On the equivalence of certain seminorms on some weighted Hölder spaces
The present paper is devoted to studying of some weighted H¨older spaces. These spaces are designed in the way to serve as a framework for studying different statements for the thin film equations in weighted classes of smooth functions in the multidimensional setting. These spaces can serve also fo...
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irk-123456789-1242412017-09-23T03:03:16Z On the equivalence of certain seminorms on some weighted Hölder spaces Degtyarev, S.P. The present paper is devoted to studying of some weighted H¨older spaces. These spaces are designed in the way to serve as a framework for studying different statements for the thin film equations in weighted classes of smooth functions in the multidimensional setting. These spaces can serve also for considering of other equations with the degeneration on the boundary of the domain of definition. We prove the equivalence of certain metrics on these spaces. Данная статья посвящена изучению некоторых весовых пространств Гельдера. Эти пространства являются естественными классами гладких функций для изучения уравнений типа уравнений тонких пленок в многомерном случае. Эти классы могут быть применены также для изучения других уравнений с вырождением на границе области определения. Мы доказываем эквивалентность некоторых различных метрик в этих пространствах. 2016 Article On the equivalence of certain seminorms on some weighted Hölder spaces / S.P. Degtyarev // Труды Института прикладной математики и механики. — Донецьк: ІПММ, 2016. — Т. 30. — С. 39-45. — Бібліогр.: 25 назв. — англ. 1683-4720 http://dspace.nbuv.gov.ua/handle/123456789/124241 517.9 en Труды Института прикладной математики и механики Інститут прикладної математики і механіки НАН України |
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The present paper is devoted to studying of some weighted H¨older spaces. These spaces are designed in the way to serve as a framework for studying different statements for the thin film equations in weighted classes of smooth functions in the multidimensional setting. These spaces can serve also for considering of other equations with the degeneration on the boundary of the domain of definition. We prove the equivalence of certain metrics on these spaces. |
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Degtyarev, S.P. |
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Degtyarev, S.P. On the equivalence of certain seminorms on some weighted Hölder spaces Труды Института прикладной математики и механики |
| author_facet |
Degtyarev, S.P. |
| author_sort |
Degtyarev, S.P. |
| title |
On the equivalence of certain seminorms on some weighted Hölder spaces |
| title_short |
On the equivalence of certain seminorms on some weighted Hölder spaces |
| title_full |
On the equivalence of certain seminorms on some weighted Hölder spaces |
| title_fullStr |
On the equivalence of certain seminorms on some weighted Hölder spaces |
| title_full_unstemmed |
On the equivalence of certain seminorms on some weighted Hölder spaces |
| title_sort |
on the equivalence of certain seminorms on some weighted hölder spaces |
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Інститут прикладної математики і механіки НАН України |
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2016 |
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http://dspace.nbuv.gov.ua/handle/123456789/124241 |
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On the equivalence of certain seminorms on some weighted Hölder spaces / S.P. Degtyarev // Труды Института прикладной математики и механики. — Донецьк: ІПММ, 2016. — Т. 30. — С. 39-45. — Бібліогр.: 25 назв. — англ. |
| series |
Труды Института прикладной математики и механики |
| work_keys_str_mv |
AT degtyarevsp ontheequivalenceofcertainseminormsonsomeweightedholderspaces |
| first_indexed |
2025-07-09T01:06:33Z |
| last_indexed |
2025-07-09T01:06:33Z |
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1837129475746168832 |
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ISSN 1683-4720 Труды ИПММ. 2016. Том 30
UDK 517.9
c©2016. S. P. Degtyarev
ON THE EQUIVALENCE OF CERTAIN SEMINORMS ON SOME
WEIGHTED HÖLDER SPACES
The present paper is devoted to studying of some weighted Hölder spaces. These spaces are designed
in the way to serve as a framework for studying different statements for the thin film equations in
weighted classes of smooth functions in the multidimensional setting. These spaces can serve also for
considering of other equations with the degeneration on the boundary of the domain of definition. We
prove the equivalence of certain metrics on these spaces.
