The Carnot–Carathéodory distance vis-à-vis the eikonal equation and the infinite Laplacian

In ℝn equipped with the Euclidean metric, the distance from the origin (smoothly) satisfies the eikonal equation and is (smoothly) infinite harmonic everywhere except the origin. Dragoni (Discrete Contin. Dyn. Syst. 17 (2007) 713–729) has shown that the Carnot–Carathéodory distance satisfies the eik...

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Bibliographic Details
Published in:The Bulletin of the London Mathematical Society 2010-06, Vol.42 (3), p.395-404
Main Author: Bieske, Thomas
Format: Article
Language:English
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Summary:In ℝn equipped with the Euclidean metric, the distance from the origin (smoothly) satisfies the eikonal equation and is (smoothly) infinite harmonic everywhere except the origin. Dragoni (Discrete Contin. Dyn. Syst. 17 (2007) 713–729) has shown that the Carnot–Carathéodory distance satisfies the eikonal equation in the viscosity sense outside of the origin, but Bieske, Dragoni and Manfredi (J. Geom. Anal. 19 (2009) 737–754) have shown that the distance is not viscosity infinite harmonic at all points outside the origin. We examine the behavior of the negative distance function and show that it is a viscosity solution to the eikonal equation exactly where it is viscosity infinite harmonic.
ISSN:0024-6093
1469-2120
DOI:10.1112/blms/bdp131