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

Abstract

Let Ω ⊂ R N , N ⩾ 2 , be a smooth bounded domain. It is shown that: (a) if p ∈ L ∞ ( Ω ) and ess inf x ∈ Ω p ( x ) > 1 , then the generalized Lebesgue space ( L p ( ⋅ ) ( Ω ) , ‖ ‖ p ( ⋅ ) ) is smooth; (b) if p ∈ C ( Ω ¯ ) and p ( x ) > 1 , for all x ∈ Ω ¯ , then the generalized Sobolev space ( W 0 1 , 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 p ∈ C ( Ω ¯ ) and p ( x ) ⩾ 2 , for all x ∈ Ω ¯ , then ( W 0 1 , p ( ⋅ ) ( Ω ) , ‖ ‖ 1 , p ( ⋅ ) ) is uniformly convex and smooth. To cite this article: G. Dinca, P. Matei, C. R. Acad. Sci. Paris, Ser. I 347 (2009).