On density of smooth functions in weighted Sobolev spaces with
variable exponent.
Mikhail D. Surnachev
Computational Aeroacoustics Laboratory
Keldysh Institute of Applied Mathematics RAS
Miusskaya Sq. 4, Moscow 125047, Russia
[email protected]
In this work we are concerned with the question of density of smooth functions in weighted
Sobolev-Orlicz spaces. In a bounded Lipshitz domain Ω ⊂ Rn for a weight (nonnegative
measurable function) ρ ∈ L1 (Ω) we define the weighted Sobolev-Orlicz space
Z
1,1
p(x)
|∇u| ρ dx < ∞
W := u ∈ W0 (Ω) :
Ω
with a measurable exponent p = p(x) > 1. The space W is equipped with the Luxemburg
norm
( Z )
p(x)
|∇u|
ρ dx ≤ 1 .
kukW = inf λ :
λ
Ω
0
We assume that the weight additionally satifsfies ρ−1/p ∈ Lp (Ω; dx), which guarantees
completeness of W . Next, we can define another space H as the closure of C0∞ in W .
Recently V.V. Zhikov proved the following interesting theorem.
Theorem (V.V. Zhikov). Let ρ = ωω0 , p ≡ 2 and w0 ∈ A2 . Assume that
1/t
1/t R −t
R t
ω
ω
dx
ω
ω
dx
·
0
0
Ω
< ∞.
lim inf Ω
t→∞
t2
Then H = W .
First, we obtain the generalization of this result for all constant p > 1.
Theorem 1. Let ρ = ωω0 , p = const > 1 and w0 ∈ Ap . Assume that
1/t R −t
1/t
R t
ω
ω
dx
ω
ω
dx
·
0
0
Ω
lim inf Ω
< ∞.
t→∞
tp
Then H = W .
Next, we extend this result to the case of Sobolev spaces with variable exponent. We
assume that the exponent p = p(x) : Ω → (1, ∞) satisfies
1 < α < p(x) < β < ∞,
k0
|p(x) − p(y)| ≤
|x − y| < 1.
1 ,
ln |x−y|
1
(1)
(2)
This is the widely known logarithmic condition introduced originally by V.V. Zhikov and
X.L. Fan in the first half of 1990s. To state the analogue of Theorem 1 for p = p(x) we need
to introduce a generalization of Muckenhoupt classes for variable exponents.
Definition. We say that a nonnegative locally integrable function (weight) ω ∈ Aloc
p(·) (Ω) if
there exists c0 > 0 such that
Z
1/p(x) −1/p
kω
kLp0 (Q)
< ∞,
sup sup
ω dy
|Q|
Q x∈Q
Q
where the supremum is taken over all cubes Q ⊂ Ω with faces parallel to the coordinate
hyperplanes, satisfying |Q| ≤ c0 .
A very important feature of this classes is that they inherit the self-improvement property
of the classical Muckenhoupt classes.
Theorem 2. Let the exponent p(·) satisfy the logarithmic condition (1)-(2) and ω ∈ Aloc
p(·) (Ω).
loc
Then ω ∈ Ap(·)−ε (Ω) for some ε > 0.
The main result is the following
Theorem 3. Let ρ = ωω0 , where ω0 ∈ Aloc
p(·) (Ω). Let the exponent p = p(x) satisfy the
logarithmic condition (1)-(2) and
t−1 Z
t(t+1)
Z
1/t
t
−t
−p
ω ω0 dx
lim inf
< ∞.
t ω ω0 dx
t→∞
Ω
Ω
Then H = W .
Remark. The full theory of Muckenhoupt classes with variable exponent in the whole space
was built in [4]. For our results only local behavior is essential, which makes the theory
somewhat simpler. No log-condition at infinity is required.
Acknowledgements. This work was supported by RFBR, research projects nos. 11-0100989-a and 12-01-00058-a, and also by The Ministry of Education and Science of Russian
Federation, project 14.B37.21.0362.
References
[1] V.V. Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth
conditions, Journal of Mathematical Sciences 173:5 (2011), 463–570.Translated from Problems
in Mathematical Analysis 54, February 2011, pp. 23-–112.
[2] V.V. Zhikov, Weighted Sobolev spaces, Sbornik: Mathematics 189:8 (1998), 1139—1170.
[3] V.V. Zhikov, Density of smooth functions in Sobolev-Orlicz spaces, Journal of Mathematical
Sciences 132:3 (2006), 285–294. Translated from Zapiski Nauchnykh Seminarov POMI, Vol.
310, 2004, pp. 67—81.
[4] L. Diening and P. Hästö, Muckenhoupt weights in variable exponent spaces, University of
Helsinki Preprint (2011). http://www.helsinki.fi/~hasto/pp/p75_submit.pdf
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