Continuous embedding - Local Sobolev spaces

Question

Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ and $H_{c}^{1}(\Omega)=H^1(\Omega)\cap \mathcal{E}'(\Omega)$, where $\mathcal{E}'(\Omega)$ is the space of the distributions with compact support.

Let $H_{0}^{1}(\Omega)=\overline{\mathcal{D}(\Omega)}^{\|\cdot\|_{H^1}}$. It's true that $H^{1}_{c}(\Omega) \hookrightarrow H_{0}^{1}(\Omega)$?

It is a classic result that $H^{1}_{c}(\Omega) \hookrightarrow H^{1}(\Omega)$. If this statement is true, this result can be improved, in the sense $H^{1}_{c}(\Omega) \hookrightarrow H_{0}^{1}(\Omega)\hookrightarrow H^{1}(\Omega)$.

Answer 1

See Lemma 2.8.5 on page 202 of the book: Wagschal, C. Distribution, Analyse Microlocale, Équations aux Dérivées Partielles.