Given the classical version of Green's Formula:
Let $\Omega$ be an open bounded subset of $\mathbb{R}^{n}$ and $u, v \in C^{2}(\overline{\Omega})$. \begin{equation} \int_{\Omega} Dv \cdot Du \ dx = -\int_{\Omega} u \Delta v \ dx + \int_{\partial \Omega} \frac{\partial v}{\partial \nu}u \ d\gamma. \end{equation} I need to use the next generalized version where either the domain $\Omega$ and the identity remains as in the classical version but $u \in H^{2}(\Omega)$ and $v \in H^{1}(\Omega)$. Note that in these new spaces, for $u$ and $v$, the identity make sense. But, what guarantee that in these spaces the left hand side equals the right hand side.
Do I need to go to the proof in order to check that the theorem is true in these spaces?