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 ωm, the standard symplectic form on R2m. We prove existence of a diffeomorphism φ, with optimal regularity, satisfyingφ⁎(ωm)=fand〈dφ♭;ωm〉=0. 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:u=χ∘ψwithψ⁎(ωm)=ωm,〈dχ♭;ωm〉=0and∇χ+(∇χ)t>0; moreover there exists a closed 2-form Φ such that χ=(δΦ⌟ωm)♯. Here, ♯ is the musical isomorphism and ♭ its inverse. We connect the above result to an L2-projection problem.