Gagliardo–Nirenberg inequalities in regular Orlicz spaces involving nonlinear expressions
We consider a triple of N-functions ( M , H , J ) that satisfy the Δ ′ -condition, μ = | x | α d x and suppose that an additive variant of interpolation inequality holds ∫ R n M ( | ∇ u | ) μ ( d x ) ⩽ C ( ∫ R n H ( | u | ) μ ( d x ) + ∫ R n J ( | ∇ ( 2 ) u | ) μ ( d x ) ) , where u ∈ R ⊆ W loc 2 ,...
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| Vydané v: | Journal of mathematical analysis and applications Ročník 362; číslo 2; s. 460 - 470 |
|---|---|
| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Amsterdam
Elsevier Inc
15.02.2010
Elsevier |
| Predmet: | |
| ISSN: | 0022-247X, 1096-0813 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | We consider a triple of
N-functions
(
M
,
H
,
J
)
that satisfy the
Δ
′
-condition,
μ
=
|
x
|
α
d
x
and suppose that an additive variant of interpolation inequality holds
∫
R
n
M
(
|
∇
u
|
)
μ
(
d
x
)
⩽
C
(
∫
R
n
H
(
|
u
|
)
μ
(
d
x
)
+
∫
R
n
J
(
|
∇
(
2
)
u
|
)
μ
(
d
x
)
)
,
where
u
∈
R
⊆
W
loc
2
,
1
(
R
n
)
,
R
is an arbitrary set invariant with respect to external and internal dilations. We show that the above inequality implies its certain nonlinear variant involving the expressions
∫
R
n
H
(
|
u
|
)
μ
(
d
x
)
and
∫
R
n
J
(
|
∇
(
2
)
u
|
)
μ
(
d
x
)
. Various generalizations of this inequality to the more general class of
N-functions, measures and to higher order derivatives are also discussed and the examples are presented. |
|---|---|
| ISSN: | 0022-247X 1096-0813 |
| DOI: | 10.1016/j.jmaa.2009.08.028 |