I have a question about a subspace of a Sobolev space.
Let $D$ be a domain of $\mathbb{R}^{d}$. That is, $D$ is a connected open subset. For a open subset $E \subset D$, $H^{1}(E)$ denotes first order $L^{2}$-Sobolev space on $E$ with Neumann boundary condition.
We define \begin{align*} \widehat{C}(E)&=\{f \in C^{1}(E) \mid \|f\|_{H^{1}(E)}<\infty, f=0 \text{ on } \partial E \cap D \},\\ \widehat{H}(E)&=\text{the completion of }\widehat{C}(E)\text{ with respect to the norm } \|\cdot\|_{H^{1}(E)}, \end{align*} where $\|f\|_{H^{1}(E)}=\left\{\int_{E}|\nabla f|^{2}\,dx+\int_{E}f^{2}\,dx \right\}^{1/2}$.
My question
Is it natural that $\widehat{H}(E)$ is regarded as a subspace of $H^{1}(D)$? In other words, $f \in \widehat{H}(E)$ can be extended to a function on $D$ by putting $f=0$ on $D \setminus E$.