Density of compactly supported smooth functions CC∞(Rd) in Musielak-Orlicz-Sobolev spaces W1,Φ(Ω)

We investigate here the density of the set of the restrictions from $C_C^{\infty }\left(R^d\right)$ to $C_C^{\infty }\left(\Omega \right)$ in the Musielak-Orlicz-Sobolev space $W^{1,\Phi }\left(\Omega \right)$. It is a continuation of article [15], where we have studied density of $C_C^{\infty }\left(R^d\right)$ in $W^{k,\Phi }\left(R^d\right)$ for $k\in N$. The main theorem states that for an open subset $\Omega \subset R^d$ with its boundary of class $C^1$, and Musielak-Orlicz function Φ satisfying condition (A1) which is a sort of log-Hölder continuity and the growth condition ${\Delta }_2$, the set of restrictions of functions from $C_C^{\infty }\left(R^d\right)$ to Ω is dense in $W^{1,\Phi }\left(\Omega \right)$. We obtain a corresponding result in variable exponent Sobolev space $W^{1,p(\cdot )}\left(\Omega \right)$ under the assumption that the exponent $p\left(x\right)$ is essentially bounded on Ω and $\Phi (x,t)=t^{p\left(x\right)}$, $t\ge 0$, $x\in \Omega$, satisfies the log-Hölder condition.