Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 179068, 12 pages
doi:10.1155/2011/179068
Research Article
Function Spaces with a Random Variable Exponent
Boping Tian, Yongqiang Fu, and Bochi Xu
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Correspondence should be addressed to Bochi Xu, xubochi@126.com
Received 11 February 2011; Accepted 12 April 2011
Academic Editor: Wolfgang Ruess
Copyright q 2011 Boping Tian et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The spaces with a random variable exponent Lpω D ×Ω and W k,pω D ×Ω are introduced. After
discussing the properties of the spaces Lpω D × Ω and W k,pω D × Ω, we give an application of
these spaces to the stochastic partial differential equations with random variable growth.
1. Introduction
In the study of some nonlinear problems in natural science and engineering, for example,
a class of nonlinear problems with variable exponential growth, variable exponent function
spaces play an important role. In recent years, there is a great development in the field of
variable exponent analysis. In 1, basic properties of the spaces Lpx Ω and W k,px Ω have
been discussed by Kováčik and Rákosnı́k. Some theories of variable exponent spaces can also
be found in 2, 3. In 4, Harjulehto et al. present an overview about applications of variable
exponent spaces to differential equations with nonstandard growth. In 5, Diening et al.
summarize a lot of the existing literature of theory of function spaces with variable exponents
and applications to partial differential equations. In 6, Aoyama discusses the properties of
Lebesgue spaces with variable exponent on a probability space.
Motivated by 6, 7, we first introduce Lpω D × Ω and W k,pω D × Ω in Section 2,
which are function spaces with a random variable exponent. We also discuss some properties
of these spaces. Stochastic partial differential equations have many applications in finance,
such as option pricing. In Section 3, an application of the random variable exponent spaces to
the stochastic partial differential equations with random variable growth is given. We discuss
the existence and uniqueness of weak solution for the following equation:
− div Ax, ω, u, ∇u Bx, ω, u, ∇u fx, ω,
u 0,
x, ω ∈ ∂D × Ω.
x, ω ∈ D × Ω,
1.1
2
Abstract and Applied Analysis
A and B are Carathéodory functions, which are integrable on D × Ω and continuous
for u and ∇u. fx, ω is an integrable function on D × Ω. Ω, F, P is a complete probability
space, and D is a bounded open subset of Rn n > 1. The random variable p : Ω → 1, ∞
satisfies 1 < p− ≤ pω ≤ p < ∞. Furthermore,
A : Rn × Ω × R × Rn −→ Rn ,
B : Rn × Ω × R × Rn −→ R,
f : Rn × Ω −→ R
1.2
satisfy the following growth conditions:
H1 |Ax, ω, s, ξ|β1 ω|ξ|pω−1 β2 ω|s|pω−1 K1 x, ω, |Bx, ω, s, ξ| ≤ β3 ω|ξ|pω−1 β4 ω|s|pω−1 K2 x, ω,
H2 EAx, ω, s1 , ξ − Ax, ω, s2 , ηξ − η Bx, ω, s1 , ξ − Bx, ω, s2 , ηs1 − s2 >
s2 ,
0, ξ /
η or s1 /
H3 EAx, ω, s, ξξ Bx, ω, s, ξs ≥ βE|ξ|pω |s|pω , a.e. in D,
where K1 x, ω, K2 x, ω ∈ Lp ω D × Ω, β > 0, βi ω i 1, 2, 3, 4 are nonnegative bounded
random variables, f ∈ Lp ω D × Ω, and 1/pω 1/p ω 1.
2. Some Properties of Function Spaces with a Random
Variable Exponent
Let λ be a product measure on D × Ω and ux, ω a Lebesgue measurable function on D × Ω.
In this section, p : Ω → 1, ∞ is a random variable.
On the set of all functions defined on D × Ω, the functional ρpω is defined by
|ux, ω|pω dx .
ρpω u E
2.1
D
Definition 2.1. The space Lpω D × Ω is the set of Lebesgue measurable functions u on D × Ω
such that D×Ω |ux, ω|pω dλ < ∞, and it is endowed with the following norm:
u
≤1 .
upω inf λ > 0 : ρpω
λ
2.2
Definition 2.2. The space W k,pω D × Ω is the set of functions such that Dα u ∈ Lpω D × Ω,
|α| ≤ k, and it is endowed with the following norm:
uk,pω |α|≤k
Dα upω .
