# Function Spaces with a Random Variable Exponent

Authors: Boping Tian; Yongqiang Fu; Bochi Xu
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

The spaces with a random variable exponentLp(ω)(D×Ω) and
Wk,p(ω)(D×Ω) are introduced. After discussing the properties of the spaces
Lp(ω)(D×Ω) and Wk,p(ω)(D×Ω), we give an application of these spaces to the stochastic partial differential equations with random variable growth.


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## Body

## 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 Lp(x)(Ω) and Wk,p(x)(Ω) have been discussed by Kováčik and Rákosník. 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 Lp(ω)(D×Ω) and Wk,p(ω)(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:(1.1)-div⁡A(x,ω,u,∇u)+B(x,ω,u,∇u)=f(x,ω),(x,ω)∈D×Ω,u=0,(x,ω)∈∂D×Ω.A and B are Carathéodory functions, which are integrable on D×Ω and continuous for u and ∇u. f(x,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-≤p(ω)≤p+<∞. Furthermore, (1.2)A:Rn×Ω×R×Rn⟶Rn,B:Rn×Ω×R×Rn⟶R,f:Rn×Ω⟶R
satisfy the following growth conditions: (H1)
|A(x,ω,s,ξ)|β1(ω)|ξ|p(ω)-1+β2(ω)|s|p(ω)-1+K1(x,ω), |B(x,ω,s,ξ)|≤β3(ω)|ξ|p(ω)-1+β4(ω)|s|p(ω)-1+K2(x,ω),(H2)
E((A(x,ω,s1,ξ)-A(x,ω,s2,η))(ξ-η)+(B(x,ω,s1,ξ)-B(x,ω,s2,η))(s1-s2))>0,
ξ≠ηors1≠s2,(H3)
E(A(x,ω,s,ξ)ξ+B(x,ω,s,ξ)s)≥βE(|ξ|p(ω)+|s|p(ω)),a.e.inD,whereK1(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/p(ω)+1/p′(ω)=1.


