On the Г-compactness of integral functionals with degenerate locally Lipschitz integrands and varying domains of definition
In this Note, we announce a new Г-compactness result for a sequence of integral functionals defined on varying weighted Sobolev spaces. The result concerns the case where the degenerate integrands of the functionals satisfy a local Lipschitz condition but in general may not be convex with respect to...
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irk-123456789-1241192017-09-21T03:03:03Z On the Г-compactness of integral functionals with degenerate locally Lipschitz integrands and varying domains of definition Kovalevsky, A.A. Rudakova, O.A. In this Note, we announce a new Г-compactness result for a sequence of integral functionals defined on varying weighted Sobolev spaces. The result concerns the case where the degenerate integrands of the functionals satisfy a local Lipschitz condition but in general may not be convex with respect to the variable corresponding to the gradient of functions from domains of definition of the functionals. В заметке анонсирован новый результат о Г-компактности последовательности интегральных функционалов, определенных на переменных весовых пространствах Соболева. Этот результат относится к случаю, когда вырождающиеся интегранты функционалов удовлетворяют локальному условию Липшица, но, вообще говоря, могут не быть выпуклыми относительно переменной, соответствующей градиенту функций из областей определения функционалов. В замiтцi анонсований новий результат про Г-компактнiсть послiдовностi iнтегральних функцiоналiв, визначених на змiнних вагових просторах Соболєва. Цей результат належить випадку, коли вироднi iнтегранти функцiоналiв задовольняють локальну умову Лiпшiца, але, взагалi кажучи, можуть не бути опуклими вiдносно змiнної, що вiдповiдає градiєнту функцiй з областей визначення функцiоналiв. 2012 Article On the Г-compactness of integral functionals with degenerate locally Lipschitz integrands and varying domains of definition / A.A. Kovalevsky, O.A. Rudakova // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2012. — Т. 25. — С. 113-117. — Бібліогр.: 9 назв. — англ. 1683-4720 http://dspace.nbuv.gov.ua/handle/123456789/124119 517.9 en Труды Института прикладной математики и механики Інститут прикладної математики і механіки НАН України |
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In this Note, we announce a new Г-compactness result for a sequence of integral functionals defined on varying weighted Sobolev spaces. The result concerns the case where the degenerate integrands of the functionals satisfy a local Lipschitz condition but in general may not be convex with respect to the variable corresponding to the gradient of functions from domains of definition of the functionals. |
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Article |
| author |
Kovalevsky, A.A. Rudakova, O.A. |
| spellingShingle |
Kovalevsky, A.A. Rudakova, O.A. On the Г-compactness of integral functionals with degenerate locally Lipschitz integrands and varying domains of definition Труды Института прикладной математики и механики |
| author_facet |
Kovalevsky, A.A. Rudakova, O.A. |
| author_sort |
Kovalevsky, A.A. |
| title |
On the Г-compactness of integral functionals with degenerate locally Lipschitz integrands and varying domains of definition |
| title_short |
On the Г-compactness of integral functionals with degenerate locally Lipschitz integrands and varying domains of definition |
| title_full |
On the Г-compactness of integral functionals with degenerate locally Lipschitz integrands and varying domains of definition |
| title_fullStr |
On the Г-compactness of integral functionals with degenerate locally Lipschitz integrands and varying domains of definition |
| title_full_unstemmed |
On the Г-compactness of integral functionals with degenerate locally Lipschitz integrands and varying domains of definition |
| title_sort |
on the г-compactness of integral functionals with degenerate locally lipschitz integrands and varying domains of definition |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/124119 |
| citation_txt |
On the Г-compactness of integral functionals with degenerate locally Lipschitz integrands and varying domains of definition / A.A. Kovalevsky, O.A. Rudakova // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2012. — Т. 25. — С. 113-117. — Бібліогр.: 9 назв. — англ. |
| series |
Труды Института прикладной математики и механики |
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2025-07-09T00:52:16Z |
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2025-07-09T00:52:16Z |
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| fulltext |
ISSN 1683-4720 Труды ИПММ НАН Украины. 2012. Том 25
UDK 517.9
c©2012. A.A. Kovalevsky, O.A. Rudakova
ON THE Γ-COMPACTNESS OF INTEGRAL FUNCTIONALS WITH
DEGENERATE LOCALLY LIPSCHITZ INTEGRANDS
AND VARYING DOMAINS OF DEFINITION
In this Note, we announce a new Γ-compactness result for a sequence of integral functionals defined
on varying weighted Sobolev spaces. The result concerns the case where the degenerate integrands of
the functionals satisfy a local Lipschitz condition but in general may not be convex with respect to the
variable corresponding to the gradient of functions from domains of definition of the functionals.