2.3
Here Dα u is the derivative of u with respect to x in the distribution sense.
k,pω
Definition 2.3. The space W0
D × Ω is the closure of CD × Ω {u ∈ W k,pω D × Ω :
∞
u·, ω ∈ C0 D for each ω ∈ Ω} in W k,pω D × Ω.
Abstract and Applied Analysis
3
Theorem 2.4. If pω satisfies 1 < p− ≤ pω ≤ p < ∞, then the inequality
fx, ωgx, ωdx
E
D
≤ Cf pω g p ω
2.4
holds for every f ∈ Lpω D × Ω and g ∈ Lp ω D × Ω with the constant C dependent on pω
only.
Proof. By Young inequality, we have
fx, ω gx, ω
fx, ω pω
gx, ω p ω
1
1
≤
.
f g pω f pω
p ω g p ω
pω
p ω
2.5
Integrating over D × Ω, we obtain
E
fx, ω gx, ω
D
fpω
gp ω
dx
≤1
1
1
− .
−
p
p
2.6
So
fx, ωgx, ωdx
E
≤
D
1
1 1 − − f pω g p ω .
p
p
2.7
Now the proof is completed.
Theorem 2.5. Suppose that pω satisfies 1 < p− ≤ pω ≤ p < ∞. If uk , u ∈ Lpω D × Ω, then
p−
p
p
p−
1 if upω ≥ 1, then upω ≤ ρpω u ≤ upω ,
2 if upω ≥ 1, then upω ≤ ρpω u ≤ upω ,
3 limk → ∞ uk pω 0 if and only if limk → ∞ ρpω uk 0,
4 limk → ∞ uk pω ∞ if and only if limk → ∞ ρpω uk ∞.
Proof. 1 By upω ≥ 1 and the definition of the norm,
⎛
E⎝
|u|pω
D
p
upω
⎞
⎛
dx⎠ ≤ E⎝
D
|u|
upω
pω
⎞
dx⎠ ≤ 1.
2.8
p
So ρpω u ≤ upω . As
p− /pω
upω
≤ upω ,
2.9
4
Abstract and Applied Analysis
we also have
⎛
⎜
E⎝
⎛
⎝
D
⎞
⎞pω
|u|
⎟
dx⎠ ≥ 1.
⎠
p− /pω
upω
2.10
p−
That is to say upω ≤ ρpω u. The proof is completed. 2, 3, and 4 can be easily proved
with similar methods.
Theorem 2.6. If pω satisfies 1 < p− ≤ pω ≤ p < ∞, the space Lpω D × Ω is complete.
Proof. Let un be a Cauchy sequence in Lpω D × Ω. Then, by Theorem 2.4,
|um x, ω − un x, ω|dx
E
D
≤ Cum − un pω χD p ω ,
2.11
where C is constant. That is to say un is a Cauchy sequence in L1 D × Ω. In view of the
completeness of L1 D × Ω, un converges in L1 D × Ω. Suppose that un → u, u ∈ L1 D × Ω
and further suppose that un x, ω → ux, ω a.e. in D × Ω subtracting a subsequence if
necessary. For each 0 < ε < 1, there exists n0 such that um − un < ε for m, n ≥ n0 . Fix n, by
Fatou’s lemma
E
D
|un x, ω − ux, ω|
ε
pω
dx
D
|un x, ω − um x, ω|
ε
D
|un x, ω − um x, ω|
ε
≤ E lim sup
m→∞
≤ lim sup E
m→∞
pω
dx
pω
dx
≤ 1.
2.12
p−
−
So un − upω ≤ ε, and further ρpω un − u ≤ un − upω ≤ εp ; that is to say, un − u ∈
Lpω D×Ω. Next as un ∈ Lpω D×Ω, we have u ∈ Lpω D×Ω. Now the proof is completed.
Theorem 2.7. If pω satisfies 1 < p− ≤ pω ≤ p < ∞, then the space Lpω D × Ω is reflexive.
Proof. First, suppose that v ∈ Lp ω D × Ω is fixed, and let
Lv u E
D
for every u ∈ Lpω D × Ω.
uv dx ,
2.13
Abstract and Applied Analysis
5
Note that
|Lv u| ≤ Cupω vp ω .
