On the pullback equation φ*(g) = f

We discuss the existence of a diffeomorphism phi : R-n -> R-n such that phi*(g) = f where f, g : R-n -> A(k) are closed differential forms and 2 <= k <= n. Our main results (the case k = n having been handled by Moser [J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965) 286-294] and Dacorogna and Moser [B. Dacorogna, J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincare Anal. Non Lineaire 7 (1990) 1-26]) are that - when n is even and k = 2, under some natural non-degeneracy condition, we can prove the existence of such diffeomorphism satisfying Dirichlet data on the boundary of a bounded open set and the natural Holder regularity; at the same time we get Darboux theorem with optimal regularity; - we are also able to handle the degenerate cases when k = 2 (in particular when n is odd), k = n - 1 and some cases where 3 <= k <= n - 2. (C) 2008 Published by Elsevier Masson SAS.