Abstract

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.