Abstract
In ℝn 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-Carathéodory 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.
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Communicated by Jiaping Wang.
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Bieske, T., Dragoni, F. & Manfredi, J. The Carnot-Carathéodory Distance and the Infinite Laplacian. J Geom Anal 19, 737–754 (2009). https://doi.org/10.1007/s12220-009-9087-6
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DOI: https://doi.org/10.1007/s12220-009-9087-6