Density of <i>C</i>^∞(Ω) in<i>W</i>^{1,<i>p</i> (<i>x</i> )}(Ω) with discontinuous exponent <i>p</i> (<i>x</i> )

Abstract

Abstract In this paper, we present a sufficient condition for the density of C ∞ (Ω) in W 1, p ( x ) (Ω) with p ( x ) being discontinuous. Roughly speaking, we can get the density under the following assumptions: 1) Ω can be divided into some open pieces Ω i such that p ( x ) is “fine”on each Ω i , for example, p ( x ) satisfies the Dini–Lipschitz condition on each Ω i . 2) The corresponding pieces “get along with each other regularly ”. (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)