Green's identity formula

Question

Instead, if we assume that $v\in C^\infty_c(U), u\in C^2(U)$. Will the second equality still hold? Since $v$ have compact support in U, the boundary term will just vanish and $-\int_Uu\Delta v dx$ should be well defined.

Answer 1

Yes it still holds. Note that you can find an open subset $\hat{U}$ such that $$supp(v) \subset \hat{U} \subset U.$$ Now v and all its derivatives vanish outside of $\hat{U}$ and $u \in C^2(\bar{\hat{U}}).$ Note that the integration over $U$ is now the same as the integration over $\hat{U}$, hence $$\int_U Dv \cdot Du\, dx =\int_{\hat{U}} Dv \cdot Du\, dx = -\int_{\hat{U}} u \Delta v \, dx + \int_{\partial\hat{U}}\frac{\partial v}{\partial \nu} u \, dS \\= -\int_{\hat{U}} u \Delta v \, dx = -\int_U u \Delta v \, dx.$$