Suppose $s\in (0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define $$ H^s=(1-\Delta)^{-s/2} L^2(\mathbb{R}^d), $$
$$ H_D^s=\{f\in H^s: f=0 \ \ a.e. on \ D^c\}. $$
Q: Is $C_c^\infty(D)$ dense in $H_D^s$(with norm $\|\cdot\|_{H^s}$) for any open set $D$?
Is there any element reference?