Symplectic factorization, Darboux theorem and ellipticity

O. Kneuss; B. Dacorogna; W. Gangbo

doi:10.1016/j.anihpc.2017.04.005

JournalsaihpcVol. 35, No. 2pp. 327–356

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

Download PDF

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 $R^{2 m}$ . 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.

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