We know the theorem:
- If $\Omega$ is a star domain in $\Bbb R^n$, then $C^{\infty}(\overline{\Omega})$ is dense in $W^{k,p}(\Omega)$.
It means that $\overline{C^{\infty}(\overline{\Omega})}=W^{k,p}(\Omega)$
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- I have a question (I have trouble when I learn about Sobolev space):
$a.$ For example:
Let $\Omega=[-1,1]\times [-1,1] \setminus\{x=0,y \ge \dfrac{1}{2}\}$.
Can anyone post a function $u$ such that:
$$\exists u \in W^{1,p}(\Omega) \not \to \exists \{u_n\} \subset C^{\infty}(\Omega)$$ $$\|u_n-u\|_{W^{1,p}(\Omega)} \to 0$$
$b.$ If $\Omega$ is not a star domain in $\Bbb R^n$ then a domain $\Omega$ needs the properties so that $$\overline{C^{\infty}(\Omega)}=W^{k,p}(\Omega)$$ ?
Any help will be appreciated! Thanks!