Calculus of Variations and Geometric Measure Theory

T. Bieske - F. Dragoni - J. Manfredi

The Carnot-Caratheodory distance and the infinite Laplacian equation

created by dragoni on 18 Sep 2008
modified on 07 Sep 2009


Published Paper

Inserted: 18 sep 2008
Last Updated: 7 sep 2009

Journal: Journal of Geometric Analysis
Volume: 19
Number: 4
Pages: 737-754
Year: 2009


In $R^n$ equipped with the Euclidean metric, the distance from the origin is smooth and infinite harmonic everywhere except the origin. Using geodesics, we find a geometric characterization of when the distance from the origin in an arbitrary Carnot-Carathéodory space is viscosity infinite harmonic at a point outside the origin. We specifically show that at points in the Heisenberg group and Grushin plane where this condition fails, the distance from the origin is not a viscosity subsolution. We also show that at the origin, the distance function is not a viscosity supersolution.