A global version of the Darboux theorem with optimal regularity and Dirichlet condition

Abstract

Let $n>2$ be even; $r\geq1$ be an integer; $0<\alpha<1$; $\Omega$ be a bounded, connected, smooth, open set in $\mathbb{R}^{n}$; and $\nu$ be its exterior unit normal. Let $f,g\in C^{r,\alpha}(\overline{\Omega};\Lambda^{2})$ be two symplectic forms (i.e., closed and of rank $n$) such that $f-g$ is orthogonal to the harmonic fields with vanishing tangential part, $\nu\wedge f,\nu\wedge g\in C^{r+1,\alpha}(\partial\Omega;\Lambda^{3})$ and $\nu\wedge f=\nu\wedge g$ on $\partial\Omega.$ Moreover assume that $tg+(1-t)f$ has rank $n$ for every $t\in\lbrack0,1].$ We will then prove the existence of a $\varphi \in\operatorname{Diff}^{r+1,\alpha}(\overline{\Omega};\overline{\Omega})$ satisfying \[ \left\{ \begin{array} [c]{cl} \varphi^{\ast}(g)=f & \text{in $\Omega$}\\ \varphi=\operatorname{id} & \text{on $\partial\Omega.$} \end{array} \right. \]