On the pullback equation [formula omitted

Abstract

We discuss the existence of a diffeomorphism ϕ:Rn→Rn such thatϕ∗(g)=f where f,g:Rn→Λk 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. Poincaré Anal. Non Linéaire 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 Hölder 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.