## 2. Some Properties of Function Spaces with a Random Variable Exponent

Letλ be a product measure on D×Ω and u(x,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 onD×Ω, the functional ρp(ω) is defined by (2.1)ρp(ω)(u)=E(∫D|u(x,ω)|p(ω)dx).Definition 2.1.
The spaceLp(ω)(D×Ω) is the set of Lebesgue measurable functions u on D×Ω such that ∫D×Ω|u(x,ω)|p(ω)dλ<∞, and it is endowed with the following norm:
(2.2)‖u‖p(ω)=inf⁡{λ>0:ρp(ω)(uλ)≤1}.Definition 2.2.
The spaceWk,p(ω)(D×Ω) is the set of functions such that Dαu∈Lp(ω)(D×Ω), |α|≤k, and it is endowed with the following norm:
(2.3)‖u‖k,p(ω)=∑|α|≤k‖Dαu‖p(ω).
Here Dαu is the derivative of u with respect to x in the distribution sense.Definition 2.3.
The spaceW0k,p(ω)(D×Ω) is the closure of C(D×Ω)={u∈Wk,p(ω)(D×Ω):u(·,ω)∈C0∞(D)foreachω∈Ω} in Wk,p(ω)(D×Ω).Theorem 2.4.
Ifp(ω) satisfies 1<p-≤p(ω)≤p+<∞, then the inequality
(2.4)E(∫D|f(x,ω)g(x,ω)|dx)≤C‖f‖p(ω)‖g‖p′(ω)
holds for every f∈Lp(ω)(D×Ω) and g∈Lp′(ω)(D×Ω) with the constant C  dependent on p(ω) only.Proof.
By Young inequality, we have(2.5)|f(x,ω)|‖f‖p(ω)|g(x,ω)|‖g‖p′(ω)≤1p(ω)(|f(x,ω)|‖f‖p(ω))p(ω)+1p′(ω)(|g(x,ω)|‖g‖p′(ω))p′(ω).
Integrating over D×Ω, we obtain
(2.6)E(∫D|f(x,ω)|∥f∥p(ω)|g(x,ω)|∥g∥p′(ω)dx)≤1+1p--1p+.
So
(2.7)E(∫D|f(x,ω)g(x,ω)|dx)≤(1+1p--1p+)‖f‖p(ω)‖g‖p′(ω).
Now the proof is completed.Theorem 2.5.
Suppose thatp(ω) satisfies 1<p-≤p(ω)≤p+<∞. If uk, u∈Lp(ω)(D×Ω), then(1)
if∥u∥p(ω)≥1, then ∥u∥p(ω)p-≤ρp(ω)(u)≤∥u∥p(ω)p+,(2)
if∥u∥p(ω)≥1, then ∥u∥p(ω)p+≤ρp(ω)(u)≤∥u∥p(ω)p-,(3)
limk→∞∥uk∥p(ω)=0 if and only if limk→∞ρp(ω)(uk)=0,(4)
limk→∞∥uk∥p(ω)=∞ if and only if limk→∞ρp(ω)(uk)=∞.Proof.
(1) By∥u∥p(ω)≥1 and the definition of the norm,
(2.8)E(∫D|u|p(ω)‖u‖p(ω)p+dx)≤E(∫D(|u|‖u‖p(ω))p(ω)dx)≤1.
So ρp(ω)(u)≤∥u∥p(ω)p+. As
(2.9)‖u‖p(ω)p-/p(ω)≤‖u‖p(ω),
we also have
(2.10)E(∫D(|u|‖u‖p(ω)p-/p(ω))p(ω)dx)≥1.
That is to say ∥u∥p(ω)p-≤ρp(ω)(u). The proof is completed. (2), (3), and (4) can be easily proved with similar methods.Theorem 2.6.
Ifp(ω) satisfies 1<p-≤p(ω)≤p+<∞, the space Lp(ω)(D×Ω) is complete.Proof.
Letun be a Cauchy sequence in Lp(ω)(D×Ω). Then, by Theorem 2.4,
(2.11)E(∫D|um(x,ω)-un(x,ω)|dx)≤C‖um-un‖p(ω)‖χD‖p′(ω),
