Sobolev-type Inequality for Spaces L p(x) (R N )

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1 In. J. Conemp. Mah. Sciences, Vol. 2, 27, no. 9, Sobolev-ype Inequaliy for Spaces L p(x ( R. Mashiyev and B. Çekiç Universiy of Dicle, Faculy of Sciences and Ars Deparmen of Mahemaics, 228-Diyarbakir, Turkey mrabil@dicle.edu.r & bilalc@dicle.edu.r Absrac Our aim in his paper is o prove Sobolev ype inequaliy for measurable funcions by using serial expansion for he variable exponen Lebesgue spaces and he corresponding Sobolev spaces. We also deermine pre-limi exponen for measurable (may be disconinuous funcions in Sobolev ype inequaliy. Mahemaics Subjec Classificaion: 46E35, 26D Keywords: Variable exponen, Sobolev-ype inequaliy. Inroducion The variable exponen Lebesgue space L p(x (Ω and he corresponding Sobolev space W,p(x (Ω have been a subjec of acive research simulaed by developmen of he sudies of problems in elasiciy, fluid dynamics, calculus of variaions, and differenial equaions wih p(x growh over he las decades [7]. We refer o [6,8] for fundamenal properies of hese spaces, o [,2,3,4,5] for Sobolev ype inequaliies. O.Kováčik and J.Rákosník [6] obained Sobolev-ype inequaliy u q (x, Ω C u p (x, Ω ; u C ( Ω ( for coninuous p(x funcion, where a posiive consan C is independen of u and q ( x p( x p ( x ε = p (x ε, <ε<, and hey also showed ha, in general, he Sobolev space W,p(x (Ω is no embedding in L p (x (Ω.

2 424 R. Mashiyev and B. Çekiç Laer D.E.Edmunds and J.Rákosník [3,4] proved W,p( x (Ω L p (x (Ω for bounded lipschiz domain, where p(x C, ( Ω and p(x q<. X.L.Fan, J.Shen and D.Zhao [5] invesigaed Sobolev-ype embedding for W k,p(x (Ω, where Ω is an open domain in wih cone propery, and p(x is a Lipschiz coninuous funcion defined on saisfying <p p + < k. Recenly, L.Diening [] has obained embedding above for unbounded domain assuming ha p (x is consan a infiniy and saisfies (w-lip. p (x p (y C log x y ; x, y, x y Definiions and Basic Facs Le p : [, be a measurable funcion called he variable exponen on ( 2. We wrie p + := sup p(x and p := inf p(x. We define he x R x variable exponen Lebesgue space L p(x ( o consis of measurable funcions f : R such ha he modular I p (f := f(x p(x dx is finie. If p + <, hen f p(x = inf { ( λ>: I f } p λ, defines a norm on L p(x (. This makes L p(x ( a Banach space. Moreover, if f p(x and only if I p (f. If p is consan, hen L p(x coincides wih he classical Lebesgue space L p and so he noaion can give rise o no confusion. The corresponding Sobolev space W,p(x ( is he class of all funcions f L p(x ( such ha all generalized derivaives D i f, i =,...,n, belongs o L p(x (. Endowed wih he norm f,p(x = f p(x + f p(x i forms a Banach space. Le w be a locally inegrable funcion on such ha w(x > for a.e. x. Suppose furher ha w saisfies. 2 w(xdx C w(xdx for w A, C>and ball Br x Br x 2. sup r> x r [ ] w(ydy C inf w (x where Br x = { y : y x <r }. Then Gagliardo ype inequaliy, can be derived from [9], ( u(x w(x dx C u(x w(x dx (2 holds where u C (R and C is independen of he funcion u.

3 Sobolev-ype Inequaliy Main Resul Theorem. Le p : [, be a measurable funcion such ha < p p(x p + <. Define q : [, by q(x = p(x βp(x( (p(x <p (x where < β. Suppose furher ha he se of funcions { q k+ (x } saisfies i q k (ydy σ k q k (ydy, σ, ii B x 2r [ Br x q k (ydy Br x ] k β inf q(y, k =, 2,...; β>. Then here exiss C> such ha ( holds for u C (R. Proof. Le u C (R and assume ha u q(x =. Then = u(x q(x dx = ( u(x q(x q(x d dx = d ( q(x q(x dx = d ( G q(x q(x dx G (3 where G = { x : u(x > }. ow, using Taylor expansion q(x = e q(xln = (q(xln k (x,> for funcion q(x and subsiuing ino k= (3, we obain = = [ d d G [ k= q(x ln k ] (q(x ln k dx k= ( G q k+ (x dx ] (4

