Premium
The Carnot–Carathéodory distance vis‐à‐vis the eikonal equation and the infinite Laplacian
Author(s) -
Bieske Thomas
Publication year - 2010
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdp131
Subject(s) - eikonal equation , mathematics , harmonic function , carnot cycle , mathematical analysis , viscosity , viscosity solution , laplace operator , harmonic , physics , quantum mechanics
In ℝ n 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.