Proof of integral identity

Question

I was looking for a proof of this identity: \begin{equation} \int_{\mathbb{R}^n}\phi \Delta\phi = - \int_{\mathbb{R}^n}|\nabla\phi|^2 \end{equation} given $\phi \in C_0^{\infty}(\mathbb{R}^n)$.

Answer 1

The difference is a surface term,$$\int_{\Bbb R^n}(\phi\nabla^2\phi+\nabla\phi\cdot\nabla\phi)d^nx=\int_{\Bbb R^n}\nabla\cdot(\phi\nabla\phi)d^nx.$$