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


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.


Carnot-Carathéodory spaces Infinite Laplacian Viscosity solutions 

Mathematics Subject Classification (2000)

53C17 22E25 35H20 53C22 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beatrous, F., Bieske, T., Manfredi, J.: The maximum principle for vector fields. In: The p-Harmonic Equation and Recent Advances in Analysis. Contemp. Math., vol. 370, pp. 1–9. Am. Math. Soc., Providence (2005) Google Scholar
  2. 2.
    Bellaïche, A.: The tangent space in sub-Riemannian geometry. In: Bellaïche, A., Risler, J.-J. (eds.) Sub-Riemannian Geometry. Progress in Mathematics, vol. 144, pp. 1–78. Birkhäuser, Basel (1996) Google Scholar
  3. 3.
    Bieske, T.: On ∞-harmonic functions on the Heisenberg group. Commun. PDE 27(3&4), 727–761 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bieske, T.: Lipschitz extensions on generalized Grushin spaces. Mich. Math. J. 53(1), 3–31 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bieske, T.: Properties of infinite harmonic functions on Grushin-type spaces. Rocky Mt. J. Math. 39(3), 729–756 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bieske, T.: Properties of infinite harmonic functions in Riemannian vector fields. Le Matematiche LXIII(2), 19–37 (2008) MathSciNetGoogle Scholar
  7. 7.
    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33. Am. Math. Soc., Providence (2001) zbMATHGoogle Scholar
  8. 8.
    Champion, T., De Pascale, L.: A principle of comparison with distance functions for absolute minimizers. J. Convex Anal. 14(3), 515–541 (2007) zbMATHMathSciNetGoogle Scholar
  9. 9.
    Crandall, M.G., Evans, L.C., Gariepy, R.F.: Optimal Lipschitz extensions and the infinity Laplacian. Calc. Var. Partial Differ Equ. 13(2), 123–139 (2001) zbMATHMathSciNetGoogle Scholar
  10. 10.
    Dragoni, F.: Metric Hopf-Lax formula with semicontinuous data. Discrete Contin. Dyn. Syst. 17(4), 713–729 (2007) zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Heinonen, J.: Calculus on Carnot groups. In: Fall School in Analysis, Jyväskylä, 1994, pp. 1–31. Univ. Jyväskylä, Jyväskylä (1995). Report No. 68 Google Scholar
  12. 12.
    Jensen, R.: Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Ration. Mech. Anal. 123, 51–74 (1993) zbMATHCrossRefGoogle Scholar
  13. 13.
    Lu, G., Manfredi, J., Stroffolini, B.: Convex functions on the Heisenberg group. Calc. Var. Partial Differ Equ. 19(1), 1–22 (2004) CrossRefMathSciNetGoogle Scholar
  14. 14.
    Monti, R.: Some properties of Carnot-Carathéodory balls in the Heisenberg group. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11(3), 155–167 (2000) zbMATHMathSciNetGoogle Scholar
  15. 15.
    Monti, R., Serra Cassano, F.: Surface measures in Carnot-Carathéodory spaces. Calc. Var. Partial Differ Equ. 13, 339–376 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Peres, Y., Schramm, O., Sheffield, S., Wilson, D.B.: Tug-of-war and the infinity Laplacian. In: Rend. Semin. Mat. Univ. Politec. Torino 1989, pp. 15–68 (1991). Special issue on topics in nonlinear PDEs Google Scholar
  17. 17.
    Wang, C.: The Aronsson equation for absolute minimizers of L -functionals associated with vector fields satisfying Hörmander’s condition. Trans. Am. Math. Soc. 359, 91–113 (2007) zbMATHCrossRefGoogle Scholar

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

Personalised recommendations