If $M$ is a $C^{1}$ manifold and $\omega_i$, $i=1,2$ are two continuous volume forms with the same mass, does there exist a $C^1$ diffeomorphism sending one to the other?
The answer is no. The equivalent problem is: given a positive and continuous function $g$, can we find a diffeomorphism with the Jacobian equal to $g$, $\det Df=g$? A counterexample is Theorem 1.2 in:
D. Burago, B. Kleiner, Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps. Geom. Funct. Anal. 8 (1998), 273–282. (MathSciNet review.)