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)→u(x,ω) a.e. in D×Ω (subtracting a subsequence if necessary). For each 0<ɛ<1, there exists n0 such that ∥um-un∥<ɛ for m,n≥n0. Fix n, by Fatou’s lemma
(2.12)E(∫D(|un(x,ω)-u(x,ω)|ɛ)p(ω)dx)≤E(limsup⁡m→∞∫D(|un(x,ω)-um(x,ω)|ɛ)p(ω)dx)≤limsup⁡m→∞E(∫D(|un(x,ω)-um(x,ω)|ɛ)p(ω)dx)≤1.
So ∥un-u∥p(ω)≤ɛ, and further ρp(ω)(un-u)≤∥un-u∥p(ω)p-≤ɛp-; that is to say, un-u∈Lp(ω)(D×Ω). Next as un∈Lp(ω)(D×Ω), we have u∈Lp(ω)(D×Ω). Now the proof is completed.Theorem 2.7.
Ifp(ω) satisfies 1<p-≤p(ω)≤p+<∞, then the space Lp(ω)(D×Ω) is reflexive.Proof.
First, suppose thatv∈Lp′(ω)(D×Ω) is fixed, and let
(2.13)Lv(u)=E(∫Duvdx),
for every u∈Lp(ω)(D×Ω).
Note that(2.14)|Lv(u)|≤C‖u‖p(ω)‖v‖p′(ω).SoLv(u)∈[Lp(ω)(D×Ω)]′.
Second, we show that each bounded linear functional onLp(ω)(D×Ω) is of the form Lv(u)=E(∫Duvdx) for every v∈Lp′(ω)(D×Ω). Let L∈[Lp(ω)(D×Ω)]′ be given. Let S be a subset of D×Ω. We define
(2.15)μ(S)=L(χS),
where χS  is the characteristic function of S; then
(2.16)|μ(S)|≤‖L‖‖χS‖p(ω)≤‖L‖max⁡{ρp(ω)(χS)1/p-,ρp(ω)(χS)1/p+}=‖L‖max⁡{meas⁡(S)1/p-,meas⁡(S)1/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 ṽ on D×Ω such that
(2.17)μ(S)=∫Sṽdλ=E(∫DṽχSdx).
Now L(u)=E(∫Duṽdx) holds for simple functions u. If u∈Lp(ω)(D×Ω), there is a sequence of simple function uj, converging a.e. to u and |uj(x,ω)|≤|u(x,ω)| on D×Ω. By Fatou’s lemma
(2.18)|E(∫Duṽdx)|≤limsup⁡j→∞E(∫D|ujṽ|dx)=limsup⁡j→∞L(|uj|sgn⁡ṽ)≤‖L‖limsup⁡j→∞‖uj‖p(ω)≤‖L‖‖u‖p(ω).
Then
(2.19)Lṽ(u)=E(∫Duṽdx)
is also a bounded linear functional on Lp(ω)(D×Ω). By Lebesgue theorem
(2.20)lim⁡j→∞E(∫D|uj-u|p(ω)dx)=E(∫Dlim⁡j→∞|uj-u|p(ω)dx)=0.
By Theorem 2.5, we have that
(2.21)lim⁡j→∞‖uj-u‖p(ω)=0,
that is, uj→u. As L(uj)=Lṽ(uj), by letting j→∞, we have that L(u)=Lṽ(u).
At last, we show thatṽ∈Lp′(ω)(D×Ω). Let
(2.22)El={(x,ω)∈D×Ω:|ṽ(x,ω)|≤l}.
As
(2.23)∫D×Ω|ṽχEl|p′(ω)dλ≤∫El|l|p′(ω)dλ≤∞,ṽχEl∈Lp′(ω)(D×Ω). Suppose that ∥ṽχEl∥>0, and take
(2.24)u=χEl(|ṽ|‖ṽχEl‖p′(ω))1/(p(ω)-1)sgn⁡ṽ.
Then
(2.25)|E(∫Duṽdx)|=|E(∫DχEl(|ṽ|‖ṽχEl‖p′(ω))p′(ω)‖ṽχEl‖p′(ω)dx)|≥‖ṽχEl‖p′(ω)2p+E(∫DχEl(|ṽ|(1/2)‖ṽχEl‖p′(ω))p′(ω)dx)>‖ṽχEl‖p′(ω)2p+.
As
