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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.

John
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  • I suppose you can restrict $u$ to the line segment from the north to the south pole with a gap at the center. This already can't be approximated by $C^\infty$ functions on the whole line segment. – Christoph Mar 01 '14 at 18:25
  • Alright, well I figured it out. It was rather messy unfortunately. Thanks. – John Mar 02 '14 at 22:20

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