I have also posted the question [here](http://math.stackexchange.com/questions/1369182/regularity-of-a-weak-solution). 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.

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**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.