This is a [cross-post][2].

Let $D \subseteq \mathbb{R}^2$ be the **closed** unit disk. Let $f:D \to D$ be a diffeomorphism.

>Does there exist a smooth $h \in C^{\infty}(D)$ such that $h\cdot f$ is an **area-preserving** diffeomorphism of $D$?

Clearly, such an $h$ must send the entire boundary $\partial D$ either to $1$ or to $-1$. Another necessary condition is $\det\big(d (h\cdot f)\big) = 1$. Since

$$
d (h\cdot f)=h df+dh \otimes f=h df+f \cdot (\nabla h)^T,
$$

the [matrix determinant lemma][1] implies that 
$$
1=\det\big(d (h\cdot f)\big)=h^2\det(df)+(\nabla h)^T \operatorname{adj}( hdf ) f.  \tag{1}
$$

In particular, at all points where $h \neq 0$, we have
$$
1=h^2\det(df) \big( 1+h^{-1}(\nabla h)^T ((df)^{-1} \cdot f)\big).
$$

Does the PDE $(1)$ have a solution for every diffeomorphism $f$?


$$\det(df) \cdot \big( h^2+h(\nabla h)^T ((df)^{-1} \cdot f)\big)=1.$$


[1]:https://en.wikipedia.org/wiki/Matrix_determinant_lemma
[2]:https://math.stackexchange.com/questions/3577369/is-every-diffeomorphism-conformally-equivalent-to-a-volume-preserving-diffeomorp