Symplectic factorization, Darboux theorem and ellipticity

Dacorogna, B.; Gangbo, W.; Kneuss, O.

doi:10.1016/j.anihpc.2017.04.005

Dacorogna, B. ; Gangbo, W. ; Kneuss, O.

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 2, pp. 327-356.

Résumé

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

φ^{⁎} (ω_{m}) = f and 〈 d φ^{♭}; ω_{m} 〉 = 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 \circ ψ with ψ^{⁎} (ω_{m}) = ω_{m}, 〈 d χ^{♭}; ω_{m} 〉 = 0 and \nabla χ + {(\nabla χ)}^{t} > 0;

moreover there exists a closed 2-form Φ such that

χ = {(δ Φ ⌟ ω_{m})}^{♯}

. Here, ♯ is the musical isomorphism and ♭ its inverse. We connect the above result to an

L^{2}

-projection problem.

MR Zbl

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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     author = {Dacorogna, B. and Gangbo, W. and Kneuss, O.},
     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},
     number = {2},
     year = {2018},
     doi = {10.1016/j.anihpc.2017.04.005},
     mrnumber = {3765545},
     zbl = {1476.58037},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2017.04.005/}
}

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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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