I found this exercise but I am not able to solve it: let $n > 1$ a positive integer .Prove that for any $u \; \colon \, \mathbb{R}^n \to \mathbb{R}$ with $ u \in C^2(\mathbb{R}^n)$ with compact support the following equality holds:
$\sum_{1 \leq i,j \leq n} \int _{\mathbb{R}^n}|\frac{\partial^2 u}{\partial x_i \partial x_j}|^2\, dx = \int_{\mathbb{R}^n} |\Delta u| ^2 \, dx$
where $\Delta u $ denotes the laplacian of $u$. I would need a solution or at least a hint for this because I really do not know how to do it.