$u\in H^{1}_{0}$ and $\Delta u=1$ on $\Omega= (0,1)\times(0,1)$ then $u\notin C^{2}(\overline{\Omega})$

Question

I am stuck with the following question:

Let $\Omega:=(0,1)\times (0,1)$ and let $u\in H^{1}_{0}(\Omega)$ satisfying $\Delta u=1$. Prove that $u\notin C^{2}(\overline{\Omega})$. Determine the best possible regularity of $u$ on $\Omega$ or $\overline{\Omega}$, for example, find the greatest number $s$ such that $u\in H^{s}(\Omega)$.

My vague attempt: I think first I will try to find a special solution and then turn to the Laplacian equation with $0$ on the right hand side.