R\'egularit\'e du rayon hyperbolique

TL;DR

This paper proves that the hyperbolic radius function associated with a maximal solution of a specific elliptic PDE is smooth up to the boundary of a domain, using new Schauder estimates for Fuchsian equations.

Contribution

It establishes boundary regularity of the hyperbolic radius for solutions of a nonlinear elliptic PDE, introducing new Schauder estimates for Fuchsian elliptic equations.

Findings

Hyperbolic radius is of class C^{2+α} up to the boundary.

New Schauder estimates developed for Fuchsian elliptic equations.

Boundary regularity results for solutions of nonlinear elliptic PDEs.

Abstract

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