Suppose $ \phi \in W^{k,p} (U)$ where $p \neq \infty$ and $U$ is a bounded $C^1$ open set in $\mathbb{R}^n$. Suppose moreover that $\phi \in C(\bar{U})$. It is a standard fact(cf. Partial Differential Equations by Evans) that there exists a sequence $u_m \in C^\infty(\bar{U})$ such that $u_m \rightarrow \phi$ in $ W^{k,p} (U)$.
Is it true that we can choose $u_m$ so that the convergence is also uniform(on $\bar{U}$)? Are there any references?