Suppose $a(u,v):V\times V$ is a bounded, symmetric and coercive bilinear form. $V$ is eg. $H^k(\Omega).$
Let $g \in C^\infty(\Omega)$ be such that $A \leq g^{(k)} \leq B$ for positive $A$, $B$ and $k$.
Is there any way to show that the bilinear form $b(u,v) := a(u, vg)$ is coercive? i.e., that $a(v,vg) \geq C\lVert v \rVert^2?$
I tried a lot of things for my particular problem but they all give me coercivity IF certain constants satisfy some condition. Perhaps there is a way to do it abstractly without going into the details?
I ask this because I have not yet solved Existence of solution for this parabolic PDE in the affirmative.