Régularité du rayon hyperbolique

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