In Evans' book Partial Differential Equations, appendix C.2 Theorem 3, Green's formulas are established: Let $u, v \in C^{2}( \overline{U})$. Then $$ \int_U Du \cdot Dv dx = - \int_U u \Delta v dx + \int_{\partial U } \dfrac{\partial v }{\partial \nu } u dS .$$
My question is, is this formula valid requiring less about the $u$ function? For example, if $u \in H^1(U)$ or in some other sobolev space? or some other space? I'm interested in the case $U = \mathbb{R}^N$, if something is only valid in $\mathbb{R}^N$.