Régularité Gevrey des solutions de l'équation de Monge-Ampère réelle

Kallel-Jallouli, Saoussen

Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 3, pp. 629-656

Résumé

We consider in this work the Dirichlet problem associated to a general Monge-Ampère equation: \begin{equation*}\tag{$0.1$} \begin{cases} \det (u_{ij}+a_{ij}(x,u, \nabla u))=K(x) f(x,u, \nabla u) \text{in } \Omega \subset \mathbb{R}^{n} \\ u|_{\partial\Omega} = \varphi \end{cases} \end{equation*} where the curvature $K$ satisfies: $K>0$ in $\Omega$, $K=0$$dK \neq0$ on $\partial\Omega$, and $f$ is strictly positive. We prove that if the data $\Omega$, $a_{ij}$, $K$, $f$, $\varphi$ are in a Gevrey class, every $C^{3}$ solution ($C^{2}$ if $n=2$) of problem $(0.1)$ is in the same Gevrey class on $\overline{\Omega}$.