Research Article Function Spaces with a Random Variable Exponent

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1 Abstract and Applied Analysis Volume 211, Article I 17968, 12 pages doi:1.1155/211/17968 Research Article Function Spaces with a Random Variable Exponent Boping Tian, Yongqiang Fu, and Bochi Xu epartment of Mathematics, Harbin Institute of Technology, Harbin 151, China Correspondence should be addressed to Bochi Xu, xubochi@126.com Received 11 February 211; Accepted 12 April 211 Academic Editor: Wolfgang Ruess Copyright q 211 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 L Ω and W k, Ω are introduced. After discussing the properties of the spaces L Ω and W k, 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 L p x Ω and W k,p x Ω have been discussed by Kováčik and Rákosník. Sometheories of variableexponent spacescan 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, iening 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, wefirstintroducel Ω and W k, 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 A x, ω, u, u B x, ω, u, u f x, ω, x, ω Ω, u, x, ω Ω. 1.1

2 2 Abstract and Applied Analysis A and B are Carathéodory functions, which are integrable on Ω and continuous for u and u. f x, ω is an integrable function on Ω. Ω, F,P is a complete probability space, and is a bounded open subset of R n n>1. The random variable p : Ω 1, satisfies 1 <p p <.Furthermore, A : R n Ω R R n R n, B : R n Ω R R n R, f : R n Ω R 1.2 satisfy the following growth conditions: H1 A x, ω, s, ξ β 1 ω ξ 1 β 2 ω s 1 K 1 x, ω, B x, ω, s, ξ β 3 ω ξ 1 β 4 ω s 1 K 2 x, ω, H2 E A x, ω, s 1,ξ A x, ω, s 2,η ξ η B x, ω, s 1,ξ B x, ω, s 2,η s 1 s 2 >, ξ/ η or s 1 / s 2, H3 E A x, ω, s, ξ ξ B x, ω, s, ξ s βe ξ s, a.e. in, where K 1 x, ω,k 2 x, ω L p ω Ω, β>, β i ω i 1, 2, 3, 4 are nonnegative bounded random variables, f L p ω Ω, and1/ 1/p ω Some Properties of Function Spaces with a Random Variable Exponent Let λ be a product measure on Ω and u x, ω a Lebesgue measurable function on Ω. In this section, p : Ω 1, is a random variable. On the set of all functions defined on Ω, the functional ρ is defined by ρ u E u x, ω dx. 2.1 efinition 2.1. The space L Ω is the set of Lebesgue measurable functions u on Ω such that Ω u x, ω dλ <, and it is endowed with the following norm: { ( u } u inf λ>:ρ λ efinition 2.2. The space W k, Ω is the set of functions such that α u L Ω, α k, and it is endowed with the following norm: u k, α u. α k 2.3 Here α u is the derivative of u with respect to x in the distribution sense. efinition 2.3. The space W k, Ω is the closure of C Ω {u W k, Ω : u,ω C for each ω Ω} in W k, Ω.

3 Abstract and Applied Analysis 3 Theorem 2.4. If satisfies 1 <p p <, then the inequality E f x, ω g x, ω dx C f g p ω 2.4 holds for every f L Ω and g L p ω Ω with the constant C dependent on only. Proof. By Young inequality, we have f x, ω f g x, ω gp ω 1 ( f x, ω f 1 p ω ( g x, ω p ω. 2.5 gp ω Integrating over Ω,weobtain E f x, ω g x, ω dx 1 1 p 1 p. f g p ω 2.6 So ( E f x, ω g x, ω dx 1 1 p 1 f gp p ω. 2.7 Now the proof is completed. Theorem 2.5. Suppose that satisfies 1 <p p <.Ifu k, u L Ω, then 1 if u 1,then u p ρ u u p, 2 if u 1,then u p ρ u u p, 3 lim k u k ifandonlyiflim k ρ u k, 4 lim k u k if and only if lim k ρ u k. Proof. 1 By u 1 and the definition of the norm, E u u p dx E ( u u dx So ρ u u p.as u p / u, 2.9

4 4 Abstract and Applied Analysis we also have E u u p / dx That is to say u p ρ u. The proof is completed. 2, 3,and 4 can be easily proved with similar methods. Theorem 2.6. If satisfies 1 <p p <, thespacel Ω is complete. Proof. Let u n be a Cauchy sequence in L Ω. Then, by Theorem 2.4, E u m x, ω u n x, ω dx C u m u n χp ω, 2.11 where C is constant. That is to say u n is a Cauchy sequence in L 1 Ω. Inviewofthe completeness of L 1 Ω, u n converges in L 1 Ω. Suppose that u n u, u L 1 Ω and further suppose that u n x, ω u x, ω a.e. in Ω subtracting a subsequence if necessary. Foreach<ε<1, there exists n such that u m u n <εfor m, n n. Fix n, by Fatou s lemma E ( un x, ω u x, ω ( dx E lim sup ε m lim sup E m 1. ( un x, ω u m x, ω dx ε ( un x, ω u m x, ω dx ε 2.12 So u n u ε, and further ρ u n u u n u p ;thatistosay,u εp n u L Ω.Nextasu n L Ω,wehaveu L Ω. Now the proof is completed. Theorem 2.7. If satisfies 1 <p p <, then the space L Ω is reflexive. Proof. First, suppose that v L p ω Ω is fixed, and let L v u E uv dx, 2.13 for every u L Ω.

