# CONFLUENTES MATHEMATICI

The role of the Hilbert metric in a class of singular elliptic boundary value problems in convex domains
Confluentes Mathematici, Tome 9 (2017) no. 1, pp. 105-117.

In a recent paper [7], we were led to consider a distance over a bounded open convex domain. It turns out to be the so-called Thompson metric, which is equivalent to the Hilbert metric. It plays a key role in the analysis of existence and uniqueness of solutions to a class of elliptic boundary-value problems that are singular at the boundary.

Reçu le : 2017-02-02
Révisé le : 2017-09-05
Accepté le : 2017-10-05
Publié le : 2017-09-14
DOI : https://doi.org/10.5802/cml.38
Classification : 35J75,  52A99
Mots clés: Elliptic PDEs, convex domain, Hilbert metric, singular BVP
@article{CML_2017__9_1_105_0,
author = {Denis Serre},
title = {The role of the Hilbert metric in a class of  singular elliptic boundary value problems in convex domains},
journal = {Confluentes Mathematici},
publisher = {Institut Camille Jordan},
volume = {9},
number = {1},
year = {2017},
pages = {105-117},
doi = {10.5802/cml.38},
language = {en},
url = {cml.centre-mersenne.org/item/CML_2017__9_1_105_0/}
}
Serre, Denis. The role of the Hilbert metric in a class of  singular elliptic boundary value problems in convex domains. Confluentes Mathematici, Tome 9 (2017) no. 1, pp. 105-117. doi : 10.5802/cml.38. https://cml.centre-mersenne.org/item/CML_2017__9_1_105_0/

[1] M. T. Anderson. Complete minimal varieties in hyperbolic space. Inventiones mathematicae, 69:477–494, 1982.

[2] D. Gilbarg, N. Trudinger. Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer-Verlag, Heidelberg, 2001.

[3] D. Hilbert. Ueber die gerade Linie als kürzeste Verbindung zweier Punkte. Mathematische Annalen, 46:91–96, 1895.

[4] Fang Hua Lin. On the Dirichlet problem for minimal graphs. Inventiones mathematicae, 96:593–612, 1989.

[5] L. Marquis. Géométrie de Hilbert. Images des Mathématiques, CNRS (2015). http://images.math.cnrs.fr/Geometrie-de-Hilbert.html.

[6] D. Serre. Multi-dimensional shock interaction for a Chaplygin gas. Arch. Rational Mech. Anal., 191:539–577, 2009.

[7] D. Serre. Gradient estimate in terms of a Hilbert-like distance, for minimal surfaces and Chaplygin gas. Comm. Partial Diff. Equ., 41:774–784, 2016.

[8] C. Walsh. Gauge-reversing maps on cones, and Hilbert and Thompson isometries. Preprint arXiv:1312.7871 [math.MG] (December 2013).