Classical solution satisfies weak formulation of Poisson equation

Question

I have a domain $\Omega \subset \mathbb{R}^2$ and the Poisson equation with Dirichelet boundary condition: $$ \begin{cases} -\nabla^2 u &= f\qquad in\ \Omega \\ u &= 0\qquad on\ \partial\Omega \end{cases} $$ and I wanto to show that a solution $u$ satisfies the weak formulation, which to my understanding is $$ \int_{\Omega} \nabla u \cdot \nabla v = \int_{\Omega} vf $$ for all $v \in V = \{v \in \mathcal{H}^1(\Omega)| v = 0\ on\ \partial\Omega \}$, where $\mathcal{H}^1(\Omega)$ is the Sobolev space of functions with square integrable first derivative on $\Omega$. The only thing I see is that $u \in V$, but I do not understand how having a classical solution can be used in the second equation. I thought I could substitute $-\nabla^2 u = f$, but I don't know how to get $-\nabla^2 u$ in the weak formulation. Any help would be much appreciated.

Answer 1

Use $$\text{div} (v\nabla u) = \nabla v \nabla u + v \Delta u$$ and partial integration $$\int_\Omega \text{div}X = \int_{\partial \Omega} nX$$ together with $v=0 $ on $\partial \Omega$

(in case $\Omega$ does not satisfy the prerequisites of the divergence theorem you'll have to use an approximation argument)