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

Dragoni, Federica; Manfredi, Juan; Vittone, Davide

doi:10.1090/S0002-9947-2012-05612-1

Weak Fubini property and infinity harmonic functions in Riemannian and sub-Riemannian manifolds
HTML articles powered by AMS MathViewer

by Federica Dragoni, Juan J. Manfredi and Davide Vittone PDF

Trans. Amer. Math. Soc. 365 (2013), 837-859 Request permission

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

Vladimir I. Arnold, Ordinary differential equations, Universitext, Springer-Verlag, Berlin, 2006. Translated from the Russian by Roger Cooke; Second printing of the 1992 edition. MR 2242407
Gunnar Aronsson, Michael G. Crandall, and Petri Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 4, 439–505. MR 2083637, DOI 10.1090/S0273-0979-04-01035-3
Frank H. Beatrous, Thomas J. Bieske, and Juan J. Manfredi, The maximum principle for vector fields, The $p$-harmonic equation and recent advances in analysis, Contemp. Math., vol. 370, Amer. Math. Soc., Providence, RI, 2005, pp. 1–9. MR 2126697, DOI 10.1090/conm/370/06825
André Bellaïche, The tangent space in sub-Riemannian geometry, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 1–78. MR 1421822, DOI 10.1007/978-3-0348-9210-0_{1}
Thomas Bieske, Lipschitz extensions on generalized Grushin spaces, Michigan Math. J. 53 (2005), no. 1, 3–31. MR 2125531, DOI 10.1307/mmj/1114021082
Thomas Bieske, Federica Dragoni, and Juan Manfredi, The Carnot-Carathéodory distance and the infinite Laplacian, J. Geom. Anal. 19 (2009), no. 4, 737–754. MR 2538933, DOI 10.1007/s12220-009-9087-6
Thierry Champion and Luigi De Pascale, Principles of comparison with distance functions for absolute minimizers, J. Convex Anal. 14 (2007), no. 3, 515–541. MR 2341302
M. G. Crandall, L. C. Evans, and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations 13 (2001), no. 2, 123–139. MR 1861094, DOI 10.1007/s005260000065
Federica Dragoni, Metric Hopf-Lax formula with semicontinuous data, Discrete Contin. Dyn. Syst. 17 (2007), no. 4, 713–729. MR 2276470, DOI 10.3934/dcds.2007.17.713
B. Franchi, P. Hajłasz, and P. Koskela, Definitions of Sobolev classes on metric spaces, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 6, 1903–1924. MR 1738070, DOI 10.5802/aif.1742
Nicola Garofalo and Duy-Minh 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, DOI 10.1007/BF02819446
Juha Heinonen, Calculus on Carnot groups, Fall School in Analysis (Jyväskylä, 1994) Report, vol. 68, Univ. Jyväskylä, Jyväskylä, 1995, pp. 1–31. MR 1351042, DOI 10.1530/jrf.0.0680001
Robert Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal. 123 (1993), no. 1, 51–74. MR 1218686, DOI 10.1007/BF00386368
Petri Juutinen, Absolutely minimizing Lipschitz extensions on a metric space, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 1, 57–67. MR 1884349
Petri Juutinen and Nageswari 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, DOI 10.1002/mana.200510411
E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837–842. MR 1562984, DOI 10.1090/S0002-9904-1934-05978-0
Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. MR 1867362, DOI 10.1090/surv/091
Roberto Monti and Matthieu Rickly, Geodetically convex sets in the Heisenberg group, J. Convex Anal. 12 (2005), no. 1, 187–196. MR 2135806
Roberto Monti and Francesco Serra Cassano, Surface measures in Carnot-Carathéodory spaces, Calc. Var. Partial Differential Equations 13 (2001), no. 3, 339–376. MR 1865002, DOI 10.1007/s005260000076
Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
Pierre Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60 (French, with English summary). MR 979599, DOI 10.2307/1971484
Yuval Peres, Oded Schramm, Scott Sheffield, and David B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), no. 1, 167–210. MR 2449057, DOI 10.1090/S0894-0347-08-00606-1
Pierpaolo Soravia, On Aronsson equation and deterministic optimal control, Appl. Math. Optim. 59 (2009), no. 2, 175–201. MR 2480779, DOI 10.1007/s00245-008-9048-7
V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics, vol. 102, Springer-Verlag, New York, 1984. Reprint of the 1974 edition. MR 746308, DOI 10.1007/978-1-4612-1126-6
Changyou 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), no. 1, 91–113. MR 2247884, DOI 10.1090/S0002-9947-06-03897-9
Hassler Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89. MR 1501735, DOI 10.1090/S0002-9947-1934-1501735-3