Originally Posted by

**Jose27** If $\displaystyle \Omega \subset \mathbb{R} ^n$ open and connected , $\displaystyle u \in L_{loc}^1 ( \Omega) = \{ u: \Omega \longrightarrow \mathbb{R}$ : $\displaystyle u$ integrable in $\displaystyle \Omega _1$ for every open $\displaystyle \Omega _1 \hspace{2 mm} such \hspace{2 mm} that \hspace{2 mm} \overline{ \Omega_1} \subset \Omega \hspace{2 mm} is \hspace{2 mm} compact \hspace{2 mm} in \hspace{2 mm} \Omega \}$ is weakly differentiable in $\displaystyle \Omega $and $\displaystyle D_{i} u =0$ $\displaystyle \forall i=1, ... , n$ then $\displaystyle u$ is constant a.e. in $\displaystyle \Omega$

I don't even know where to begin with this one, the usual proof for when $\displaystyle u$ is differentiable does not apply since we don't really have a mean value theorem for this derivatives, so... I'm stuck. Any help is appreciated.