Show that $\mathcal C^1(\bar \Omega )$ is dense in $W^{1,p}(\Omega )$.

Question

Let $\Omega \subset \mathbb R^d$ a bounded domain. I want to show that $\mathcal C^1(\bar \Omega )$ is dense in $W^{1,p}(\Omega )$.

Attempts

Let $u\in W^{1,p}(\Omega )$. Then $u$ can be extended to $\bar u\in W^{1,p}(\mathbb R^d)$. We know that $\mathcal C_0^\infty (\mathbb R^d)$ is dense in $W^{1,p}(\mathbb R^d)$. Therefore, there is $(u_n)\subset \mathcal C_0^\infty (\mathbb R^d)$ s.t. $$\|u_n-\bar u\|_{W^{1,p}(\mathbb R^d)}\to 0.$$ Since $$\|u_n-u\|_{W^{1,p}(\Omega )}\leq \|u_n-\bar u\|_{W^{1,p}(\mathbb R^d)}\to 0$$ we have that $(u_n|_{\Omega })\subset \mathcal C^\infty (\bar \Omega)$ that converge to $u$ in $W^{1,p}(\Omega )$. The claim follow.

Question Is it correct ? If no, why ?

Answer 1

As noted by Jonas Lenz, both your claim ($C^1(\bar \Omega)$ is dense in $W^{1,p}(\Omega)$) and the extension theorem are not true for general domains. The reasoning is otherwise correct.