The definition of Hilbert space $H^k_0(\Omega)$

Question

Some time I see the $H^k_0(\Omega)$ space is the completion of $C^\infty_0(\Omega)$ in $\|\cdot\|_{H^k}$ norm. But when we solve pde, we use $H^k_0(\Omega)=\left\{u\in{}H^k(\Omega):u^{(\alpha)}\vert_{\partial{\Omega}}=0 \quad\forall\vert\alpha\vert<k\right\}$, where $u^{(\alpha)}$ indicates the $\vert\alpha\vert$th weak derivative in the normal direction of the trace. How could I show the equivalence of those two definitions?