(2.26)‖u‖p(ω)=inf⁡{λ>0:E(∫DχEl(|ṽ|λ‖ṽχEl‖p′(ω))p′(ω)dx)≤1}=1,
we have that
(2.27)‖ṽχEl‖p′(ω)≤2p+‖L‖,E(∫D(|ṽχEl|2p+‖L‖)p′(ω)dx)≤1.
By Fatou’s lemma
(2.28)E(∫D(|ṽ|2p+‖L‖)p′(ω)dx)≤limsup⁡l→∞E(∫D(|ṽχEl|2p+∥L∥)p′(ω)dx)≤1.
Soṽ∈Lp′(ω)(D×Ω). Now we reach the conclusion that Lp′(ω)(D×Ω)=[Lp(ω)(D×Ω)]′, and, furthermore, Lp(ω)(D×Ω) is reflexive.Theorem 2.8.
Ifp(ω) satisfies 1<p-≤p(ω)≤p+<∞, then the space Wk,p(ω)(D×Ω) is a reflexive Banach space.Proof.
Wk,p(ω)(D×Ω) can be treated as a subspace of the product space
(2.29)∏mLp(ω)(D×Ω),
where m is the number of multi-indices α with |α|≤k.
Then we need only to show thatWk,p(ω)(D×Ω) is a closed subspace of ∏mLp(ω)(D×Ω). Let {un}⊂Wk,p(ω)(D×Ω) be a convergent sequence. Then {un} is a convergent sequence in Lp(ω)(D×Ω), so there exists u∈Lp(ω)(D×Ω) such that un→u in Lp(ω)(D×Ω) by Theorem 2.6.
Similarly, there existsuα∈Lp(ω)(D×Ω) such that Dαun→uα in Lp(ω)(D×Ω) for |α|≤k. As, for φ∈C(D×Ω),
(2.30)(-1)|α|E(∫DDαunφdx)=E(∫DunDαφdx),
we have
(2.31)(-1)|α|E(∫Duαφdx)=E(∫DuDαφdx),
as n→∞. By the definition of weak derivative, Dαu=uα holds for each |α|≤k. So Dαu∈Lp(ω)(D×Ω). Then Wk,p(ω)(D×Ω) is a closed subspace of ∏mLp(ω)(D×Ω).Theorem 2.9.
Suppose that the sequenceun∈Lp(ω)(D×Ω) is bounded in Lp(ω)(D×Ω). If un→u a.e. in D×Ω, then un⇀u in Lp(ω)(D×Ω).Proof.
By Theorem2.7, we need only to show that
(2.32)E(∫Dungdx)⟶E(∫Dugdx)
for each g∈Lp′(ω)(D×Ω).
Let∥un∥p(ω)≤C for each n∈N. By Fatou’s lemma,
(2.33)E(∫D|uC|p(ω)dx)≤limsup⁡n→∞E(∫D|unC|p(ω)dx)≤1,
so ∥u∥p(ω)≤C. By the absolute continuity of Lebesgue integral,
(2.34)lim⁡meas⁡(E)→0∫D×Ω|gχE|p′(ω)dλ=0,
where g∈Lp′(ω)(D×Ω) and E⊂D×Ω. By Theorem 2.5, limmeas(E)→0∥gχE∥p′(ω)=0. So there exists δ>0, such that
(2.35)‖gχE‖p′(ω)<14Cε(1+1p--1p+)-1,
for meas(E)<δ. By Egorov theorem, there exists a set B⊂D×Ω such that un→u uniformly on B with meas(Ω∖B)<δ. Choose n0 such that n>n0 implies
(2.36)max⁡(x,ω)∈B|u-un|‖g‖p′(ω)‖χB‖p′(ω)(1+1p--1p+)<ε2.
Taking E=D×Ω∖B,
(2.37)|E(∫Dugdx)-E(∫Dungdx)|≤∫B|un-u||g|dλ+∫E|un-u||g|dλ≤max⁡(x,ω)∈B|u-un|E(∫D|gχB|dx)+E(∫D|un-u||gχE|dx)≤max⁡(x,ω)∈B|u-un|‖g‖p′(ω)‖χB‖p′(ω)(1+1p--1p+)+‖un-u‖p(ω)‖gχE‖p′(ω)(1+1p--1p+)<ε.
That is to say un⇀u weakly in Lp(ω)(D×Ω).