5 Abstract and Applied Analysis 5 Note that L v u C u v p ω So L v u L Ω. Second, we show that each bounded linear functional on L Ω is of the form L v u E uvdx for every v Lp ω Ω. LetL L Ω be given. Let S be a subset of Ω. Wedefine μ S L ( χ S, 2.15 where χ S is the characteristic function of S; then μ S L χs ( 1/p ( 1/p } L max {ρ χs, ρ χs { L max meas S 1/p, meas S 1/p }, 2.16 so μ is absolutely continuous with respect to the measure λ. Alsowehavethatμ is σ finite measure. By Radon-Nikodym theorem, there is an integrable function ṽ on Ω such that μ S S ṽdλ E ṽχ S dx Now L u E uṽdx holds for simple functions u.ifu L Ω, there is a sequence of simple function u j, converging a.e. to u and u j x, ω u x, ω on Ω. ByFatou s lemma E uṽdx lim sup E uj ṽ dx j lim sup L ( uj sgn ṽ j L lim sup j L u. uj 2.18 Then Lṽ u E uṽdx 2.19

6 6 Abstract and Applied Analysis is also a bounded linear functional on L Ω. By Lebesgue theorem lim E j uj u dx E lim uj u dx. j 2.2 By Theorem 2.5, wehavethat lim uj u, j 2.21 that is, u j u.asl u j Lṽ u j, by letting j,wehavethatl u Lṽ u. At last, we show that ṽ L p ω Ω. Let E l { x, ω Ω : ṽ x, ω l} As Ω ṽχel p ω dλ E l l p ω dλ, 2.23 ṽχ El L p ω Ω. Suppose that ṽχ El >, and take ( 1/ 1 ṽ u χ El sgn ṽ ṽχelp ω Then E uṽdx E ṽχ p El ω E 2 p > ṽχel p ω 2 p. ( ṽ p ω ṽχel χ El ṽχ p p ω dx El ω χ El ( ṽ 1/2 ṽχel p ω p ω dx 2.25 As u inf λ>:e χ El ( ṽ λ ṽχel p ω p ω dx 1 1, 2.26

7 Abstract and Applied Analysis 7 we have that E ṽχel p ω 2p L, ( ṽχ El 2 p L p ω dx By Fatou s lemma E ( ṽ p ω dx 2 p L lim sup l E ( ṽχel 2 p L p ω dx So ṽ L p ω Ω. Now we reach the conclusion that L p ω Ω L Ω, and, furthermore, L Ω is reflexive. Theorem 2.8. If satisfies 1 <p p <, then the space W k, Ω is a reflexive Banach space. Proof. W k, Ω can be treated as a subspace of the product space L Ω, m 2.29 where m is the number of multi-indices α with α k. Then we need only to show that W k, Ω is a closed subspace of m L Ω. Let {u n } W k, Ω be a convergent sequence. Then {u n } is a convergent sequence in L Ω,sothereexistsu L Ω such that u n u in L Ω by Theorem 2.6. Similarly, there exists u α L Ω such that α u n u α in L Ω for α k.as,forϕ C Ω, 1 α E α u n ϕdx E u n α ϕdx, 2.3 we have 1 α E u α ϕdx E u α ϕdx, 2.31 as n. By the definition of weak derivative, α u u α holds for each α k. So α u L Ω. ThenW k, Ω is a closed subspace of m L Ω. Theorem 2.9. Suppose that the sequence u n L Ω is bounded in L Ω.Ifu n u a.e. in Ω,thenu n uin L Ω.

8 8 Abstract and Applied Analysis Proof. By Theorem 2.7, we need only to show that E u n gdx E ug dx 2.32 for each g L p ω Ω. Let u n C for each n N. By Fatou s lemma, E u C dx lim sup E n u n C dx 1, 2.33 so u C. By the absolute continuity of Lebesgue integral, lim gχ E p ω dλ, meas E Ω 2.34 where g L p ω Ω and E Ω. ByTheorem 2.5, lim meas E gχ E p ω. So there exists δ>, such that gχ p E ω < 1 ( 4C ε 1 1 p 1 1 p, 2.35 for meas E <δ. By Egorov theorem, there exists a set B Ω such that u n u uniformly on B with meas Ω \ B <δ. Choose n such that n>n implies max u u n gp x,ω B ω χbp ω (1 1p 1p < ε Taking E Ω \ B, E ug dx E u n gdx u n u g dλ u n u g dλ B E max x,ω B u u n E gχb dx E max u u n g x,ω B p χ p ω B (1 1p ω 1p (1 1p 1p u n u gχe dx 2.37 u n u gχ E p ω <ε.

