An example of open set $\Omega$ in $\mathbb{R^n}$ for which $C^{\infty}_c(\Omega)$ is dense in $W^{l,p}(\Omega)$ We know that the statement $C^{\infty}_c(\mathbb{R^n})$ is dense in $W^{l,p}(\mathbb{R^n})$ is always true for any $l\in \mathbb{N}$ and $p\geq \infty$, $p\neq \infty$. My professor told me that it also holds for special open sets $\Omega$ in $\mathbb{R^n}$, but I cannot find an example of such $\Omega$.
Do anyone know something about this? Thanks.

You seem to be looking for domains $\Omega$ for which $W^{1,p}_0(\Omega) = W^{1,p}(\Omega)$. This will be the case if $\partial \Omega$ has $p$-capacity equal to $0$. Look e.g. in Measure Theory and Fine Properties of Functions by Evans and Gariepy, or Function Spaces and Potential Theory by Adams and Hedberg.
