Régularité du rayon hyperbolique

Satyanad Kichenassamy

doi:10.1016/j.crma.2003.10.037

Équations aux dérivées partielles

Régularité du rayon hyperbolique
[Smoothness of hyperbolic radius]

Presented by: Haïm Brezis

Satyanad Kichenassamy ¹

¹ Laboratoire de mathématiques (UMR 6056), CNRS & Université de Reims Champagne-Ardenne, Moulin de la Housse, BP 1039, 51687 Reims cedex 2, France

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

Abstract
Résumé

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

Soit $Ω \subset ℝ^{2}$ un domaine borné de classe $C^{2 + α}, 0 < α < 1$ . On montre que si u est la solution maximale de Δu=4exp(2u), qui tend vers +∞ si $(x, y) \to \partial Ω$ , 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.

Received: 2003-10-26
Published online: 2003-12-21

DOI: 10.1016/j.crma.2003.10.037

Author's affiliations:

Satyanad Kichenassamy ¹

¹ Laboratoire de mathématiques (UMR 6056), CNRS & Université de Reims Champagne-Ardenne, Moulin de la Housse, BP 1039, 51687 Reims cedex 2, France

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     title = {R\'egularit\'e du rayon hyperbolique},
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     pages = {13--18},
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     number = {1},
     year = {2004},
     doi = {10.1016/j.crma.2003.10.037},
     language = {fr},
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Satyanad Kichenassamy. Régularité du rayon hyperbolique. Comptes Rendus. Mathématique, Volume 338 (2004) no. 1, pp. 13-18. doi : 10.1016/j.crma.2003.10.037. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.10.037/

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