Symplectic factorization, Darboux theorem and ellipticity

Abstract

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 ${\mathbb{R}}^{2m}$. We prove existence of a diffeomorphism φ, with optimal regularity, satisfying

We establish uniqueness of φ 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:

moreover there exists a closed 2-form Φ such that $\chi ={\left(\delta \mathrm{\Phi }┘{\omega }_m\right)}^{♯}$. Here, ♯ is the musical isomorphism and ♭ its inverse. We connect the above result to an $L^2$-projection problem.