). Due to (21) we have
lim
s→∞ ‖ūs − qsu‖Lp(ν,Ωs) = 0.
Then by virtue of assertions (18) and (19),
lim
s→∞Φs(ūs) = Φσ(u), (22)
lim inf
s→∞ Is(ūs) > I(u). (23)
Due to (22)
lim
j→∞
Φsj (usj ) = Φσ(u).
Taking into account that for every s ∈ N, Φs(us) 6 1, from the latter equality we derive
that Φσ(u) 6 1. Consequently, u ∈ V σ. Moreover, from (23) we obtain
lim inf
j→∞
Isj (usj ) > I(u). (24)
Further, we fix v ∈ V σ. Let us show that
lim sup
s→∞
Is(us) 6 I(v). (25)
From (19) it follows that there exists a sequence vs ∈ W̃ 1,p
0 (ν,Ωs) such that
lim
s→∞ ‖vs − qsv‖Lp(ν,Ωs) = 0, (26)
lim
s→∞ Is(vs) = I(v). (27)
For every s ∈ N we set τs = (1 + |Φs(vs) − Φσ(v)|)−1. From (18) and (26) it follows
that
lim
s→∞ τs = 1. (28)
177
O.A. Rudakova
For every s ∈ N we define ws = τsvs. Clearly, ws ∈ W̃ 1,p
0 (ν,Ωs), s ∈ N. Moreover,
using (11) and (12), and the inclusion v ∈ V σ, we establish that for every s ∈ N,
Φs(ws) 6 τsΦs(vs) 6 τs(1 + |Φs(vs)− Φσ(v)|) 6 1.
This implies that for every s ∈ N, ws ∈ Vs. Then, taking into account that for every
s ∈ N the function us minimizes the functional Is on Vs, we get
∀ s ∈ N, Is(us) 6 Is(ws). (29)
Using (4), (5), (7) and (8), we obtain that for every s ∈ N,
Is(ws) 6 τsIs(vs) + (1− τs)(‖ψs‖L1(Ωs) + ‖ψ‖L1(Ω)).
This along with (2) and (27)-(29) implies that inequality (25) is valid. From this inequality
and inequality (24) we infer that the function u minimizes the functional I on V σ.
We observe that, due to (4), (7), condition (∗4) and the fact that the function σ is
positive, the functional I is strictly convex. Therefore, since the set V σ is convex, the
function u is the unique minimizer of the functional I on V σ. Thus, assertion (15) holds
true.
Next, let us show that assertion (16) holds true. For every s ∈ N we define αs =
‖us−qsu‖Lp(ν,Ωs). Suppose that the sequence {αs} does not converge to zero. Then there
exist ε > 0 and an increasing sequence {s̄k} ⊂ N such that
∀ k ∈ N, αs̄k
> ε. (30)
Taking into account (20) and conditions (∗1) and (∗2), we establish that there exist an
increasing sequence {s̃j} ⊂ {s̄k} and a function w ∈
◦
W 1,p(ν, Ω) such that
lim
j→∞
‖us̃j − qs̃jw‖Lp(ν,Ωs̃j
) = 0. (31)
Thus, by analogy with the above result for the function u, we obtain that w ∈ V σ and w
minimizes the functional I on V σ. This fact along with the uniqueness of the minimizer
of the functional I on V σ allows us to deduce that w = u a. e. in Ω. Hence, by (31),
αs̃j → 0. However, this contradicts (30). The contradiction obtained proves that αs → 0.
Thus, assertion (16) holds true.
Now, from (19) and (16) we get
lim inf
s→∞ Is(us) > I(u).
This and (25), with v = u, imply that assertion (17) holds true. The theorem is proved.
We note that the convergence of minimizers and minimum values of variational
problems with certain pointwise constraints for a sequence of functionals like Js + Gs
was studied in [12]. Moreover, in [12] a rather extensive review of the works related to
the topic is contained.
178
On the convergence of solutions of the variational problems with integral constraints
In the nondegenerate case, the questions concerning convergence of minimizers of
variational problems with general sets of constraints, and in particular sets of constraints
of an integral kind, for integral functionals defined on varying Sobolev spaces were studied
in [8]. In this connection see also Subsections 1.4 and 2.5 of [10] where convergenсe
of solutions of variational inequalities with different kinds of sets of constraints was
investigated.
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О.А. Рудакова
О сходимости решений вариационных задач с интегральными ограничениями и вы-
рождением в переменных областях.
В настоящей статье для последовательности интегральных функционалов, определенных на весо-
вых пространствах Соболева, связанных с последовательностью n-мерных областей, рассмотрены
вариационные задачи с множествами ограничений интегрального вида. Установлены достаточные
условия сходимости минимизантов и минимальных значений рассматриваемых вариационных за-
дач.
Ключевые слова: переменные весовые пространства Соболева, вариационная задача, инте-
179
O.A. Rudakova
гральный функционал, вырождение, интегральное ограничение, Γ-сходимость.
О.А. Рудакова
Про збiжнiсть розв’язкiв варiацiйних задач з iнтегральними обмеженнями i вироджен-
ням в змiнних областях.
Для послiдовностi iнтегральних функцiоналiв, визначених на вагових просторах Соболєва, пов’я-
заних з послiдовнiстю n-вимiрних областей, розглянуто варiацiйнi задачi з множинами обмежень
iнтегрального вигляду. Встановлено достатнi умови збiжностi мiнiмiзантiв i мiнiмальних значень
розглянутих варiацiйних задач.
Ключовi слова: змiннi ваговi простори Соболєва, варiацiйна задача, iнтегральний функцiонал,
виродження, iнтегральне обмеження, Γ-збiжнiсть.
Institute of Applied Mathematics and Mechanics of NAS of Ukraine
rudakova@iamm.ac.donetsk.ua
Received 03.06.13
180
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