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2000 | 143 | 3 | 267-293
Tytuł artykułu

Sobolev embeddings with variable exponent

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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
143
Numer
3
Strony
267-293
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-10-25
poprawiono
2000-09-01
Twórcy
  • Centre for Mathematical Analysis, and Its Applications, School of Mathematical Sciences, University of Sussex, Brighton BN1 9QH, United Kingdom
  • 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.
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  • [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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  • [M1] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non-standard growth conditions, Arch. Rational Mech. Anal. 105 (1989), 267-284.
  • [M2] P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differential Equations 105 (1993), 296-333.
  • [Ma] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Univ. Press, Cambridge, 1999.
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  • [MS] N. G. Meyers and J. Serrin, H=W, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 1055-1056.
  • [Mu] J. Musielak, Orlicz Spaces and Modular Spaces, Springer, Berlin, 1983.
  • [R1] M. Růžička, Electrorheological fluids: mathematical modelling and existence theory, Habilitationsschrift, Universität Bonn, 1998.
  • [R2] M. Růžička, Flow of shear dependent electrorheological fluids, C. R. Acad. Sci. Paris Sér. I 329 (1999), 393-398.
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  • [Z] W. P. Ziemer, Weakly Differentiable Functions, Grad. Texts in Math. 120, Springer, New York, 1989.
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Bibliografia
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