Geometry of Sobolev spaces with variable exponent: smoothness and uniform convexity

George Dinca; Pavel Matei

doi:10.1016/j.crma.2009.04.028

Partial Differential Equations/Functional Analysis

Geometry of Sobolev spaces with variable exponent: smoothness and uniform convexity

Presented by: Philippe G. Ciarlet

George Dinca ¹; Pavel Matei 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

Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 885-889.

Abstract
Résumé

Let $Ω \subset R^{N}$ , $N ⩾ 2$ , be a smooth bounded domain. It is shown that: (a) if $p \in L^{\infty} (Ω)$ and $ess \inf_{x \in Ω} p (x) > 1$ , then the generalized Lebesgue space $(L^{p (\cdot)} (Ω), ‖ ‖_{p (\cdot)})$ is smooth; (b) if $p \in C (\bar{Ω})$ and $p (x) > 1$ , for all $x \in \bar{Ω}$ , then the generalized Sobolev space $(W_{0}^{1, p (\cdot)} (Ω), ‖ ‖_{1, p (\cdot)})$ 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 $p \in C (\bar{Ω})$ and $p (x) ⩾ 2$ , for all $x \in \bar{Ω}$ , then $(W_{0}^{1, p (\cdot)} (Ω), ‖ ‖_{1, p (\cdot)})$ is uniformly convex and smooth.

Soit $Ω \subset R^{N}$ , $N ⩾ 2$ , un domain borné et régulier. On demontre que : (a) si $p \in L^{\infty} (Ω)$ et $ess \inf_{x \in Ω} p (x) > 1$ , alors l'espace de Lebesgue généralisé $(L^{p (\cdot)} (Ω), ‖ ‖_{p (\cdot)})$ est lisse ; (b) si $p \in C (\bar{Ω})$ et $p (x) > 1$ , pour tout $x \in \bar{Ω}$ , alors l'espace de Sobolev généralisé $(W_{0}^{1, p (\cdot)} (Ω), ‖ ‖_{1, p (\cdot)})$ 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 $p \in C (\bar{Ω})$ et $p (x) ⩾ 2$ , pour tout $x \in \bar{Ω}$ , alors $(W_{0}^{1, p (\cdot)} (Ω), ‖ ‖_{1, p (\cdot)})$ est uniformément convexe et lisse.

Received: 2009-02-17
Accepted: 2009-03-13
Published online: 2009-06-02

DOI: 10.1016/j.crma.2009.04.028

Author's affiliations:

George Dinca ¹; Pavel Matei 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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     author = {George Dinca and Pavel Matei},
     title = {Geometry of {Sobolev} spaces with variable exponent: smoothness and uniform convexity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {885--889},
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     doi = {10.1016/j.crma.2009.04.028},
     language = {en},
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George Dinca; Pavel Matei. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.028/

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