9 Abstract and Applied Analysis 9 That is to say u n uweakly in L Ω. 3. An Application to Partial ifferential Equations with Random Variable Growth efinition 3.1. u W 1, Ω is said to be the weak solution of 1.1 if E A x, ω, u, u ϕ B x, ω, u, u ϕdx E f x, ω ϕdx 3.1 for all ϕ W 1, Ω. efinition 3.2. Let X be a reflexive Banach space with dual X,andlet, denote a pairing between X and X.IfK X is a closed convex set, then a mapping L : K X is called monotone if Lu Lv, u v forallu, v K. Further,L is called coercive on K if there exists φ K such that Luj Lϕ, u j ϕ uj ϕ X, 3.2 whenever u j is sequence in K with u j ϕ X. Theorem 3.3 see 8. Let K be a nonempty closed convex subset of X, andletl : K X be monotone, coercive and strong-weakly continuous on K. Then there exists an element u in K such that Lu, v u for all v K. In the following, let K W 1, Ω. ThenitisobviousthatK is a closed convex subset of X W 1, Ω. LetL : K X, Lu, v E A x, ω, u, u v B x, ω, u, u vdx f x, ω vdx, 3.3 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,forallu K,by H1, H3, and Young Equality, L u L ( ϕ,u ϕ E A x, ω, u, u u B x, ω, u, u udx A ( x, ω, ϕ, ϕ ϕ B ( x, ω, ϕ, ϕ ϕdx A x, ω, u, u ϕ A ( x, ω, ϕ, ϕ u

10 1 Abstract and Applied Analysis B x, ω, u, u ϕ B ( x, ω, ϕ, ϕ udx ( β ( 4β 1 ε ( ρ u ρ u ( β ( 4β 1 C ε ( ρ ( ϕ ρ ( ϕ C ε ε ( ρ p ω K 1 x, ω ρ p ω K 2 x, ω, 3.4 where β is the bound of nonnegative bounded random variable β i ω, i 1, 2, 3, 4, ε is small enough such that β 4β 1 ε >, and C ε is constant only dependent on ε.then L u L ( ϕ,u ϕ u ϕw 1, Ω ( ( β 4β 1 ε ( ρ u ρ u u W 1, Ω ϕ W 1, Ω C( ϕ, ϕ, K 1,K 2,ε. u ϕw 1, Ω 3.5 For ε 1,ε 2 > small enough, we have that ρ u ρ u u ( u E u ε 1 ( u ε 2 dx ( u ε 1 ( u ε 2 min{ ( u ε 1 p ( } p ( p (, u ε 1 min{ } p u ε 2, u ε Thus, we have that L u L ( ϕ,u ϕ u ϕ W 1, Ω, 3.7 as u W 1, Ω. The proof is completed. Lemma 3.5. L is strong-weakly continuous. Proof. Let u n W 1, Ω be a sequence that converges to u W 1, Ω. Then u n C for some constant C, and there exists a subsequence such that u nk u, u nk u, 3.8

11 Abstract and Applied Analysis 11 a.e. in Ω. A, B are Caratheodory functions, so A x, ω, u nk, u nk A x, ω, u, u, B x, ω, u nk, u nk B x, ω, u, u, 3.9 a.e. in Ω. ByTheorem 2.9, A x, ω, u nk, u nk A x, ω, u, u, B x, ω, u nk, u nk B x, ω, u, u, 3.1 and A x, ω, u nk, u nk and B x, ω, u nk, u nk are bounded. For all ϕ W 1, Ω, we have that L unk,ϕ E A x, ω, u nk, u nk ϕ B x, ω, u nk, u nk ϕdx E L u,ϕ, A x, ω, u, u ϕ B x, ω, u, u ϕdx f x, ω ϕdx f x, ω ϕdx 3.11 that is to say L is strong-weakly continuous. Theorem 3.6. Under conditions (H1 (H3, there exists a unique weak solution u W 1, Ω to 1.1 for any f L p ω Ω. Proof. From Theorem 3.3, Lemmas 3.4,and3.5, thereexistsu W 1, Ω such that Lu, v u, 3.12 for all v W 1, Ω. Letϕ W 1, Ω, thenu ϕ, u ϕ W 1, Ω. So Lu, ϕ That is to say u is the weak solution of 1.1.Ifu 1,u 2 W 1, Ω are both weak solutions, then E A x, ω, u 1, u 1 A x, ω, u 2, u 2 u 1 u 2 B x, ω, u 1, u 1 B x, ω, u 2, u 2 u 1 u 2 dx, 3.14 so u 1 u 2,a.e.in Ω. If not, it is contradicting to H2. Thus, the weak solution is unique. Now the proof is completed.

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