I am interested in $H^s$ Sobolev spaces in $\mathbb R^n$ which have functions with support in a given closed set $K$ , denoted by $H^s_K$. Here $K$ is the complement of some bounded open set $\Omega$. For instance, $\Omega$ may satisfy the uniform cone condition or have Lipschitz boundary. An example is the quotient isometric isomorphism $H^s(\Omega) = H^s/{H^s_{\mathbb R^n\setminus\Omega}}$ .
A simple argument shows that these spaces are Hilbert spaces.
But what about dense spaces of smooth functions? For instance, is $C^{\infty}_K = \{f \in C^{\infty}_0 : \operatorname{supp}f \subset K\} $ dense in $H^s_K$? I think continuous functions on closed sets may introduce significant complications but I have seen papers which study them.
Just references will do.