Keywords: weighted Hölder spaces, degenerate parabolic equations, equivalent metrics.
The present paper is devoted to studying of some weighted Hölder spaces C
m+γ, m+γ
m
n,ωγ .
These spaces were introduced in [1] and they are designed in the way to serve as
a framework for consideration of different statements for the thin film equations in
weighted classes of smooth functions in the multidimensional setting. These spaces can
serve also for considering of other equations with the degeneration on the boundary of
the domain of definition, for example, in the spirit of [2].
The literature on the subject of the thin film equations is very numerous but almost
all results with sufficient regularity are devoted to the case of one spatial variable. As a
possible target for an application of the spaces C
m+γ, m+γ
m
n,ωγ we only mention the papers
[3]–[17].
The spaces C
m+γ, m+γ
m
n,ωγ arise at the considering linearised version of the thin film
equations. Let us explain this on the example for the thin film equation in the case
of partial wetting (see, for example, [3] for the accurate statement). Consider the thin
film equation of fourth order for an unknown function h(x, t) (compare [18])
∂h
∂t
+∇ (hn∇∆h− β∇h) = f(x, t) in Ω, (1)
where n > 0 is fixed, Ω is a half space Ω = {(x, t) : x = (x′, xN ) ∈ RN , xN > 0, t > 0}.
Consider also partial wetting conditions at xN = 0
h(x′, 0, t) = 0,
∂h
∂xN
(x′, 0, t) = 1 (2)
and an initial condition
h(x, 0) = w(x). (3)
From (2) it follows that we must have for w(x)
w(x′, 0) = 0,
∂w
∂xN
(x′, 0) = 1. (4)
39
S. P. Degtyarev
Consequently, we have
w(x) ∼ xN , xN → 0. (5)
The linearization of equation (1) at the initial datum w(x) means that we denote in (1)
h = w + u and extract linear with respect to u part (we also drop lower order terms).
Formally, one can just replace hn by wn in (1) and replace h by u in other places of
this equation. Taking into account (5) and replacing w by just xN , we arrive at
∂u
∂t
+∇(xn
N∇∆u− β∇u) = f(x, t) in Ω. (6)
For second order equations this procedure is described in details in, for example, [19],
[20], [2], and for fourth order see [17], [3], [4] formula (13), [5] formula (7).
If we are going to consider equations (6) (and correspondingly (1)) in classes of
Hölder functions we have to consider f(x, t) in (6) from some (may be weighted) Hölder
class. This leads to the consideration of ∇(xn
N∇∆u) from the same weighted Hölder
class. In our definition below this will be the class C
m+γ, m+γ
m
n,(n/4)γ . In the case of second
order equations such classes were used in fact in [21]–[23], [2], where the papers [21]–[23]
are based on the Carnot–Carathéodory metric and the paper [2] is based on classes
C
m+γ, m+γ
m
n,ωγ . Note that we consider the framework of classes C
m+γ, m+γ
m
n,ωγ as an alternative
for considering the Carnot–Carathéodory metric for studying degenerate equations in
classes of smooth functions – [21]–[23], [17]. Therefor, in this paper we are going to
prove the equivalence of the Carnot–Carathéodory metric in spaces C
m+γ, m+γ
m
n,ωγ to some
another weighted metric in these spaces.
Note that in the case of elliptic equations more simple weighted Hölder classes with
unweighted Hölder constants can be used – [24], [25]. The reason is that in the elliptic
case no agreement between smoothness in x-variables and t- variable is needed.
Let us turn now to exact definitions and to the main results.
Denote H = {x = (x′, xN ) ∈ RN : xN > 0}, Q = {(x, t) : x ∈ H,−∞ < t < ∞}.
And we note at once that all the reasoning and statement below are valid in evident
way also for Q+ = {(x, t) : x ∈ H, t ≥ 0} instead of Q. Let m be a positive integer and
let n be a positive number, n < m. Denote
ω = n/m < 1.
