Symplectic factorization, Darboux theorem and ellipticity
Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 2, pp. 327-356.

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, satisfying

φ(ωm)=fanddφ;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=χψwithψ(ωm)=ωm,dχ;ωm=0andχ+(χ)t>0;
moreover there exists a closed 2-form Φ such that χ=(δΦωm). Here, ♯ is the musical isomorphism and ♭ its inverse. We connect the above result to an L2-projection problem.

DOI : 10.1016/j.anihpc.2017.04.005
Mots clés : Symplectic decomposition, Darboux theorem for symplectic forms, Elliptic systems, Optimal transport of symplectic forms
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     title = {Symplectic factorization, {Darboux} theorem and ellipticity},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {327--356},
     publisher = {Elsevier},
     volume = {35},
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Dacorogna, B.; Gangbo, W.; Kneuss, O. Symplectic factorization, Darboux theorem and ellipticity. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 2, pp. 327-356. doi : 10.1016/j.anihpc.2017.04.005. http://www.numdam.org/articles/10.1016/j.anihpc.2017.04.005/

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