Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8898140 | Annales de l'Institut Henri Poincare (C) Non Linear Analysis | 2018 | 30 Pages |
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 Ïm, the standard symplectic form on R2m. We prove existence of a diffeomorphism Ï, with optimal regularity, satisfyingÏâ(Ïm)=fandãdÏâ;Ïmã=0. We establish uniqueness of Ï 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=ÏâÏwithÏâ(Ïm)=Ïm,ãdÏâ;Ïmã=0andâÏ+(âÏ)t>0; moreover there exists a closed 2-form Φ such that Ï=(δΦâÏm)â¯. Here, ⯠is the musical isomorphism and â its inverse. We connect the above result to an L2-projection problem.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
B. Dacorogna, W. Gangbo, O. Kneuss,