The Carnot-Caratheodory distance and the infinite Laplacian

Thomas Bieske, Federica Dragoni, and Juan J. Manfredi

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Submission date: 20. Oct. 2008
Pages: 22
published in: The journal of geometric analysis, 19 (2009) 4, p. 737-754 
DOI number (of the published article): 10.1007/s12220-009-9087-6
Bibtex
MSC-Numbers: 53C17, 22E25, 35H20
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Abstract:
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 for when the distance from the origin in an arbitrary Carnot-Caratheodory space is a viscosity infinite harmonic function at a point outside the origin. We show that at points in the Heisenberg group and Grushin plane where this condition fails, the distance from the origin is not a viscosity infinite harmonic subsolution. In addition, the distance function is not a viscosity infinite harmonic supersolution at the origin.