$H^1(\Omega) \subset L^2(\Omega)$ dense for $\Omega$ a $C^l$ hypersurface with boundary?

Let $\Omega$ be a bounded open set. It is well know that $H^1(\Omega) \subset L^2(\Omega)$ is dense. The proof is: $C_c^1(\Omega)$ is dense in $L^2(\Omega)$ and $C_c^1(\Omega) \subset H^1(\Omega).$

Is this same argument true if $\Omega$ were a $C^l$ hypersurface in $\mathbb{R}^n$ with boundary $\partial \Omega?$ What assumptions are required on the boundary and on $l?$

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