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## Studia Mathematica

2000 | 143 | 3 | 267-293
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### Sobolev embeddings with variable exponent

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Let Ω be a bounded open subset of $ℝ^{n}$ with Lipschitz boundary and let $p:\overline{Ω} → [1,∞)$ be Lipschitz-continuous. We consider the generalised Lebesgue space $L^{p(x)}(Ω)$ and the corresponding Sobolev space $W^{1,p(x)}(Ω)$, consisting of all $f ∈ L^{p(x)}(Ω)$ with first-order distributional derivatives in $L^{p(x)}(Ω)$. It is shown that if 1 ≤ p(x) < n for all x ∈ Ω, then there is a constant c > 0 such that for all $f∈ W^{1,p(x)}(Ω)$, $|f|_{M,Ω} ≤ c|f|_{1,p,Ω}$. Here $|·|_{M,Ω}$ is the norm on an appropriate space of Orlicz-Musielak type and $|·|_{1,p,Ω}$ is the norm on $W^{1, p(x)}(Ω)$. The inequality reduces to the usual Sobolev inequality if $sup_Ω p Słowa kluczowe Czasopismo Rocznik Tom Numer Strony 267-293 Opis fizyczny Daty wydano 2000 otrzymano 1999-10-25 poprawiono 2000-09-01 Twórcy autor • Centre for Mathematical Analysis, and Its Applications, School of Mathematical Sciences, University of Sussex, Brighton BN1 9QH, United Kingdom autor • Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, 11567 Praha 1, Czech Republic Bibliografia • [BMS] L. Boccardo, P. Marcellini and C. Sbordone,$L^∞$-regularity for variational problems with sharp nonstandard growth conditions, Boll. Un. Mat. Ital. A (7) 4 (1990), 219-225. • [DL] J. Deny et J.-L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier (Grenoble) 5 (1953/54), 305-370. • [ER] D. E. Edmunds and J. Rákosník, Density of smooth functions in$W^{k,p(x)}(Ω)$, Proc. Roy. Soc. London Ser. A 437 (1992), 229-236. • [EvR] W. D. Evans and J. Rákosník, Anisotropic Sobolev spaces and a quasidistance function, Bull. London Math. Soc. 23 (1991), 59-66. • [FS] N. Fusco and C. Sbordone, Some remarks on the regularity of minima of anisotropic integrals, Comm. Partial Differential Equations 18 (1993), 153-167. • [G] M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math. 59 (1987), 245-248. • [Gu] M. de Guzmán, A covering lemma with application to differentiability of measures and singular integral operators, Studia Math. 34 (1970), 299-317. • [H] M. Hestenes, Extension of the range of a differentiable function, Duke Math. J. 8 (1941), 183-192. • [Hu] H. Hudzik, The problems of separability, duality, reflexivity and of comparison for generalized Orlicz-Sobolev space$W^k_M(Ω)$, Comment. Math. Prace Mat. 21 (1980), 315-324. • [KR] O. Kováčik and J. Rákosník, On spaces$L^{p(x)}(Ω)$and$W^{k,p(x)}(Ω)\$, Czechoslovak Math. J. 41 (116) (1991), 592-618.
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