2.14
So Lv u ∈ Lpω D × Ω .
Second, we show that each bounded linear functional on Lpω D × Ω is of the form
Lv u E D uvdx for every v ∈ Lp ω D × Ω. Let L ∈ Lpω D × Ω be given. Let S be a
subset of D × Ω. We define
μS L χS ,
2.15
where χS is the characteristic function of S; then
μS ≤ LχS pω
1/p−
1/p ≤ L max ρpω χS
, ρpω χS
2.16
−
L max meas S1/p , meas S1/p ,
so μ is absolutely continuous with respect to the measure λ. Also we have that μ is σ finite
measure. By Radon-Nikodym theorem, there is an integrable function v on D × Ω such that
μS v χS dx .
v dλ E
S
2.17
D
Now Lu E D u
vdx holds for simple functions u. If u ∈ Lpω D × Ω, there is a sequence
of simple function uj , converging a.e. to u and |uj x, ω| ≤ |ux, ω| on D × Ω. By Fatou’s
lemma
E
uj v dx
≤ lim sup E
u
v
dx
j→∞
D
D
lim sup L uj sgn v
j→∞
≤ Llim supuj j →∞
2.18
pω
≤ Lupω .
Then
Lv u E
u
v dx
D
2.19
6
Abstract and Applied Analysis
is also a bounded linear functional on Lpω D × Ω. By Lebesgue theorem
uj − upω dx
lim E
j →∞
E
pω
lim uj − u
dx
D j →∞
D
0.
2.20
By Theorem 2.5, we have that
lim uj − upω 0,
2.21
j →∞
that is, uj → u. As Luj Lv uj , by letting j → ∞, we have that Lu Lv u.
At last, we show that v ∈ Lp ω D × Ω. Let
El {x, ω ∈ D × Ω : |
vx, ω| ≤ l}.
2.22
As
v χE p ω dλ≤
l
D×Ω
|l|p ω dλ ≤ ∞,
2.23
El
v χEl ∈ Lp ω D × Ω. Suppose that vχEl > 0, and take
u χEl
v|
|
v χE l
1/pω−1
sgn v .
2.24
p ω
Then
⎛
⎞
p ω
v|
v χE dx⎠
E
E⎝ χE |
u
v
dx
l
l p ω
v χEl D
D
p ω
⎛
p ω ⎞
v χEl v
| |
p ω
≥
E⎝ χEl
dx⎠
2p
1/2v χEl p ω
D
>
v χE l p ω
2p
2.25
.
As
upω
⎧
⎛
⎨
v|
|
inf λ > 0 : E⎝ χEl
⎩
λ
v
χ
El
D
p ω
p ω
⎞
dx⎠ ≤ 1
⎫
⎬
⎭
1,
2.26
Abstract and Applied Analysis
7
we have that
v χE ≤ 2p L,
⎞
⎛ v χEl p ω
dx⎠ ≤ 1.
E⎝
2p L
D
l
p ω
2.27
By Fatou’s lemma
E
D
v|
|
2p L
⎛
p ω
≤ lim sup E⎝
dx
l→∞
p ω
v χE l
2p L
D
⎞
dx⎠ ≤ 1.
2.28
So v ∈ Lp ω D ×Ω. Now we reach the conclusion that Lp ω D ×Ω Lpω D × Ω ,
and, furthermore, Lpω D × Ω is reflexive.
Theorem 2.8. If pω satisfies 1 < p− ≤ pω ≤ p < ∞, then the space W k,pω D×Ω is a reflexive
Banach space.
Proof. W k,pω D × Ω can be treated as a subspace of the product space
Lpω D × Ω,
2.29
m
where m is the number of multi-indices α with |α| ≤ k.
Then we need only to show that W k,pω D×Ω is a closed subspace of m Lpω D×Ω.
Let {un } ⊂ W k,pω D × Ω be a convergent sequence. Then {un } is a convergent sequence in
Lpω D×Ω, so there exists u ∈ Lpω D×Ω such that un → u in Lpω D×Ω by Theorem 2.6.
