Let $E\subset\mathbb{R}^n$ be a bounded domain with $\partial E\in C^1$.
I don't know how to prove that its characteristic function $\chi_E$ is NOT in $W^{1,p}(\mathbb{R^n})$.
It's obvious that $\chi_E$ is in $L^p$ and that:
$\langle\nabla T_{\chi_E},X\rangle=-\int_E \text{div}( X) \ dx$,
but i don't know how to continue.