Partial Differential Equations/Functional Analysis
Geometry of Sobolev spaces with variable exponent: smoothness and uniform convexity
Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 885-889.

Let ΩRN, N2, be a smooth bounded domain. It is shown that: (a) if pL(Ω) and essinfxΩp(x)>1, then the generalized Lebesgue space (Lp()(Ω),p()) is smooth; (b) if pC(Ω¯) and p(x)>1, for all xΩ¯, then the generalized Sobolev space (W01,p()(Ω),1,p()) is smooth. In both situations, the formulae giving the Gâteaux derivative of the norm, corresponding to each of the above spaces, are given; (c) if pC(Ω¯) and p(x)2, for all xΩ¯, then (W01,p()(Ω),1,p()) is uniformly convex and smooth.

Soit ΩRN, N2, un domain borné et régulier. On demontre que : (a) si pL(Ω) et essinfxΩp(x)>1, alors l'espace de Lebesgue généralisé (Lp()(Ω),p()) est lisse ; (b) si pC(Ω¯) et p(x)>1, pour tout xΩ¯, alors l'espace de Sobolev généralisé (W01,p()(Ω),1,p()) est lisse. Dans les deux cas, les formules de la dérivée au sens de Gâteaux de chaque norme des espaces ci-dessus sont données ; (c) si pC(Ω¯) et p(x)2, pour tout xΩ¯, alors (W01,p()(Ω),1,p()) est uniformément convexe et lisse.

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Published online:
DOI: 10.1016/j.crma.2009.04.028
Dinca, George 1; Matei, Pavel 2

1 Faculty of Mathematics and Computer Science, 14, Academiei St, 010014 Bucharest, Romania
2 Department of Mathematics, Technical University of Civil Engineering, 124, Lacul Tei Blvd., 020396 Bucharest, Romania
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Dinca, George; Matei, Pavel. Geometry of Sobolev spaces with variable exponent: smoothness and uniform convexity. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 885-889. doi : 10.1016/j.crma.2009.04.028. http://www.numdam.org/articles/10.1016/j.crma.2009.04.028/

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