I was reading Brezis functional analysis,Sobolev space PDE book.There is a remark in page 287 which says:If $\Bbb{R}^N\setminus \Omega$ is "sufficient thin" and $p<N$ then $W^{1,p}_0(\Omega) = W^{1,p}(\Omega)$ I try to prove the theorem that:
Prove $H^1(\Omega) = H^1_0(\Omega)$ for $\Omega = \Bbb{R}^N\setminus \{0\}$ with $N\ge 2$
I'm not sure whether this is a simple exercise or not.The only approach is the use the $C^\infty_c(\Omega)$ to approximate $H^1(\Omega)$?Is there some reference?
Very, very, rough sketch of an approach for $N>2$: Decompose $u=u_1+u_2$, where $u_1$ is supported near the origin, and u_2 away from it.
By mimicking the proof of the $\Omega=\mathbb{R}^N$ case you can show that $u_2$ can be approximated by $C_c^\infty(\mathbb{R}^N\setminus\{0\})$ functions.
On the other hand if you choose $u_1(x)=\eta(x/r)u(x)$ with $\eta$ a nice cut-off function supported on a ball of radius $1$, then you can show that $\| u_1\|_{H^1(\mathbb{R}^N\setminus\{0\})}\to 0$ as $r\to 0$. For this step you need Sobolev embedding on an annulus/ball (to bound the integral of $|\nabla \eta_r|^2|u|^2$ by Hölder's inequality), and it's why I'm not sure the approach works in the $N=2$ case. Maybe a compensated compactness type argument works, but I haven't thought too much about it.
With these facts combined, then you first choose $r$ small enough, then approximate the difference $u-u_1=u_2$.