I start saying that it's clearly an easy problem, but I'm stupidly stuck, so be kind please.
The problem is:
$ \mbox{Given}\ \Omega \in \mathbb{R}^n\ \mbox{bounded and }\ v \in C^2(\Omega)\cap C^1(\bar{\Omega})\ \mbox{such that } -\Delta v=2 \mbox{ in }\Omega \mbox{ and } v=0 \mbox{ in } \partial \Omega $
Prove that $v \in C^\infty (\Omega)$
It's clear that it's true. My idea was to use Green's representation theorem but I get stuck with the $\partial_\nu v$ term.
Thanks for the help.