## 3. An Application to Partial Differential Equations with Random Variable Growth

Definition 3.1.
u∈W01,p(ω)(D×Ω) is said to be the weak solution of (1.1) if
(3.1)E(∫DA(x,ω,u,∇u)∇φ+B(x,ω,u,∇u)φdx)=E(∫Df(x,ω)φdx)
for all φ∈W1,p(ω)(D×Ω).Definition 3.2.
LetX 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
(3.2)〈Luj-Lφ,uj-φ〉‖uj-φ‖X⟶∞,
whenever uj is sequence in K with ∥uj-φ∥X→∞.Theorem 3.3 (see [8]).
LetK 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.In the following, letK=W01,p(ω)(D×Ω). Then it is obvious that K is a closed convex subset of X=W1,p(ω)(D×Ω). Let ℒ:K→X′, (3.3)〈Lu,v〉=E(∫DA(x,ω,u,∇u)∇v+B(x,ω,u,∇u)vdx-∫Df(x,ω)vdx),
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,
(3.4)〈L(u)-L(φ),u-φ〉=E(∫DA(x,ω,u,∇u)∇u+B(x,ω,u,∇u)udx+∫DA(x,ω,φ,∇φ)∇φ+B(x,ω,φ,∇φ)φdx-∫DA(x,ω,u,∇u)∇φ+A(x,ω,φ,∇φ)∇u+B(x,ω,u,∇u)φ+B(x,ω,φ,∇φ)udx∫DA(x,ω,u,∇u)∇φ)≥(β-(4β0+1)ε)(ρp(ω)(|∇u|)+ρp(ω)(u))+(β-(4β0+1)C(ε))(ρp(ω)(|∇φ|)+ρp(ω)(φ))-(C(ɛ)+ɛ)(ρp′(ω)(K1(x,ω))+ρp′(ω)(K2(x,ω))),
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+1)ε)>0, and C(ε) is constant only dependent on ε. Then
(3.5)〈L(u)-L(φ),u-φ〉‖u-φ‖W01,p(ω)(D×Ω)≥(β-(4β0+1)ε)(ρp(ω)(|∇(u)|)+ρp(ω)(u))‖u‖W01,p(ω)(D×Ω)+‖φ‖W01,p(ω)(D×Ω)+C(φ,∇φ,K1,K2,ε)‖u-φ‖W01,p(ω)(D×Ω).
For ε1,ε2>0 small enough, we have that
(3.6)ρp(ω)(|∇(u)|)+ρp(ω)(u)=E(∫D(|∇u|p(ω)(‖∇u‖p(ω)-ε1)p(ω)(∥∇u∥p(ω)-ɛ1)p(ω)+|u|p(ω)(‖u‖p(ω)-ε2)p(ω)(‖u‖p(ω)-ε2)p(ω))dx)≥min⁡{(‖∇u‖p(ω)-ε1)p-,(‖∇u‖p(ω)-ε1)p+}+min⁡{(‖u‖p(ω)-ε2)p-,(‖u‖p(ω)-ε2)p+}.
Thus, we have that
(3.7)〈L(u)-L(φ),u-φ〉‖u-φ‖W01,p(ω)(D×Ω)⟶∞,
as ∥u∥W01,p(ω)(D×Ω)→∞. The proof is completed.Lemma 3.5.
ℒ is strong-weakly continuous.Proof.
Letun∈W01,p(ω)(D×Ω) be a sequence that converges to u∈W01,p(ω)(D×Ω). Then ∥un∥≤C for some constant C, and there exists a subsequence such that
(3.8)unk⟶u,∇unk⟶∇u,
a.e. in D×Ω. A,B are Caratheodory functions, so
(3.9)A(x,ω,unk,∇unk)⟶A(x,ω,u,∇u),B(x,ω,unk,∇unk)⟶B(x,ω,u,∇u),
a.e. in D×Ω. By Theorem 2.9,
(3.10)A(x,ω,unk,∇unk)⇀A(x,ω,u,∇u),B(x,ω,unk,∇unk)⇀B(x,ω,u,∇u),
and A(x,ω,unk,∇unk) and B(x,ω,unk,∇unk) are bounded. For all φ∈W1,p(ω)(D×Ω), we have that
(3.11)〈L(unk),φ〉=E(∫DA(x,ω,unk,∇unk)∇φ+B(x,ω,unk,∇unk)φdx-∫Df(x,ω)φdx)⟶E(∫DA(x,ω,u,∇u)∇φ+B(x,ω,u,∇u)φdx-∫Df(x,ω)φdx)=〈L(u),φ〉,
that is to say ℒ is strong-weakly continuous.Theorem 3.6.
Under conditions (H1)–(H3), there exists a unique weak solutionu∈W01,p(ω)(D×Ω) to (1.1) for any f∈Lp′(ω)(D×Ω).Proof.
From Theorem3.3, Lemmas 3.4, and 3.5, there exists u∈W01,p(ω)(D×Ω) such that
(3.12)〈Lu,v-u〉≥0,
for all v∈W01,p(ω)(D×Ω). Let φ∈W01,p(ω)(D×Ω), then u-φ,u+φ∈W01,p(ω)(D×Ω). So
(3.13)〈Lu,φ〉=0.
That is to say u is the weak solution of (1.1). If u1,u2∈W01,p(ω)(D×Ω) are both weak solutions, then
(3.14)E(∫D(A(x,ω,u1,∇u1)-A(x,ω,u2,∇u2))∇(u1-u2)+(B(x,ω,u1,∇u1)-B(x,ω,u2,∇u2))(u1-u2)dx)=0,
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.



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*Source: 179068-2011-06-05.xml*