For $ f \in L^2(\Omega) $ u is called a weak solution of \begin{cases}-\Delta u=f&\text{in $\Omega$} \\ u=0&\text{in $\partial \Omega$}\end{cases} if:
1.$u\in W_0^{1,2}(\Omega)$
2.$\int_{\Omega}(\nabla u(x)\nabla \phi(x)-f(x)\phi(x))dx=0 $ for $\phi \in W_0^{1,2}(\Omega)$.
Now to prove is :Let $\Omega \subset\mathbb{R^n} $ be a bounded domain and $\partial \Omega \in C^{\infty}$
$u \in C^2(\overline \Omega) $ is a weak solution $\Rightarrow$u is a classical solution
Can someone give me a hint , do I have to integrate 2. ?