Proposition 9.18 from Brezis' book mentions that if $u\in W_{0}^{1,p}(\Omega)$, then the extension of $u$ outside $\Omega$ by zero belongs to $W^{1,p}(\mathbb{R}^n)$ for $1
Is there a counterexample where this does not hold in the case $p=1$?
The proof as given in Brezis uses the fact that $D(\mathbb{R}^n)$ is dense in $L^p(\mathbb{R}^n)$ for $1\leq p<\infty$.
Is my line of reasoning valid? We need only show that $D_{e_1}\bar{u}=\overline{(D_{e_1}u)}$
$\int_{\mathbb{R^n}}\bar{u}D_{e_1}\phi=\int_{\Omega}{u}D_{e_1}\phi=\mathrm{lim}_{m\to\infty}\int_{\Omega}{u_m}D_{e_1}\phi=\mathrm{lim}_{m\to\infty}-\int_{\Omega}{D_{e_1}u_m}\phi=-\int_{\Omega}({D_{e_1}u})\phi=-\int_{\mathbb{R^n}}(\overline{{D_{e_1}u}})\phi$
where $D_{e_i}$ represents the weak derivative on the i-th variable, $u_m$ is the sequence in $D(\Omega)$ converging to $u$ in the norm of $W^{1,p}(\Omega)$ and $\phi$ belongs to $D(\mathbb{R^n})$. The overbar denotes extension of the subject outside its domain by zero.