I'm working through the following problem, and I just need a hint to finish it I think.
Consider the set $\Omega = B(0,1) \backslash \left\{x\in \mathbb{R}^N : x_N = 0 \right\}$. We are given the function \begin{equation*} u(x):= \left\{ \begin{array}{ll} 1, x_N > 0 \\ 0, x_N < 0 \end{array} \right. \end{equation*} I'm to show that $u \in W^{1,p}(\Omega)$, but cannot be approximated by function in $C^\infty \left(\overline{\Omega} \right)$.
$u \in W^{1,p}(\Omega)$ is pretty simple. It's been a while since I did multivariable calculus so I'm having a little bit of trouble showing the last part in $N$ dimensions. My approach was to assume $w_n \rightarrow u$. I showed that there is a point $y_1$ in the upper hemisphere where $w_n(y_1) \geq \frac{3}{4}$ and there is a point $y_2$ in the lower hemisphere where $w_n(y_2) \leq \frac{1}{4}$ for all $n$ sufficiently large. I'd like to use this to show that $\int_{\overline{\Omega}} (w_n)_{x_N} dx$ is bounded below by a positive constant for all large $n$. In the one dimensional case this is pretty trivial. The multidimensional case probably is as well, but I'm having trouble with it none the less. Any help would be greatly appreciated.