Similarly, there exists uα ∈ Lpω D × Ω such that Dα un → uα in Lpω D × Ω for
|α| ≤ k. As, for ϕ ∈ CD × Ω,
|α|
−1 E
α
D un ϕ dx
E
α
un D ϕ dx ,
D
2.30
D
we have
−1|α| E
uα ϕ dx
D
E
uDα ϕ dx ,
2.31
D
as n → ∞. By the definition of weak derivative, Dα u uα holds for each |α| ≤ k. So Dα u ∈
Lpω D × Ω. Then W k,pω D × Ω is a closed subspace of m Lpω D × Ω.
Theorem 2.9. Suppose that the sequence un ∈ Lpω D × Ω is bounded in Lpω D × Ω. If un → u
a.e. in D × Ω, then un u in Lpω D × Ω.
8
Abstract and Applied Analysis
Proof. By Theorem 2.7, we need only to show that
un g dx
E
−→ E
D
ug dx
2.32
D
for each g ∈ Lp ω D × Ω.
Let un pω ≤ C for each n ∈ N. By Fatou’s lemma,
pω pω u
un E
dx ≤ lim sup E
dx ≤ 1,
C
n→∞
D
D C
2.33
so upω ≤ C. By the absolute continuity of Lebesgue integral,
gχE p ω dλ 0,
lim
measE → 0
2.34
D×Ω
where g ∈ Lp ω D × Ω and E ⊂ D × Ω. By Theorem 2.5, limmeasE → 0 gχE p ω 0. So there
exists δ > 0, such that
gχE p ω
1
1
1 −1
ε 1 − − ,
4C
p
p
<
2.35
for measE < δ. By Egorov theorem, there exists a set B ⊂ D × Ω such that un → u uniformly
on B with measΩ \ B < δ. Choose n0 such that n > n0 implies
1
ε
1
max |u − un | g p ω χB p ω 1 − − < .
x,ω∈B
p
p
2
2.36
Taking E D × Ω \ B,
E
ug dx − E
un g dx D
≤
D
|un − u|g dλ B
E
gχB dx E
|un − u|gχE dx
≤ max |u − un |E
x,ω∈B
|un − u|g dλ
D
D
1
1
≤ max |u − un | g p ω χB p ω 1 − − x,ω∈B
p
p
1
1
un − upω gχE p ω 1 − − p
p
< ε.
2.37
Abstract and Applied Analysis
9
That is to say un u weakly in Lpω D × Ω.
3. An Application to Partial Differential Equations with
Random Variable Growth
1,pω
Definition 3.1. u ∈ W0
D × Ω is said to be the weak solution of 1.1 if
Ax, ω, u, ∇u∇ϕ Bx, ω, u, ∇uϕ dx
E
E
D
fx, ωϕ dx
3.1
D
for all ϕ ∈ W 1,pω D × Ω.
Definition 3.2. Let X be a reflexive Banach space with dual X , and let ·, · denote a pairing
between X and X . If K ⊂ X is a closed convex set, then a mapping L : K → X is called
monotone if Lu − Lv, u − v ≥ 0 for all u, v ∈ K. Further, L is called coercive on K if there
exists φ ∈ K such that
!
Luj − Lϕ, uj − ϕ
−→ ∞,
uj − ϕ 3.2
X
whenever uj is sequence in K with uj − ϕX → ∞.
Theorem 3.3 see 8. Let K be a nonempty closed convex subset of X, and let L : K → X be
monotone, coercive and strong-weakly continuous on K. Then there exists an element u in K such
that Lu, v − u ≥ 0 for all v ∈ K.
1,pω
D × Ω. Then it is obvious that K is a closed convex
In the following, let K W0
subset of X W 1,pω D × Ω. Let L : K → X ,
fx, ωv dx ,
Ax, ω, u, ∇u∇v Bx, ω, u, ∇uv dx −
Lu, v E
D
3.3
D
where v ∈ X.
Lemma 3.4. L is monotone and coercive on K.
Proof. In the view of H2, it is immediate that L is monotone. Next that L is coercive is
shown. Fixing ϕ ∈ K, for all u ∈ K, by H1, H3, and Young Equality,
!
