Suppose $\Omega$ is a bounded open subset of $\mathbb{R}^n$. Choose any $k\geq 1$. Do we have
- $C^k(\Omega)\cap C(\overline{\Omega})$ dense in $H^1(\Omega)$?
- $C^k(\overline{\Omega}):=\{u\in C^k(\Omega):D^\alpha u\in C(\overline{\Omega}),\;\forall 0\leq |\alpha|\leq k \}$ dense in $H^1(\Omega)$?
- Does the same hold if $k=\infty$ in both questions?
Also, what conditions on $\Omega$ are needed to guarantee such results? Does this hold true for domains with corners?
Best Answer
Indeed by convolution techniques is possible to show that for any Lipschitz domain $\Omega$, the space of infinitely smooth and compactly supported $\mathcal{C}_0^{\infty}(\bar{\Omega})$ is dense on $H^{1}(\Omega)$. The proof is quite long, see e.g (1975) Adams Sobolev spaces chp. 3 pp 51.
In view of the contention \begin{align} \mathcal{C}_0^{\infty}(\bar{\Omega})\subset C^{k}(\Omega) \cap C(\bar{\Omega}) \subset H^{1}(\Omega). \end{align} applying the closure
\begin{align} H^1(\Omega)=\overline{\mathcal{C}_0^{\infty}(\bar{\Omega})}^{\|\cdot\|_{H^1(\Omega)}} \subset \overline{ C^{k}(\Omega) \cap C(\bar{\Omega})}^{\|\cdot\|_{H^1(\Omega)}} \subset H^{1}(\Omega). \end{align}