Keywords: varying weighted Sobolev spaces, integral functional, degenerate integrand, varying domains,
Γ-convergence, Γ-compactness.
1. Introduction. In this Note, we announce a Γ-compactness theorem for a sequence
of integral functionals defined on varying weighted Sobolev spaces. The theorem concerns
the case where the degenerate integrands of the functionals satisfy a local Lipschitz
condition. The given result is a close analogue of the Γ-compactness theorem obtained in
[4, 9] in the case where the integrands of integral functionals, defined on the same spaces,
are convex with respect to the variable corresponding to the gradient of functions from
domains of definition of the functionals.
In this connection we note that besides the mentioned works, the questions related
to the Γ-convergence of integral functionals, defined on varying weighted Sobolev spaces,
and the convergence of minimizers of the corresponding variational problems were studied
in [5-8].
2. Preliminaries. Let n ∈ N, n > 2, and let Ω be a bounded domain of Rn. Let
p ∈ (1, n). 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
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
.
We observe that by virtue of Young’s inequlity 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
113
A.A. Kovalevsky, O.A. Rudakova
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 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).
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. 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:
1) 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);
2) 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. Statement of the main result. Let b ∈ L1(Ω), b > 0 in Ω, and let {ψs} be a
sequence of functions satisfying the following conditions: for every s ∈ N, ψs ∈ L1(Ωs)
and ψs > 0 in Ωs; for every open cube Q ⊂ Rn we have
lim sup
s→∞
∫
Q∩Ωs
ψs dx 6
∫
Q∩Ω
b dx.
114
On the Γ-compactness of integral functionals with degenerate locally Lipschitz integrands
Next, let ci, i = 1, . . . , 4, be positive constants, and 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; 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); (2)
for every s ∈ N, for almost every x ∈ Ωs and for every ξ, ξ′ ∈ Rn,
|fs(x, ξ)−fs(x, ξ′)| 6 c3ν(x)(1+|ξ|+|ξ′|)p−1|ξ−ξ′|+c4[ν(x)]1/p[ψs(x)](p−1)/p|ξ−ξ′|. (3)
Obviously, for every s ∈ N, fs is a Carathéodory function, and if s ∈ N and u ∈
W 1,p(ν, Ωs), we have fs(x,∇u) ∈ L1(Ωs).
Definition 3. If s ∈ N, then Js is the functional on W̃ 1,p
0 (ν, Ωs) such that for every
function u ∈ W̃ 1,p
0 (ν, Ωs),
Js(u) =
∫
Ωs
fs(x,∇u) dx.
We denote by FLip the set of all functions f : Ω × Rn → R satisfying the following
conditions: for every ξ ∈ Rn the function f(·, ξ) is measurable in Ω; for almost every
x ∈ Ω and for every ξ ∈ Rn,
−b(x) 6 f(x, ξ) 6 c2ν(x)|ξ|p + b(x);
there exist positive constants c′ and c′′ such that for almost every x ∈ Ω and for every
ξ, ξ′ ∈ Rn,
|f(x, ξ)− f(x, ξ′)| 6 c′ν(x)(1 + |ξ|+ |ξ′|)p−1|ξ − ξ′|+ c′′[ν(x)]1/p[b(x)](p−1)/p|ξ − ξ′|.
It is easy to see that for every f ∈ FLip and for every u ∈
◦
W 1,p(ν, Ω) the function
f(x,∇u) is summable in Ω.
