I'm trying to wrap my head around this problem - the interplay between $\nabla$ and $\Delta$ is doing my head in. It says to use the divergence theorem.
Prove that
$\int_\Omega u \cdot \Delta v\, dx + \int_\Omega \nabla u \cdot \nabla v \,dx = \int_{\delta \Omega}u\nabla v \cdot n\, d\sigma$