Given $\Omega$ is a bounded, $C^1$ domain in $\mathbb{R}^n$. $\chi _{\Omega }(x)$ is the characteristic function of $\Omega$.
I have done the followings:
We can get $\chi _{\Omega }(x) \in L^2(\Omega)$ for all $n \in \mathbb{N}$ quite easily. The next thing is to show the existence of the weak derivative of $\chi_{\Omega}$.
For $n=1$, $\Omega$ is an open interval. Let $\phi \in C_{c}^{\infty }(\Omega)$ be an arbitrary test function. We have
$\int_{a}^{b}\chi _{(a,b) }\phi 'dx=\int_{a}^{b}\phi'dx =\phi(b)-\phi(a)=0$
So $\int_{a}^{b}\chi _{(a,b) }\phi 'dx=-\int_{a}^{b}0.\phi dx$, and the weak derivative of $\chi_{\Omega}$ is $0$.
For $n\geq 2$, I get stuck and don't know if $\chi_{\Omega}$ has the weak derivative or not.
Thank you very much for your help.