1) Good point about $J$ not being constant, little oversight. I guess I mean something like $||J-1||^2){L^2 \Omega or \Omega_h} \leq C_2h^2$. 2) Also a good point. I guess my question is if there exists such a map $\omega$ such that the Jacobian has this property. I think it must be true, as if $\Omega$ and $\Omega_h$ are sufficiently close then the Jacobian will almost be the identity.