Density of smooth functions in W k, p(x) ( Ω )

Abstract

Kováčik & Rákosník investigated the spaces L p(x) ( Ω ) of functions which are integrable with variable power p(x) and the corresponding counterparts of the Sobolev spaces W k , p(x) ( Ω ). We continue that investigation and describe a class of functions p(x) for which the set of smooth functions on Ω is dense in W k , p(x) ( Ω ). As a corollary we obtain in terms of the distance function a condition on elements of W k , p(x) ( Ω ) sufficient to ensure that they belong to W k 0 , p(x) ( Ω ).