Let Cγ
ωγ(H), γ ∈ (0, 1), be the weighted Hölder space of continuous functions u(x)
with the finite norm
|u|(γ)
ωγ,H
≡ ‖u‖Cγ
ωγ(H) ≡ |u|(0)
H
+ 〈u〉(γ)
ωγ,H
, (7)
where
|u|(0)
H
= max
x∈H
|u(x)|, 〈u〉(γ)
ωγ,H
= sup
x,x∈H
(x∗N )ωγ |u(x)− u(x)|
|x− x|γ , x∗N = max{xN , xN}. (8)
40
On the equivalence of certain metrics on some weighted Hölder spaces
Thus 〈u〉(γ)
ωγ,H
represents a weighted Hölder constant of the function u(x). We suppose
that
n < m, , if n is a noninteger (1− ω)γ = γ
(
1− n
m
)
< min({n}, 1− {n}), (9)
where for a real number a ,{a} is the fractional part of a, [a] is the integer part of a.
This assumption is technical and it allows us, for example, to consider the functions
xn−j
N as elements of Cγ
ωγ(H) for all integer j < n.
In the similar way we define the Hölder seminorms with respect to each variable
separately
〈u〉(γ)
ωγ,xi,H
= sup
x,x∈H
(x∗N )ωγ |u(x)− u(x)|
hγ
, x∗N = max{xN , xN}, i = 1, N, (10)
where x = (x1, ...xi, ..., xN ), x = (x1, ...xi + h, ..., xN ), h > 0.
In the standard way we denote by 〈u〉(γ)
xi,H
, 〈u〉(γ)
x′,H
, and 〈u〉(γ)
x,H
usual unweighted
Hölder seminorms with respect to each variable separately, with respect to
x′ = (x1, ..., xN−1) or with respect to all x-variables.
Note that in terms of the Carnot-Carathéodory metric seminorm (8) is equivalent
to
〈u〉(γ)
ωγ,H
' sup
x,x∈H
|u(x)− u(x)|
s(x, x)γ
,
where the Carnot–Carathéodory distance is defined as
s(x, x) =
|x− x|
|x− x|ω + xω
N + xω
N
. (11)
In the case of m = 2, n ∈ (0, 1) this was proved in [2] and in the general case we have
the following theorem which is the main result of the present paper.
Denote
[u](γ)
s ≡ sup
x,x∈H
|u(x)− u(x)|
s(x, x)γ
, (12)
where s(x, x) is defined in (11)
Theorem. Seminorm (10) is equivalent to seminorm (12). This means that there
are constants C1 and C2 with the property
[f ](γ)
s ≤ C1 〈f〉(γ)
ωγ,H
≤ C2[f ](γ)
s (13)
for any continuous in H function f(x).
Proof. Let the seminorm [f ](γ)
s is finite. We show, that then
〈f〉(γ)
ωγ,H
≤ C[f ](γ)
s . (14)
41
S. P. Degtyarev
Let ε0 ∈ (0, 1) is small and fixed. Let x = (x′, xN ) and let first
|x′ − x′| ≥ ε0xN . (15)
Then the more
|x− x| ≥ |x′ − x′| ≥ ε0xN . (16)
Under this condition
s(x, x) =
|x− x|
xω
N + xω
N + |x′ − x′|ω ≤
≤ C
|x− x|
|x− x|ω + xω
N + |x′ − x′|ω ≤ C|x− x|1−ω.
