I'm reading haim brezis's functioal analysis sobolev space and pde and have trouble in understanding the following proof in the case of $\Gamma=\partial\Omega$ unbounded.
the boundary of $\Omega\cap B(2n_0)$ is $C^1$, too? If yes, we can indeed ues extension theorem.
Our aim is to prove $\|w-u\|_{W^{1,p}(\mathbb{R}^N)}$. My understanding is \begin{align} \|w-u\|_{1,p,\Omega} &\le\|w-\zeta_{n_0}u\|_{1,p,\Omega}+\|\zeta_{n_0}u-u\|_{1,p,\Omega}\\ &\le\|w-\zeta_{n_0}u\|_{1,p,\Omega\cap B(2n_0)}+\|w-\zeta_{n_0}u\|_{1,p,\Omega\backslash B(2n_0)}+\epsilon \end{align} The first term can be controlled by extension, but how the second?
Some notation and theorem may be need.
- $\rho(x)=\begin{cases}e^{\frac{1}{|x|^{2}-1}} & \text { if }|x|<1 \\0 & \text { if }|x|>1 \end{cases}, \rho_{n}(x)=C n^{N} \rho(n x)$, where $C=\frac{1}{\int \rho}.$
- Fix $\zeta\in C_c^{\infty}(\mathbb{R}^N), 0\le\zeta\le 1,\zeta(x)=\begin{cases} 1 & |x|\le 1\\ 0 & |x|\ge 2 \end{cases}$, and $\zeta_n(x)=\zeta(\frac{x}{n}), n=1,2,\dots$
- Friedrichs approximate: $C_c^{\infty}(\mathbb{R}^N)$ is dense in $W^{1,p}(\mathbb{R}^N)$, where $1\le p<\infty$.
- Extension theorem: $\Omega$ is $C^1$ with boundary $\Gamma$ bounded, then there is a linear operator $P: W^{1,p}(\Omega)\to W^{1,p}(\mathbb{R}^N)$ s.t $Pu|_{\Omega}=u$, and $\|Pu\|_{1,p,\mathbb{R}^N}\le C\|u\|_{1,p,\Omega}$.(I think the latter is not used in the proof)