Is there a theorem deals with $\int_{\Omega} \bigtriangleup u \bigtriangleup v $

Question

I am working on defining a norm on proving a norm on $H_0^2({\Omega})$, $\Omega $ is bounded open set in $\mathbb{R}^N$. But to do that, we need to prove that

$\int_{\Omega} \bigtriangleup u \bigtriangleup v dx =\sum_{i,j=1}^N \int_{\Omega} \partial_{ij}u \;\partial_{ij}v \;dx $.

But, I could not find a formula helps me working on left hand side.