Lu − L ϕ , u − ϕ
E
Ax, ω, u, ∇u∇u Bx, ω, u, ∇uudx A x, ω, ϕ, ∇ϕ ∇ϕ
D
B x, ω, ϕ, ∇ϕ ϕdx −
D
D
Ax, ω, u, ∇u∇ϕ A x, ω, ϕ, ∇ϕ ∇u
10
Abstract and Applied Analysis
Bx, ω, u, ∇uϕ B x, ω, ϕ, ∇ϕ udx
≥ β − 4β0 1 ε ρpω |∇u| ρpω u β − 4β0 1 Cε ρpω ∇ϕ ρpω ϕ
− Cε ε ρp ω K1 x, ω ρp ω K2 x, ω ,
3.4
where β0 is the bound of nonnegative bounded random variable βi ω, i 1, 2, 3, 4, ε is
small enough such that β − 4β0 1ε > 0, and Cε is constant only dependent on ε. Then
!
Lu − L ϕ , u − ϕ
u − ϕ 1,pω
W0
D×Ω
β − 4β0 1 ε ρpω |∇u| ρpω u
C ϕ, ∇ϕ, K1, K2 , ε
≥
.
u − ϕ 1,pω
uW 1,pω D×Ω ϕW 1,pω D×Ω
W
D×Ω
0
0
3.5
0
For ε1 , ε2 > 0 small enough, we have that
ρpω |∇u| ρpω u
⎛ ⎛
⎞ ⎞
pω
pω
pω
|u|
|∇u|
pω
⎜ ⎜
⎟ ⎟
E⎝ ⎝
⎠dx⎠
pω ∇upω −ε1
pω upω −ε2
D
∇upω −ε1
upω −ε2
≥ min
"
"
p− p#
p − p#
min upω −ε2 , upω −ε2
.
∇upω −ε1 , ∇upω −ε1
3.6
Thus, we have that
!
Lu − L ϕ , u − ϕ
−→ ∞,
u − ϕ 1,pω
W0
3.7
D×Ω
as uW 1,pωD×Ω → ∞. The proof is completed.
0
Lemma 3.5. L is strong-weakly continuous.
1,pω
1,pω
Proof. Let un ∈ W0
D × Ω be a sequence that converges to u ∈ W0
un ≤ C for some constant C, and there exists a subsequence such that
unk −→ u,
∇unk −→ ∇u,
D × Ω. Then
3.8
Abstract and Applied Analysis
11
a.e. in D × Ω. A, B are Caratheodory functions, so
Ax, ω, unk , ∇unk −→ Ax, ω, u, ∇u,
3.9
Bx, ω, unk , ∇unk −→ Bx, ω, u, ∇u,
a.e. in D × Ω. By Theorem 2.9,
Ax, ω, unk , ∇unk Ax, ω, u, ∇u,
3.10
Bx, ω, unk , ∇unk Bx, ω, u, ∇u,
and Ax, ω, unk , ∇unk and Bx, ω, unk , ∇unk are bounded. For all ϕ ∈ W 1,pω D × Ω, we
have that
!
Ax, ω, unk , ∇unk ∇ϕ Bx, ω, unk , ∇unk ϕ dx −
fx, ωϕ dx
Lunk , ϕ E
D
D
−→ E
Ax, ω, u, ∇u∇ϕ Bx, ω, u, ∇uϕ dx −
fx, ωϕ dx
D
D
!
Lu, ϕ ,
3.11
that is to say L is strong-weakly continuous.
1,pω
Theorem 3.6. Under conditions (H1)–(H3), there exists a unique weak solution u ∈ W0
to 1.1 for any f ∈ Lp ω D × Ω.
1,pω
Proof. From Theorem 3.3, Lemmas 3.4, and 3.5, there exists u ∈ W0
D × Ω such that
Lu, v − u ≥ 0,
1,pω
for all v ∈ W0
1,pω
D × Ω. Let ϕ ∈ W0
3.12
1,pω
D × Ω, then u − ϕ, u ϕ ∈ W0
!
Lu, ϕ 0.
1,pω
D Ax, ω, u1 , ∇u1 D × Ω. So
3.13
That is to say u is the weak solution of 1.1. If u1 , u2 ∈ W0
then
E
D × Ω
D×Ω are both weak solutions,
− Ax, ω, u2 , ∇u2 ∇u1 − u2 Bx, ω, u1 , ∇u1 − Bx, ω, u2 , ∇u2 u1 − u2 dx 0,
3.14
so u1 u2 , a.e. in D × Ω. If not, it is contradicting to H2. Thus, the weak solution is unique.
Now the proof is completed.
12
Abstract and Applied Analysis
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