Density of smooth functions in <i>Wk, p(x)</i> ( <i>Ω</i> )

Abstract

Abstract Kováčik &amp; Rákosník investigated the spaces Lp(x) (Ω) of functions which are integrable with variable power p(x) and the corresponding counterparts of the Sobolev spaces Wk, 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 Wk, p(x) (Ω). As a corollary we obtain in terms of the distance function a condition on elements of Wk, p(x) (Ω) sufficient to ensure that they belong to Wk0, p(x) (Ω).