# Regularity conditions for Green's identity

While working on regularity theory for variational formulation of PDEs I often find ''Let $\Omega \subset \mathbb R^n$ and $u$ be sufficiently regular so that Green's identity $$\int_\Omega (-\Delta u)v = \int_\Omega \nabla u \nabla v - \int_{\partial \Omega}v \partial_n u$$ holds''. My question is: How regular does $\Omega$ has to be so that the above statement holds for $u \in H^2 (\Omega)$?

$\partial\Omega$ and $u$ have to be smooth enough in order for the integral $\int_{\partial\Omega}v\partial_n u$ to make sense.
What suffices is: Lipschitz continuity for the boundary and $\partial_n u\in L^2(\partial\Omega)$, for $u$. Even with weaker assumptions it is OK, for example $\partial_n u\in H^{-1/2}(\partial\Omega)$.