Consider a bounded domain $D \subset \mathbb{R}^n$ and the Sobolev space $H^1_{0}(D):=\overline{C_c^{\infty}(D)}^{W^{1, 2}(D)}$.
Further, consider a Sobolev function which happens to be smooth: $u\in H^{1}_{0}(D)\cap C^{\infty}(D)$ . Now remove some point $a\in D$ and restrict our function $u$ by removing the point $a$ of the domain of definition, i.e. consider the function $u_{\vert D-\lbrace{a\rbrace}}$.
I'm wondering if the restricted function is still a sobolev space, i.e. $u_{\vert D-\lbrace{a\rbrace}}\in H^{1}_0(D-\lbrace{ a\rbrace})$? Or to put it in an equivalent way: Is it possible to approximate $u_{\vert D-\lbrace{a\rbrace}}$ with respect to $\Vert\cdot \Vert_{W^{1,2}(D)}$ by smooth functions with compact support?
I appreciate any help! Best regards
No, not in general. Consider the following example:
Take $D=(-1,1)$ and $u\colon (-1,1)\to\mathbb{R},\,x\mapsto 1-x^2$. Then $u$ is in $H^1_0((-1,1))$. However, $u$ can not be approximated by smooth functions with support in $(-1,0)\cup(0,1)$.
This works more general for open intervals $I$: By the Sobolev embedding theorem, the functions in $H^1(I)$ are continuous, and it is not hard to show that the functions in $H^1_0(I)$ have boundary values $0$. So $u$ must vanish in $a$ if it can be approximated by smooth functions supported in $I\setminus \{a\}$.