Régularité du rayon hyperbolique

Abstract

Let $\hat{I}\text{c◯}\subset {\mathbb{R}}^2$ be a bounded domain of class $C^{2+\hat{I}\pm }$, $0<\hat{I}\pm <1$. We show that if $u$ is the maximal solution of $\hat{I}”u=4\exp (2u)$, which tends to $+\infty$ as $(x,y)\to \partial \hat{I}\text{c◯}$, then the hyperbolic radius $v=\exp (-u)$ is of class $C^{2+\hat{I}\pm }$ up to the boundary. The proof relies on new Schauder estimates for Fuchsian elliptic equations.