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

Abstract

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