Density of compactly supported smooth functions $C_C^{\mathrm{\infty }}({\mathbb{R}}^d)$ in Musielak-Orlicz-Sobolev spaces $W^{1,\mathrm{\Phi }}(\mathrm{\Omega })$

We investigate here the density of the set of the restrictions from $C_C^{\mathrm{\infty }}({\mathbb{R}}^d)$ to $C_C^{\mathrm{\infty }}(\mathrm{\Omega })$ in the Musielak-Orlicz-Sobolev space $W^{1,\mathrm{\Phi }}(\mathrm{\Omega })$. It is a continuation of article \cite{KamZyl3}, where we have studied density of $C_C^{\mathrm{\infty }}({\mathbb{R}}^d)$ in $W^{k,\mathrm{\Phi }}({\mathbb{R}}^d)$ for $k\in \mathbb{N}$. The main theorem states that for an open subset $\mathrm{\Omega }\subset {\mathbb{R}}^d$ with its boundary of class $C^1$, and Musielak-Orlicz function $\mathrm{\Phi }$ satisfying {\rm condition (A1)} which is a sort of log-Hölder continuity and the growth condition ${\mathrm{\Delta }}_2$, the set of restrictions of functions from $C_C^{\mathrm{\infty }}({\mathbb{R}}^d)$ to $\mathrm{\Omega }$ is dense in $W^{1,\mathrm{\Phi }}(\mathrm{\Omega })$. We obtain a corresponding result in variable exponent Sobolev space $W^{1,p(\cdot )}(\mathrm{\Omega })$ under the assumption that the exponent $p(x)$ is essentially bounded on $\mathrm{\Omega }$ and $\mathrm{\Phi }(x,t)=t^{p(x)}$, $t\ge 0$, $x\in \mathrm{\Omega }$, satisfies the log-Hölder condition.