Let $u \in W^{1,p}_0(\Omega)$ and let $\tilde{u}$ the function defined extending $u$ to zero on $\mathbb{R}^N \setminus \Omega$. I should prove that $\tilde{u}\in W^{1,p}(\mathbb{R}^N)$.
The exercise do not specify whether $\Omega$ is a bounded, open set with $C^1$ boundary, so I should take as definition of $W^{1,p}_0(\Omega)$ that it is the closure of $C^{\infty}_c(\Omega)$ with the Sobolev norm ($p<\infty$).
My attempt was to take a sequence $(u_n)\in C^{\infty}_c(\Omega)$ s.t. $u_n \to u$ in $W^{1,p}(\Omega)$, defining $\tilde{u}_n $ as above and show that $\tilde{u}_n \to \tilde{u}$ in $W^{1,p}(\mathbb{R}^N)$, and so the thesis should follow by the completeness of $W^{1,p}(\mathbb{R}^N)$. I’m having some problems to show that $\tilde{u}_n \in W^{1,p}(\mathbb{R}^N)$. I’m trying to do it using integration by part:
Taking $\phi \in C_c^{\infty}(\mathbb{R}^N)$$$ - <\partial_i \tilde{u_n}, \phi>= \int_{\Omega} u_n\partial_i\phi dx = \int_{\Omega}(\partial_i u_n) \phi dx+ \int_{\partial \Omega} u_n\phi ds$$
and the thesis follow by the fact that $u_n=0$ on $\partial \Omega$. The problema is that I don’t know if $\Omega$ is a bounded, open set with $C^1$ boundary, so I don’t know if I can use the integration by part theorem.
Can someone please help me?