Régularité du rayon hyperbolique

Kichenassamy, Satyanad

Comptes Rendus. Mathématique, Tome 338 (2004) no. 1, pp. 13-18.

Soit $\Omega \subset {ℝ}^2$ un domaine borné de classe $C^{2+\alpha },0<\alpha <1$. On montre que si u est la solution maximale de Δu=4exp(2u), qui tend vers +∞ si $(\mathrm{x},\mathrm{y})\to \partial \Omega$, alors le rayon hyperbolique v=exp(−u) est de classe C^2+α jusqu'au bord. La démonstration repose sur de nouvelles estimations de Schauder pour des équations fuchsiennes elliptiques.

Let $\Omega \subset {ℝ}^2$ be a bounded domain of class $C^{2+\alpha },0<\alpha <1$. We show that if u is the maximal solution of Δu=4exp(2u), which tends to +∞ as $(\mathrm{x},\mathrm{y})\to \partial \Omega$, 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.

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