# Symplectic factorization, Darboux theorem and ellipticity

### O. Kneuss

Department of Mathematics, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil### B. Dacorogna

Section de Mathématiques, EPFL, 1015 Lausanne, Switzerland### W. Gangbo

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, USA

## 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 $ω_{m}$, the standard symplectic form on $R_{2m}$. 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 $χ=(δΦ┘ω_{m})_{♯}$. Here, ♯ is the musical isomorphism and ♭ its inverse. We connect the above result to an $L_{2}$-projection problem.

## Cite this article

O. Kneuss, B. Dacorogna, W. Gangbo, Symplectic factorization, Darboux theorem and ellipticity. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 2, pp. 327–356

DOI 10.1016/J.ANIHPC.2017.04.005