Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Weak Fubini property and infinity harmonic functions in Riemannian and sub-Riemannian manifolds


Authors: Federica Dragoni, Juan J. Manfredi and Davide Vittone
Journal: Trans. Amer. Math. Soc. 365 (2013), 837-859
MSC (2010): Primary 53C17, 22E25, 35H20, 53C22
DOI: https://doi.org/10.1090/S0002-9947-2012-05612-1
Published electronically: September 19, 2012
MathSciNet review: 2995375
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We examine the relationship between infinity harmonic functions, absolutely minimizing Lipschitz extensions, strong absolutely minimizing Lipschitz extensions, and absolutely gradient minimizing extensions in Carnot-Carathéodory spaces. Using the weak Fubini property we show that absolutely minimizing Lipschitz extensions are infinity harmonic in any sub-Riemannian manifold.


References [Enhancements On Off] (What's this?)

  • 1. V. Arnold,
    Ordinary differential equations,
    Springer-Verlag, Berlin, 2006. MR 2242407 (2007b:34001)
  • 2. G. Aronson, M.G. Crandall, P. Juutinen,
    A tour of the theory of absolutely minimizing functions,
    Bull. Amer. Math. Soc. 41 (2004), no. 4, 439-505. MR 2083637 (2005k:35159)
  • 3. F.H. Beatrous, T.  Bieske, J. Manfredi,
    The maximum principle for vector fields,
    The $ p$-harmonic equation and recent advances in analysis. Contemp. Math. 370, Amer. Math. Soc., Providence, RI, 2005, 1-9. MR 2126697 (2005m:35021)
  • 4.
    A. Bellaïche,
    The tangent space in sub-Riemannian geometry,
    in Sub-Riemannian Geometry, (eds. J.J. Risler and A. Bellaïche),
    Birkhäuser, Basel, 1996, 1-78. MR 1421822 (98a:53108)
  • 5. T.  Bieske,
    Lipschitz extensions on generalized Grushin spaces,
    Michigan Math. J. 53 (2005), no. 1, 3-31. MR 2125531 (2006i:35032)
  • 6. T.  Bieske, F. Dragoni, J. Manfredi,
    The Carnot-Carathéodory distance and the infinite Laplacian,
    J. Geom. Anal. 19 (2009), no. 4, 737-754. MR 2538933 (2011b:53065)
  • 7. T. Champion, L. De Pascale,
    Principles of comparison with distance functions for absolute minimizers,
    Journal of Convex Analysis 14 (2007), no. 3, 515-541. MR 2341302 (2008j:49073)
  • 8. M.G. Crandall, L.C. Evans, R.F. Gariepy,
    Optimal Lipschitz extensions and the infinity Laplacian,
    Calc. Var. P.D.E. 13 (2001), no. 2, 123-139. MR 1861094 (2002h:49048)
  • 9. F. Dragoni,
    Metric Hopf-Lax formula with semicontinuous data,
    Discrete Contin. Dyn. Syst. 17 (2007), no.4, 713-729. MR 2276470 (2007k:35044)
  • 10. B. Franchi, P. Hajlasz, P. Koskela,
    Definitions of Sobolev classes on metric spaces,
    Annales de l'Institut Fourier 49 (1999), no. 6, 1903-1924 . MR 1738070 (2001a:46033)
  • 11. N. Garofalo, D.-M. Nhieu,
    Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces,
    J. Anal. Math. 74 (1998), 67-97. MR 1631642 (2000i:46025)
  • 12. J. Heinonen,
    Calculus on Carnot Groups,
    Fall School in Analysis, Jyväskylä (1994), 1-31. MR 1351042 (96j:22015)
  • 13. R. Jensen,
    Uniqueness of Lipschitz Extensions: Minimizing the Sup Norm of the Gradient,
    Arch. Ration. Mech. Anal. 123 (1993), no. 1, 51-74. MR 1218686 (94g:35063)
  • 14. P. Juutinen,
    Absolutely minimizing Lipschitz extensions on a metric space,
    Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 1, 57-67. MR 1884349 (2002m:54020)
  • 15. P. Juutinen, N. Shanmugalingam,
    Equivalence of AMLE, strong AMLE, and comparison with cones in metric measure spaces,
    Math. Nachr. 279 (2006), no. 9-10, 1083-1098. MR 2242966 (2008e:31009)
  • 16. E.J. McShane,
    Extension of range of functions,
    Bull. Amer. Math. Soc. 40 (1934), 837-842. MR 1562984
  • 17.
    R. Montgomery,
    A Tour of Subriemannian Geometries, Their Geodesics and Applications,
    American Mathematical Society, Providence, 2002. MR 1867362 (2002m:53045)
  • 18. R. Monti, M. Rickly,
    Geodetically convex sets in the Heisenberg group,
    J. Convex Anal. 12 (2005), no. 1, 187-196. MR 2135806 (2005m:53045)
  • 19. R. Monti, F. Serra Cassano,
    Surface measures in Carnot-Carathéodory spaces,
    Rend. Mat. Acc. Lincei 9 (2001), no. 3, 155-167. MR 1865002 (2002j:49052)
  • 20. B. O'Neill,
    Semi-Riemannian geometry. With applications to relativity,
    Pure and Applied Mathematics, 103. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. MR 719023 (85f:53002)
  • 21. P. Pansu.
    Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un,
    Ann. of Math. (2) 129 (1989), 1-60. MR 979599 (90e:53058)
  • 22. Y. Peres, O. Schramm, S. Sheffield, D.B. Wilson,
    Tug-of-war and the infinity Laplacian,
    J. Amer. Math. Soc. 22 (2009), no. 1, 167-210. MR 2449057 (2009h:91004)
  • 23. P. Soravia,
    On Aronsson equation and deterministic optimal control,
    Appl. Math. Optim. 59 (2009), 175-201. MR 2480779 (2009m:35167)
  • 24. V. S. Varadarajan,
    Lie groups, Lie algebras, and their representations,
    Springer-Verlag, New York, 1984. xiii+430 pp. MR 746308 (85e:22001)
  • 25. C. Wang,
    The Aronsson equation for absolute minimizers of $ {L^\infty }$-functionals associated with vector fields satisfying Hörmander's condition,
    Trans. Amer. Math. Soc. 359 (2007), 91-113. MR 2247884 (2007g:35021)
  • 26. H. Whitney,
    Analytic extension of differentiable functions defined in closed sets,
    Trans. Amer. Math. Soc. 36 (1934), no. 1, 63-89. MR 1501735

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C17, 22E25, 35H20, 53C22

Retrieve articles in all journals with MSC (2010): 53C17, 22E25, 35H20, 53C22


Additional Information

Federica Dragoni
Affiliation: School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, Wales, United Kingdom CF24 4AG

Juan J. Manfredi
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

Davide Vittone
Affiliation: Dipartimento di Matematica, University of Padova, via Trieste 63, 35121 Padova, Italy

DOI: https://doi.org/10.1090/S0002-9947-2012-05612-1
Keywords: Absolutely minimizing Lipschitz extension, infinity Laplace equation, Riemannian manifolds, Carnot-Carathéodory spaces
Received by editor(s): December 15, 2010
Received by editor(s) in revised form: April 22, 2011
Published electronically: September 19, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society