Therefore, denoting β = γ(1− ω),
|f(x, t)− f(x, t)|
|x− x|β ≤ C
|f(x, t)− f(x, t)|
s(x, x)γ
≤ C[u](γ)
s . (17)
Besides, because of (16), and then of (17),
xγω
N
|f(x, t)− f(x, t)|
|x− x|γ ≤ xγω
N
(ε0xN )γω
|f(x, t)− f(x, t)|
|x− x|γ(1−ω)
≤ C[u](γ)
s . (18)
Let now
|x′ − x′| ≤ ε0xN . (19)
Under this condition, as it easy to see,
s(x, x) ∼ Cx−ω
N |x− x|. (20)
Consequently,
xγω
N
|f(x, t)− f(x, t)|
|x− x|γ ≤ C
|f(x, t)− f(x, t)|
s(x, x)γ
≤ C[u](γ)
s . (21)
We estimate now the unweighted Hölder constant of the function f with the
exponent β.
To estimate it we consider the two cases.
If
|xN − xN | ≥ ε0xN ,
then
|x− x| ≥ |xN − xN | ≥ ε0xN
and therefore, as it was above,
s(x, x) ≤ |x− x|
(|x− x|/ε0)ω
≤ C|x− x|1−ω,
42
On the equivalence of certain metrics on some weighted Hölder spaces
so that, as above,
|f(x, t)− f(x, t)|
|x− x|β ≤ C
|f(x, t)− f(x, t)|
s(x, x)γ
≤ C[u](γ)
s . (22)
If now, under the condition (19), we have
|xN − xN | ≤ ε0xN , (23)
then in this case
|x− x| ≤ |x′ − x′|+ |xN − xN | ≤ 2ε0xN . (24)
Therefore, in the force of (20),
s(x, x) ≤ Cx−ω
N |x− x| ≤
≤ Cx−ω
N (2ε0xN )ω|x− x|1−ω = C|x− x|1−ω. (25)
Consequently, in this case
|f(x, t)− f(x, t)|
|x− x|β ≤ C
|f(x, t)− f(x, t)|
s(x, x)γ
≤ C[u](γ)
s . (26)
The estimate (14) follows now from (17), (18), (21), (22) and (26).
Further, let now the seminorm 〈f〉(γ)
ωγ,H
is finite. Let us prove the following estimate
[f ](γ)
s ≤ C 〈f〉(γ)
ωγ,H
. (27)
Let first
|x′ − x′| ≤ ε0xN , xN > 0. (28)
Then
s(x, x) ≥ ν
|x− x|
xω
N
,
and consequently
|f(x, t)− f(x, t)|
s(x, x)γ
≤ Cxγω
N
|f(x, t)− f(x, t)|
|x− x|γ ≤ C 〈f〉(γ)
ωγ,H
. (29)
In the particular case xN = 0 we have xN = 0 and therefore
s(x, x) = |x′ − x′|1−ω = |x− x|1−ω,
and so again
|f(x, t)− f(x, t)|
s(x, x)γ
=
|f(x, t)− f(x, t)|
|x− x|β ≤ C 〈f〉(γ)
ωγ,H
. (30)
Let now we have
|x′ − x′| ≥ ε0xN . (31)
43
S. P. Degtyarev
Then
s(x, x) ≥ ν
|x− x|
|x′ − x′|ω ≥ ν|x− x|1−ω, (32)
and consequently,
|f(x, t)− f(x, t)|
s(x, x)γ
≤ C
|f(x, t)− f(x, t)|
|x− x|β ≤ C 〈f〉(γ)
ωγ,H
. (33)
Thus, (27) follows from (29), (30), (32). And so the theorem is proved. ¤
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С.П. Дегтярев
Об эквивалентности некоторых полунорм в весовых пространствах Гельдера.
Данная статья посвящена изучению некоторых весовых пространств Гельдера. Эти пространства
являются естественными классами гладких функций для изучения уравнений типа уравнений
тонких пленок в многомерном случае. Эти классы могут быть применены также для изучения
других уравнений с вырождением на границе области определения. Мы доказываем эквивалент-
ность некоторых различных метрик в этих пространствах.
Ключевые слова: весовые пространства Гельдера, вырождающиеся параболические уравне-
ния, эквивалентные метрики.
ГУ «Ин-т прикл. математики и механики», г.Донецк
degtyar@i.ua
Received 17.05.16
45
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