Prove that $a(u,v)=\int_\Omega A\nabla u\cdot \nabla v$ is continuous if $A$ uniformly elliptic. Let $\Omega \subset \mathbb R^d$ smooth, bounded and connected domain. Let $A\in \mathbb R^{d\times d}$ symetric and uniformly elliptic, i.e. there is $C>0$ such that $$C^{-1}\|x\|^2\leq Ax\cdot x\leq C\|x\|^2.$$
How can I prove that $$a(u,v)=\int_\Omega A \nabla u\cdot \nabla v,$$
continuous ? I know that $$|a(u,v)|\leq \int_\Omega \|A\nabla u\|\|\nabla v\|.$$
I suppose that $|A\nabla u|\leq C|\nabla u|$, but I can't prove it since $|A\nabla u|$ is not $A\nabla u\cdot \nabla u$. I also tried as $$|A\nabla u|^2=A^2\nabla u\cdot \nabla u,$$
but is also $A^2$ uniformly elliptic ? If yes how can I prove it ? If no, how can I conclude ?

If $A$ is a symmetric matrix, then $Ax \cdot x \le C \|x\|^2$ implies that all eigenvalues of $A$ are bounded from above by $C$. Hence, $\|A x\| \le C \|x\|$.
