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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 $J$)?

Improve this question
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  • $\begingroup$ Is the $J$ inside the $H^1(\Omega)$-norm on the left hand side a typo? $\endgroup$
    – Hannes
    Commented Feb 19, 2018 at 16:38
  • $\begingroup$ Anyway, I suggest taking a look into Chapter 1.1.7 in Sobolev Spaces by Maz'ya where it is shown that Lipschitz-transformations indeed do what you want. $\endgroup$
    – Hannes
    Commented Feb 19, 2018 at 16:41
  • $\begingroup$ No it is not. It comes from the fact that $\int_{\Omega}{(f\circ\Phi)J} = \int_{\Phi(\Omega)}{f}$. $\endgroup$
    – PeteAgor
    Commented Feb 19, 2018 at 16:42
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    $\begingroup$ Isn't this a direct consequence of the Cauchy-Schwarz inequality, the chain rule, and the change of variable formula for the integral? The constant $C$ is simply the sup norm of $|\partial\Phi|$ $\endgroup$
    – Deane Yang
    Commented Feb 19, 2018 at 17:48

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