A question about extension of functions in a manifold

Question

If I have a funcion u $\in$ $H_{0}^{1}(\Omega, g)$ being $\Omega$ relatively compact on M, how can I prove that the extension $\bar{u}=1$ in $\Omega$ and $\bar{u}=0$ in M-$\Omega$ $\in$ $H_{0}^{1}$(M, g)? Thank you.