Let $\Omega=(0,1)^2$. Let $u$ be a weak solution of $\Delta u=f$ con $f \in L^2(\Omega)$ e $u \in H^1_0(\Omega)$. I would like to prove that $u \in H^2(\Omega)$.
I know that $u \in H^2_{loc}(\Omega)$ because of elliptic interior regularity. I tried to adapt the demonstration of that fact and to find compact subsets $K_n$ with $\|u\|_{H^2(K_n)}$ uniformly bounded, but I was not able to go any further.
As a reference to understand what I know about the subject(which is very few) I've studied the chapter about that of Evans book and of Brezis book.