Let $\Omega$ be a bounded Lipschitz domain with boundary $\Gamma$ and exterior normal $n$. Let $u, v\in C^2(\Omega)\cap C^1(\partial\Omega)$. Then
$ \begin{aligned} \int_{\Omega} (u\Delta v +\nabla u\cdot \nabla v)\,d\Omega &= \int_{\partial \Omega}u\frac{\partial v}{\partial n}\,d\Gamma\\ \int_{\Omega}(u\,\Delta v - \Delta u\, v)\,d\Omega &= \int_{\partial \Omega}(u\frac{\partial v}{\partial n}-v\frac{\partial u}{\partial n})\,d\Gamma \end{aligned} $
The first identity follows from the [Divergence (Gauss) Theorem](https://www.notion.so/Divergence-Gauss-Theorem-1e88e7dbdc1a403fbe2c023f9185612d) by setting $F= u\nabla v$ and using that $\text{div}\, F = u\,\Delta v+ \nabla u\cdot \nabla v$. The second formula follows by interchanging the roles of $u$ and $v$ and subtracting.
# Related Items
[[Divergence Theorem]]
# Citations
A. Kirsch, F. Hettlich, _The Mathematical Theory of Time-Harmonic Maxwell’s Equations,_ Theorem A.12