I have also posted the question here. Let me explain what difficulties I have. In fact, one may write \begin{equation} \partial_1(f-\partial_1 u)=0 \end{equation} in $\Omega$. Then one may have the formula $(f-\partial_1 u)(x,y)=g(y)$ for some one dimensional function $g$. If one wants to construct a counter example, then $g$ should have no derivative. But since $u$ is approximated by smooth functions with compact support, it looks like very impossible to construct such example.

Update: So far I have proved that if $f\in H_0^1(\Omega)$, then $\partial_1\alpha$ has to be in $H^1_{loc}(\Omega)$. The trick is to use the discrete Fourier transform.