Let $\Omega \subset R^n\, $open. Then one defines the weak derivation of the function $f \in L^2 (\Omega)$ by $$ <D^{(\alpha)}f, \phi>= (-1)^{|\alpha|}<f, D^{\alpha}\phi>, \, \, \forall \phi \in D(\Omega), $$
where $D(\Omega)$ is the subset of $C^\infty (\Omega)$ functions with compact support.
One also defines: $$ W^m(\Omega)=\{f\in L^2(\Omega):D^{(\alpha)}f\in L^2(\Omega)\, exists \,\,\forall |\alpha|\leq m \}, $$ and $$ H^m(\bar \Omega)=\overline {C^m(\bar \Omega) \,\cap \, W^m(\Omega)}.$$
Can somebody explain, what kind of elements (functions) does the set $H^m (\bar \Omega)$ comprise ? What i dont understand is how can a function be m-times continuously differentiable and at the same time weak differentiable. I am also not quite sure to understand how the multiindices work in the context of weak deifferentiability.
I need some hints or a proposal of proof for proving the continuity of the differential Operator $D^{(\alpha)}:W^{m+|\alpha|}(\Omega)\rightarrow W^m (\Omega).$
Thanks.
I will mention that I don't think your notation is very standard. We can define weak derivatives for any $f \in L^1_{loc} (\Omega)$ (the set of functions which are integrable on any compact subset of $\Omega$, not just $f \in L^2 (\Omega)$. I also think the most common notation for Sobolev spaces is
\begin{equation*} W^{k, p} (\Omega) := \{f \in L^p (\Omega) : D^\alpha f \text{ exists and belongs to } L^p (\Omega) \text{ for all } |\alpha| \leq k \}. \end{equation*}
For a function which is $m$-times continuously differentiable, the weak derivative agrees with the classical derivative. The proof of this is more or less immediate from integraion by parts and the definition of the weak derivative. $C^m (\bar{\Omega}) \cap W^m (\Omega)$ therefore consists of functions which are continuously differentiable on $\Omega$ (and can be continuously extended to the boundary), and whose classical derivatives are all in $L^2 (\Omega)$, using your notation. $H^m (\bar{\Omega})$ is therefore the set of all functions which can be approximated in the Sobolev norm by functions of this form. I'm not sure what your confusion is about multi-indices. If you understand them for classical derivatives, you should understand them for weak derivatives, since we just define weak derivatives by passing derivatives onto test functions.
Use the sequential characterization of continuity, and write out what it means for a sequence $f_n$ to converge to $f$ in in $W^{m + |\alpha|} (\Omega)$, and what you need for $D^{\alpha} f_n$ to converge in $W^m (\Omega)$.