I have a question concerning the equivalence of some norms. Let $\Omega$ be a bounded domain (with sufficiently smooth boundary). On $ \mathrm{C}^1_0(\Omega) $ there is the norm $ || \nabla v||_2 $ for $ v\in\mathrm{C}^1_0(\Omega).$ If we now consider a matrixvalued $C^1-$ function $ A: \bar{\Omega}\rightarrow \mathbb{R}^{d\times d}_{\mathrm{symm}},$ such that $ A(x) $ is a positive definite matrix, then the mapping $v\mapsto \left( \int_\Omega \nabla v(x)\cdot (A(x)\nabla v(x))dx\right)^{1/2}$ defines a norm on $ \mathrm{C}^1_0(\Omega) $ as well. Let's denote it by $||\nabla v||_A. $
My Question now is: Are the two norms equivalent on $ \mathrm{C}^1_0(\Omega) $ or does at least exist a constant c$>0$, such that the estimate $ \mathrm{c}||\nabla v||_2 \leq ||\nabla v||_A$ holds? Or do I need more assumption on A?