Definition 4. If f ∈ FLip, then Jf is the functional on
◦
W 1,p(ν, Ω) such that for every
function u ∈
◦
W 1,p(ν, Ω),
Jf (u) =
∫
Ω
f(x,∇u) dx.
Theorem 1. Assume that ν ∈ L1(Ω), and let gs : Ωs×Rn → R, s ∈ N, be a sequence
of functions satisfying the following conditions: for every s ∈ N and for every ξ ∈ Rn the
function gs(·, ξ) is measurable in Ωs; for every s ∈ N and for almost every x ∈ Ωs the
function gs(x, ·) is convex in Rn; if ε > 0, then there exists σε > 0 such that for every
s ∈ N, for almost every x ∈ Ωs and for every ξ ∈ Rn,
|fs(x, ξ)− gs(x, ξ)| 6 εν(x)|ξ|p + σεψs(x). (4)
Next, suppose that there exists a sequence of nonempty open sets Ω(k) of Rn such that:
a) for every k ∈ N, Ω(k) ⊂ Ω(k+1) ⊂ Ω;
b) lim
k→∞
meas(Ω \ Ω(k)) = 0;
115
A.A. Kovalevsky, O.A. Rudakova
c) for every k ∈ N the functions ν and b are bounded in Ω(k).
Then there exist an increasing sequence {sj} ⊂ N and a function f ∈ FLip such that
the sequence {Jsj} Γ-converges to the functional Jf .
The proof of the theorem will be published in a forthcoming authors’ article. We only
note that the proof is analogous to that of the Γ-compactness result given in [4, 9] for the
case where the integrands of integral functionals defined on the spaces W̃ 1,p
0 (ν,Ωs) are
convex with respect to the variable corresponding to the gradient of functions in these
spaces. At the same time, some additional difficulties are connected exactly with the fact
that in the present article for the integrands fs conditions (3) and (4) are used instead of
the convexity of the functions fs(x, ·) for every s ∈ N and for almost every x ∈ Ωs which
was assumed in [4, 9].
We observe that some Γ-compactness results for integral functionals with degenerate
integrands and an unvarying domain of definition were established in [1–3] for the case
where the integrands are convex with respect to the variable corresponding to the gradient
of functions from the domain of definition of the functionals.
4. An example. It is not difficult to verify that condition (2) along with the
additional requirement that
for every s ∈ N and for almost everyx ∈ Ωs the function fs(x, ·) is convex in Rn (5)
implies that conditions (3) and (4) are satisfied, and in this case the constants c3 and c4
depend only on p and c2, and for every s ∈ N, gs = fs.
The following example shows that sequences of integrands satisfying conditions (2)–
(4) may not satisfy condition (5).
In fact, let p > 2, and let for every s ∈ N and for every (x, ξ) ∈ Ωs × Rn,
fs(x, ξ) = ν(x)|ξ|p − [ν(x)](p−1)/p[ψs(x)]1/p|ξ|p−1 .
Then for every s ∈ N, for every x ∈ Ωs and for every ξ ∈ Rn inequality (2) holds with
c1 = 1/p and c2 = 1. Moreover, for every s ∈ N, for every x ∈ Ωs and for every ξ, ξ′ ∈ Rn
inequality (3) holds with c3 = 2(p− 1) and c4 = 1. Finally, defining for every s ∈ N the
function gs : Ωs ×Rn → R by gs(x, ξ) = ν(x)|ξ|p, (x, ξ) ∈ Ωs ×Rn, we establish that the
following properties hold: for every s ∈ N and for every ξ ∈ Rn the function gs(·, ξ) is
measurable in Ωs; for every s ∈ N and for every x ∈ Ωs the function gs(x, ·) is convex in
Rn; if ε > 0, then for every s ∈ N, for every x ∈ Ωs and for every ξ ∈ Rn inequality (4)
holds with σε = ε1−p . On the other hand, supposing that for every s ∈ N, ψs > 0 a. e. in
Ωs, we find that for every s ∈ N and for almost every x ∈ Ωs the function fs(x, ·) is not
convex in Rn, and therefore, property (5) does not hold.
