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Journal of Geometric Analysis

, Volume 19, Issue 4, pp 737–754 | Cite as

The Carnot-Carathéodory Distance and the Infinite Laplacian

  • Thomas Bieske
  • Federica Dragoni
  • Juan ManfrediEmail author
Article

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-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.

Keywords

Carnot-Carathéodory spaces Infinite Laplacian Viscosity solutions 

Mathematics Subject Classification (2000)

53C17 22E25 35H20 53C22 

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Copyright information

© Mathematica Josephina, Inc. 2009

Authors and Affiliations

  • Thomas Bieske
    • 1
  • Federica Dragoni
    • 2
  • Juan Manfredi
    • 3
    Email author
  1. 1.Department of MathematicsUniversity of South FloridaTampaUSA
  2. 2.Department of MathematicsImperial College LondonLondonUK
  3. 3.Department of MathematicsUniversity of PittsburghPittsburghUSA

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