Does Green's (first) identity hold for Weak Derivatives? Recall Green's First Identity:

$$\int_{\Omega}v \Delta u \  (d\Omega) =\int_{\partial \Omega}v  (\nabla u )\vec{n} \   d (\partial \Omega) - \int_{\Omega} \nabla u \nabla v \  (d \Omega)$$

Which requieres $u \in C^2(\Omega)$ and $v \in C^1(\Omega)$.
So the question is simple: Does this apply to weak derivatives? i.e. Can we weaken the conditions given to be $u \in H^2(\Omega)$ and $v \in H^1(\Omega)$?
Aditional information: I am reading Sobolev Spaces, Functional Analysis and Partial Differential Equations, By Haim Brezis, the proof of Theorem 9.25, which deals with the regularity of the weak solutions for the Laplacian problem. 
It is stated, under the assumption that $u \in H^2(\Omega)$, $f \in H^1(\Omega)$ and $\varphi \in C^{\infty}(\Omega)$ that from: $$\int_{\mathbb{R}^n} \nabla u \nabla(D\varphi)+ \int_{\mathbb{R}^n}u(D \varphi) = \int_{\mathbb{R}^n}f (D \varphi) 
$$
We deduce: $$\int_{\mathbb{R}^n} \nabla (Du) \nabla(\varphi)+ \int_{\mathbb{R}^n}(Du)\varphi = \int_{\mathbb{R}^n} (Df) \varphi 
$$
Which I could show If I could apply Green's First Identity restricting the integrals to the (compact $\Rightarrow$) bounded support of $\varphi$.
 A: An identity that is originally proved for smooth functions can be extended to a Sobolev space provided that both sides are continuous with respect to the Sobolev norm. This is a general topological fact: if two continuous functions agree on a dense subset, then they agree everywhere.  
In the case of linear expressions like $u\mapsto \int \varphi \nabla u$, or bilinear like $(u,v) \mapsto \int u\Delta v$, continuity is equivalent to boundedness. 
Consider the identity that you stated, 
$$\int_{\Omega}v \Delta u   =\int_{\partial \Omega}v  (\nabla u )\vec{n}   - \int_{\Omega} \nabla u \nabla v  $$
On the left we have a continuous function on $H^2\oplus H^1$, because 
$$\left|\int_{\Omega}v \Delta u \right|\le \|v\|_{L^2} \|\Delta u\|_{L^2} \le C\|v\|_{H^1} \| u\|_{H^2} $$ 
Similarly, $\int_{\Omega} \nabla u \nabla v $  is bounded. The boundary term  is okay on sufficiently smooth domains (Lipschitz is enough), because the trace operator is bounded from $H^1(\Omega)$ to $L^2(\partial \Omega)$.  So we get
$$
\left| \int_{\partial \Omega}v  (\nabla u )\vec{n}  \right|
\le \|v\|_{L^2(\partial \Omega)}\|\nabla u\|_{L^2(\partial \Omega)}
\le C\|v\|_{H^1} \| u\|_{H^2} 
$$
In the book the boundary term doesn't appear since $\varphi$ is compactly supported.