1. Carbone L., Sbordone C. Some properties of Γ-limits of integral functionals // Ann. Mat. Pura
Appl. (4). – 1979. – 122. – P. 1-60.
2. De Arcangelis R. Compactness and convergence of minimum points for a class of nonlinear nonequi-
coercive functionals // Nonlinear Anal. – 1990. – 15, № 4. – P. 363-380.
3. De Arcangelis R., Donato P. Convergence of minima of integral functionals and multiplicative
perturbations of the integrands // Ann. Mat. Pura Appl. (4). – 1988. – 150. – P. 341-362.
116
On the Γ-compactness of integral functionals with degenerate locally Lipschitz integrands
4. Kovalevskii A.A., Rudakova O.A. On the Γ-compactness of integral functionals with degenerated
integrands // Nelinejnye Granichnye Zadachi. – 2005. – 15. – P. 149-153. (in Russian)
5. Kovalevskii A.A., Rudakova O.A. On the strong connectedness of weighted Sobolev spaces and
the compactness of sequences of their elements // Tr. Inst. Prikl. Mat. Mekh. Nats. Akad. Nauk
Ukrainy. – 2006. – 12. – P. 85-99. (in Russian)
6. Kovalevsky A.A., Rudakova O.A. Variational problems with pointwise constraints and degeneration
in variable domains // Differ. Eqns. Appl. – 2009. – 1, № 4. – P. 517-559.
7. Kovalevsky A.A., Rudakova O.A. Γ-convergence of integral functionals with degenerate integrands
in periodically perforated domains // Tr. Inst. Prikl. Mat. Mekh. Nats. Akad. Nauk Ukrainy. – 2009.
– 19. – P. 101-109.
8. Rudakova O.A. On the coercivity of the integrand of the Γ-limit functional of a sequence of integral
functionals defined on different weighted Sobolev spaces // Tr. Inst. Prikl. Mat. Mekh. Nats. Akad.
Nauk Ukrainy. – 2007. – 15. – P. 171-180. (in Russian)
9. Rudakova O.A. On the Γ-convergence of integral functionals defined on various weighted Sobolev
spaces // Ukrainian Math. J. – 2009. – 61, № 1. – P. 121-139.
А.А. Ковалевский, О.А. Рудакова
О Γ-компактности интегральных функционалов с вырождающимися локально липши-
цевыми интегрантами и переменными областями определения.
В заметке анонсирован новый результат о Γ-компактности последовательности интегральных функ-
ционалов, определенных на переменных весовых пространствах Соболева. Этот результат относит-
ся к случаю, когда вырождающиеся интегранты функционалов удовлетворяют локальному усло-
вию Липшица, но, вообще говоря, могут не быть выпуклыми относительно переменной, соответ-
ствующей градиенту функций из областей определения функционалов.
Ключевые слова: переменные весовые пространства Соболева, интегральный функционал, вы-
рождающийся интегрант, переменные области, Γ-сходимость, Γ-компактность.
О.А. Ковалевський, О.А. Рудакова
Про Γ-компактнiсть iнтегральних функцiоналiв з виродними локально лiпшiцiвими iн-
тегрантами i змiнними областями визначення.
В замiтцi анонсований новий результат про Γ-компактнiсть послiдовностi iнтегральних функцiо-
налiв, визначених на змiнних вагових просторах Соболєва. Цей результат належить випадку, коли
вироднi iнтегранти функцiоналiв задовольняють локальну умову Лiпшiца, але, взагалi кажучи, мо-
жуть не бути опуклими вiдносно змiнної, що вiдповiдає градiєнту функцiй з областей визначення
функцiоналiв.
Ключовi слова: змiннi ваговi простори Соболєва, iнтегральний функцiонал, виродний iнте-
грант, змiннi областi, Γ-збiжнiсть, Γ-компактнiсть.
Institute of Applied Mathematics and Mechanics of NAS of Ukraine
alexkvl@iamm.ac.donetsk.ua
rudakova@iamm.ac.donetsk.ua
Received 07.12.2012
117
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