Let $\Omega_1, \Omega_2 \subset \mathbb{R^n}$ be bounded. The mapping $ F: \Omega_1 \rightarrow \Omega_2 $ shall be bijective, continuously differentiable and such that $||DF(x)||$ and $||DF(x)^{-1}||$ are bounded in suitable compatible matrix norm for all x. I need to show that $\nu \in H^1(\Omega_2)$ implies that $\nu \circ F \in H^1(\Omega_1)$. Where $H^1$ is the Sobolev space of maps with derivative in $L^2$.
Could anyone give my any hints on how to approach the problem because I am clueless.
Use the chain rule and the assumptions on $F$ to estimate the Sobolev norm of $\nu \circ F$, with respect to the Sobolev norm of $\nu$ and some constants depending on $F$.