4 426 R. Mashiyev and B. Çekiç From he assumpions of heorem and he inequaliy (2 for he se of he funcions q k+ (x, we obain G q k+ (x dx Cβ k ( k+ q (x u dx, (5 where = { x : < u(x }, = and C>is independen of k. Using he following es funcion 2 { ( u (x 2 z (x = min } +, (6 2 in he inequaliy (2, where (g + = { g(x, g(x, g(x < we have C d ( β k (ln k k= k! k+ q (x u dx from inequaliies (4 and (5. Applying Minkowski s inequliy as form: [ ( ] a a ( a c k f k (x dx c k f k (x a dx we obain c k= d and since e β ln q( = c ( k= k= d β k (ln k q k (x + β k (ln k q k ( we have k= u q(x dx (7 β q(x q (x u dx. (8 Using Young s inequaliy a.b ap(x p(x + bp (x p (x,

5 Sobolev-ype Inequaliy 427 where a, b >,p(x, + p(x p (x = is used in (8, hen we have βq(x q (x u p(x u ε p(x p(x + + ε p (x ( βq(x p (x+ p (x p(x q p (x (x, ε >. Therefore, inequaliy (8 can be rewrien as p(x c d u ε p (x p(x dx + ( d u p(x C ε p(x dx p(x + C d ( ε p (x γ(x p q (x (x dx + ε p ( x γ(x p q ( x ( x dx (9 where γ(x =( βq(x inequaliy + p(x p (x and we have applied he well known (a + b p 2 p (a p + b p, a,b >, p. ow, if we apply Minkowski s inequaliy o each inegral above, hen we ge { [ 2 u ( x ( C ε d u ( x p ( x ] } p ( x dx + p ( x + C { ε p ( x u ( x [ 2 u ( x γ(x d ] q p ( x ( x dx } or u ( x C ln 2 ε u p ( x p (x dx + p ( x { +C 2 ε p ( x u ( x γ(x q p ( x ( x dx }. ( I is easy o see ha [( βq( x + ] p ( x + p ( x = q ( x

6 428 R. Mashiyev and B. Çekiç from definiion of q(x. Since p(x and <ε<, i follows ε p(x ε. Using hese resuls in ( and aking ino accoun of boundedness of funcion q(x and u (x q(x dx =, he inequaliy ( can be rewrien as p ( x C ( ε u dx + C 2 ε p q ( x ( x u ( x q ( x dx and by choosing C 2 ε < we find ha C (ε p ( x u ( x p ( x dx = C C 2 ε 3 u(x dx ( C (ε C 2 ε where C 3 = is a posiive number. Hence, since p(x and C 3 > from inequaliy ( for he funcion u(x, we have u(x q(x C 3 which yields (. The heorem is proved. ( u u(x q(x Remark 2. If he funcion q saisfies sup condiions i and ii is held. p ( x dx q(x β inf q(x, hen above Remark 3. If p is a Lipschiz coninuous funcion, hen by aking β =, we obain ( uq k+ (x ( dx C uq k+ (x dx or ( u q k+ (xdx ( k+ u q (xdx + k+ u q (x q(x dx where C> is independen of k. Hence we have limi case q(x = p(x p(x as β =. ACKOWLEDGEMETS The auhors are hankful o he referees for heir helpful suggesions and valuable conribuions. This research was suppored by DUAPK gran o. 4 FF 4.

7 Sobolev-ype Inequaliy 429 References [] L. Diening, Riesz poenial and Sobolev embeddings of generalized Lebesgue and Sobolev spaces L p (x (Ω and W k, p (x ( Ω. Mah. achr., Ocober 8, 23, 2-5. [2] B. Çekiç, R. Mashiyev and G.T.Alisoy, On The Sobolev-ype Inequaliy for Lebesgue Spaces wih a Variable Exponen.In. Mah. Forum, Vol., no , 26, [3] D.E. Edmunds, J. Rákosník, Sobolev embedding wih variable exponen. Sudia Mah., 43 no.3 2, [4] D.E. Edmunds, J. Rákosník, Sobolev embedding wih variable exponen II. Mah. achr., , 22, [5] X.L. Fan, J.S. Shen, D. Zhao, Sobolev embedding heorems for spaces W k, p (x ( Ω. J. Mah. Anal. Appl., 262, 2, [6] O. Kováčik and J. Rákosník, On spaces L p (x (Ω and W k, p (x ( Ω. Czechoslovak Mah. J., 6, 99, [7] M. Růžička, Elecrorheological Fluids: Modeling and Mahemaical Theo., Springer-Verlag, Berlin, 2. [8] S.G. Samko, Convoluion and poenial ype operaors in L p (x (R n. Inegr. Transform. and Special Func. v.7, , [9] E. Sawyer, R.L. Wheeden, Weighed inequaliies for Fracional Inegrals on Euclidian and Homogeneous Spaces. American Journal of Mah., 4, 992, Received: April 5, 26

On The Sobolev-type Inequality for Lebesgue Spaces with a Variable Exponent

International Mathematical Forum,, 2006, no 27, 33-323 On The Sobolev-type Inequality for Lebesgue Spaces with a Variable Exponent BCEKIC a,, RMASHIYEV a and GTALISOY b a Dicle University, Dept of Mathematics,