---- > [!theorem] Theorem. ([[Global Sobolev approximation by functions smooth up to the boundary]]) > Suppose $k \geq 1$ and $1 \leq p < \infty$. If $\Omega$ satisfies [[Lipschitz boundary|the segment condition]], the set of restrictions to $\Omega$ of functions in $C_{c}^{\infty}(\mathbb{R}^{n})$ is [[dense]] [[Sobolev space|in]] $W^{k,p}(\Omega)$: $\overline{\{f |_{\Omega}: f \in C_{c}^{\infty}(\mathbb{R}^{n}) \}}=W^{k,p}(\Omega).$ ^theorem > [!proposition] Corollary 1. > For a *[[bounded set|bounded]]* [[Lipschitz boundary|Lipschitz domain]] $\Omega$, $C_{c}^{\infty}(\mathbb{R}^{n})$ is dense in $W^{k,p}(\Omega)$ in the sense that $\overline{\{ \varphi |_{\Omega} : \varphi \in C_{c}^{\infty}(\mathbb{R}^{n}) \}}^{W^{k,p}(\Omega)}=W^{k,p}(\Omega).$ In particular, $C^{\infty}(\overline{\Omega}) \cap W^{k, p}(\Omega) \text{ is dense in } W^{k, p}(\Omega)$ since $C^{\infty}_{c}(\mathbb{R}^{n}) |_{\Omega} \subset C^{\infty}(\overline{\Omega}) \cap W^{k,p}(\Omega)$. > Here, $C^{\infty}(\overline{\Omega})$ denotes the collection of functions [[continuously differentiable|smooth up to the boundary]] [[boundary|of]] $\Omega$. ^proposition > [!proposition] Corollary 2. > For $k \geq 1$ and $1 \leq p< \infty$, the space $C_{c}^{\infty}(\mathbb{R}^{n})$ is [[dense]] in $W^{k,p}(\mathbb{R}^{n})$, that is, $W^{k,p}(\mathbb{R}^{n})=W_{0}^{k,p}(\mathbb{R}^{n})$. ^proposition > [!proof]- Proof. ([[Global Sobolev approximation by functions smooth up to the boundary]]) > The full proof is omitted from our course. Do see Q6 on Sheet 2, however, which proves Corollary 1 directly for $\Omega$ an open [[star-convex set]]. ---- #### ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```