Symplectic factorization, Darboux theorem and ellipticity

This manuscript identifies a maximal system of equations which renders the classical Darboux problem elliptic, thereby providing a selection criterion for its well posedness. Let f be a symplectic form close enough to omega(m), the standard symplectic form on R-2m. We prove existence of a diffeomorphism phi, with optimal regularity, satisfying phi* (omega(m)) = f and < d phi(b); omega(m)> = 0. We establish uniqueness of phi when the system is coupled with a Dirichlet datum. As a byproduct, we obtain, what we term symplectic factorization of vector fields, that any map u, satisfying appropriate assumptions, can be factored as: u = chi o psi with psi* (omega(m)) = omega(m) < d chi(b); omega(m)> = 0 and del chi + (del chi)(t) > 0; moreover there exists a closed 2-form Phi such that chi = (delta Phi right perpendicular omega(m))(#). Here, # is the musical isomorphism and b its inverse. We connect the above result to an L-2-projection problem. (C) 2017 Elsevier Masson SAS. All rights reserved.