How to show that $\int_\Omega \sum\limits_{i,j=1}^n u_{x_i}v_{x_j} dx\le C\int_\Omega |Du||Dv| dx$?

Question

$\Omega\subset \mathbb R^n$ is bounded and open. $u,v\in H_0^1(\Omega)$. $Du$ is gradient of $u$.

How to show that $\int_\Omega \sum\limits_{i,j=1}^n u_{x_i}v_{x_j} dx\le C\int_\Omega |Du||Dv| dx$ ?

Answer 1

$|Du|=\sqrt{u_{x_1}^2 +\cdots+u_{x_n}^2}\geq c( |u_{x_1}|+\cdots+|u_{x_n}|)$ so $|Du||Dv|\geq c( |u_{x_1}|+\cdots+|u_{x_n}|)(|v_{x_1}|+\cdots+|v_{x_n}|)\geq|\sum u_{x_i}v_{x_j}|$