I have found some posts about equivalent definitions of $C^k(\overline \Omega)$:
Equivalent definitions of $C^r(\Omega)$?
Definition of convergence in $C^\infty(\Omega)$
However, they all haven't answered the following question:
Let $\Omega$ be an open, bounded subset in $\mathbb{R}^N$. Denote $$C^k(\overline{\Omega}):= \{ u \in C^k(\Omega): D^\alpha f \text{ is uniformly continuous on } \Omega, \text{ for all }|\alpha|\leq k \}$$ and $$D^k(\overline {\Omega}):= \{ u: \text{ there exists }\Omega' \supset \overline{\Omega} \text{ and }g \in C^k(\Omega') \text{ such that }u = g|_{\Omega}\}.$$ Under which conditions (of $\Omega$), one has $C^k(\overline{\Omega)} = D^k(\overline{\Omega})$?
Thanks for your helps.
