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
$$ d (h\cdot f)=h df+dh \otimes f=h df+f \cdot (\nabla h)^T, $$
by the matrix determinant lemma, at all points where $h \neq 0$, we have
$$ \det\big(d (h\cdot f)\big)=h^2\det(df) \big( 1+h^{-1}(\nabla h)^T ((df)^{-1} \cdot f)\big), $$
so we need to solve the following PDE for $h$
$$\det(df) \cdot \big( h^2+h(\nabla h)^T ((df)^{-1} \cdot f)\big)=1.$$