July  2014, 34(7): 2779-2793. doi: 10.3934/dcds.2014.34.2779

On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties

1. 

Dipartimento di Matematica dell'Università di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy

2. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, United States, United States

Received  August 2013 Revised  October 2013 Published  December 2013

We characterize $p-$harmonic functions in the Heisenberg group in terms of an asymptotic mean value property, where $1 < p <\infty$, following the scheme described in [16] for the Euclidean case. The new tool that allows us to consider the subelliptic case is a geometric lemma, Lemma 3.2 below, that relates the directions of the points of maxima and minima of a function on a small subelliptic ball with the unit horizontal gradient of that function.
Citation: Fausto Ferrari, Qing Liu, Juan Manfredi. On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2779-2793. doi: 10.3934/dcds.2014.34.2779
References:
[1]

N. Arcozzi and F. Ferrari, Metric normal and distance function in the Heisenberg group, Math. Z., 256 (2007), 661-684. doi: 10.1007/s00209-006-0098-8.

[2]

N. Arcozzi and F. Ferrari, The Hessian of the distance from a surface in the Heisenberg group, Ann. Acad. Sci. Fenn. Math., 33 (2008), 35-63.

[3]

N. Arcozzi, F. Ferrari and F. Montefalcone, CC-distance and metric normal of smooth hypersurfaces in sub-Riemannian two-step Carnot groups, preprint, arXiv:0910.5648v1.

[4]

T. Bieske, Equivalence of weak and viscosity solutions to the $p$-Laplace equation in the Heisenberg group, Ann. Acad. Sci. Fenn. Math., 31 (2006), 363-379.

[5]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007.

[6]

J.-M. Bony, Principe du maximum et inégalité de Harnack pour les opérateurs elliptiques dégénérés, in 1969 Séminaire de Théorie du Potentiel, dirigé par M. Brelot, G. Choquet et J. Deny: 1967/1968, Exp. 10, Secrétariat Mathématique, Paris, 1969, 20 pp.

[7]

J.-M. Bony, Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble), 19 (1969), 277-304. doi: 10.5802/aif.319.

[8]

L. Capogna and G. Citti, Generalized mean curvature flow in Carnot groups, Comm. Partial Differential Equations, 34 (2009), 937-956. doi: 10.1080/03605300903050257.

[9]

L. Capogna, D. Danielli, S. D. Pauls and J. T. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Progress in Mathematics, 259, Birkhäuser Verlag, Basel, 2007.

[10]

B. Franchi, R. Serapioni and F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann., 321 (2001), 479-531. doi: 10.1007/s002080100228.

[11]

F. Ferrari, Q. Liu and J. J. Manfredi, On the horizontal mean curvature Flow for axisymmetric surfaces in the Heisenberg group, to appear in Commun. Contemp. Math. doi: 10.1142/S0219199713500272.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[13]

C. Gutiérrez and E. Lanconelli, Classical viscosity and average solutions for PDE's with nonnegative characteristic form, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 15 (2004), 17-28.

[14]

P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear elliptic equation, SIAM J. Math. Anal., 33 (2001), 699-717. doi: 10.1137/S0036141000372179.

[15]

H. Liu and X. Yang, Asymptotic mean value formula for sub-$p$-harmonic functions on the Heisenberg group, J. Funct. Anal., 264 (2013), 2177-2196. doi: 10.1016/j.jfa.2013.02.009.

[16]

J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for p-harmonic functions, Proc. Amer. Math. Soc., 138 (2010), 881-889. doi: 10.1090/S0002-9939-09-10183-1.

[17]

Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210. doi: 10.1090/S0894-0347-08-00606-1.

[18]

C. Pucci and G. Talenti, Elliptic (second-order) partial differential equations with measurable coefficients and approximating integral equations, Advances in Math., 19 (1976), 48-105. doi: 10.1016/0001-8708(76)90022-0.

show all references

References:
[1]

N. Arcozzi and F. Ferrari, Metric normal and distance function in the Heisenberg group, Math. Z., 256 (2007), 661-684. doi: 10.1007/s00209-006-0098-8.

[2]

N. Arcozzi and F. Ferrari, The Hessian of the distance from a surface in the Heisenberg group, Ann. Acad. Sci. Fenn. Math., 33 (2008), 35-63.

[3]

N. Arcozzi, F. Ferrari and F. Montefalcone, CC-distance and metric normal of smooth hypersurfaces in sub-Riemannian two-step Carnot groups, preprint, arXiv:0910.5648v1.

[4]

T. Bieske, Equivalence of weak and viscosity solutions to the $p$-Laplace equation in the Heisenberg group, Ann. Acad. Sci. Fenn. Math., 31 (2006), 363-379.

[5]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007.

[6]

J.-M. Bony, Principe du maximum et inégalité de Harnack pour les opérateurs elliptiques dégénérés, in 1969 Séminaire de Théorie du Potentiel, dirigé par M. Brelot, G. Choquet et J. Deny: 1967/1968, Exp. 10, Secrétariat Mathématique, Paris, 1969, 20 pp.

[7]

J.-M. Bony, Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble), 19 (1969), 277-304. doi: 10.5802/aif.319.

[8]

L. Capogna and G. Citti, Generalized mean curvature flow in Carnot groups, Comm. Partial Differential Equations, 34 (2009), 937-956. doi: 10.1080/03605300903050257.

[9]

L. Capogna, D. Danielli, S. D. Pauls and J. T. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Progress in Mathematics, 259, Birkhäuser Verlag, Basel, 2007.

[10]

B. Franchi, R. Serapioni and F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann., 321 (2001), 479-531. doi: 10.1007/s002080100228.

[11]

F. Ferrari, Q. Liu and J. J. Manfredi, On the horizontal mean curvature Flow for axisymmetric surfaces in the Heisenberg group, to appear in Commun. Contemp. Math. doi: 10.1142/S0219199713500272.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[13]

C. Gutiérrez and E. Lanconelli, Classical viscosity and average solutions for PDE's with nonnegative characteristic form, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 15 (2004), 17-28.

[14]

P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear elliptic equation, SIAM J. Math. Anal., 33 (2001), 699-717. doi: 10.1137/S0036141000372179.

[15]

H. Liu and X. Yang, Asymptotic mean value formula for sub-$p$-harmonic functions on the Heisenberg group, J. Funct. Anal., 264 (2013), 2177-2196. doi: 10.1016/j.jfa.2013.02.009.

[16]

J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for p-harmonic functions, Proc. Amer. Math. Soc., 138 (2010), 881-889. doi: 10.1090/S0002-9939-09-10183-1.

[17]

Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210. doi: 10.1090/S0894-0347-08-00606-1.

[18]

C. Pucci and G. Talenti, Elliptic (second-order) partial differential equations with measurable coefficients and approximating integral equations, Advances in Math., 19 (1976), 48-105. doi: 10.1016/0001-8708(76)90022-0.

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