Resolutions of the prescribed volume form equation

For a given volume form fdx on a bounded regular domain Ω in ℝⁿ, we are looking for a transformation u of Ω, keeping the boundary fixed and which sends the Lebesgue measure dx into fdx (i.e. we solve det(∇u) = f). For f in various spaces, we propose two different constructions which ensure the existence of u with some gain of regularity. Our methods permit the recovery Dacorogna and Moser's results [4], but also, we prove the existence of such u in Hölder spaces for f in C⁰, or even in L^∞.