Closedness of the range of differential operator first order

Question

The fact that the range of gradient from $H_0^1$ to $L_2$ is closed is well known. In general we can define some kind of weak derivative in the form \begin{equation} Du=\sum_{i,j}a_{ij}\partial_i u_j \end{equation} using an analogous definition to the definition of weak derivative related to Sobolev space, where $a_{ij}$ are constants. Also, we can define a Hilbert space $V$ as the closure of $C_0^\infty$ by the norm $|u|=|u|_{L_2}+|Du|_{L_2}$. Then my question is, is the range of $D:V\rightarrow L_2$ in general closed in $L_2$ or is there a counter example?