Let $n>0$ and $\Omega$ be a bounded domain of $\mathbb{R}^n$. Consider a smooth enough mapping $\Phi$, from $\Omega$ into $\Phi(\Omega)\subset\mathbb{R}^n$, that is orientation-preserving and injective. We denote by $J = det(\nabla \Phi)$ the jacobian of $\Phi$.

Let $f \in (H^1(\Phi(\Omega)))^n$ a vector valued Sobolev function.

Can we control the $H^1$-norm of $f$ in $\Omega$ after a change of variable ?

Actually, can we have an inequality of the following form :

$$ \|(f\circ\Phi)J\|_{H^1(\Omega)} \leq C \|f\|_{H^1(\Phi(\Omega))}, $$

where $C>0$ is a constant (which possibly depends on $\Omega$, $\Phi$ and $J$)?