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 & Continuous Dynamical Systems - A, 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.  doi: 10.1007/s00209-006-0098-8.  Google Scholar

[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.   Google Scholar

[3]

N. Arcozzi, F. Ferrari and F. Montefalcone, CC-distance and metric normal of smooth hypersurfaces in sub-Riemannian two-step Carnot groups,, preprint, ().   Google Scholar

[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.   Google Scholar

[5]

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

[6]

J.-M. Bony, Principe du maximum et inégalité de Harnack pour les opérateurs elliptiques dégénérés,, in 1969 Séminaire de Théorie du Potentiel, (1967).   Google Scholar

[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.  doi: 10.5802/aif.319.  Google Scholar

[8]

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

[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, (2007).   Google Scholar

[10]

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

[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.  Google Scholar

[12]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,, Reprint of the 1998 edition, (1998).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/S0036141000372179.  Google Scholar

[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.  doi: 10.1016/j.jfa.2013.02.009.  Google Scholar

[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.  doi: 10.1090/S0002-9939-09-10183-1.  Google Scholar

[17]

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

[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.  doi: 10.1016/0001-8708(76)90022-0.  Google Scholar

show all references

References:
[1]

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

[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.   Google Scholar

[3]

N. Arcozzi, F. Ferrari and F. Montefalcone, CC-distance and metric normal of smooth hypersurfaces in sub-Riemannian two-step Carnot groups,, preprint, ().   Google Scholar

[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.   Google Scholar

[5]

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

[6]

J.-M. Bony, Principe du maximum et inégalité de Harnack pour les opérateurs elliptiques dégénérés,, in 1969 Séminaire de Théorie du Potentiel, (1967).   Google Scholar

[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.  doi: 10.5802/aif.319.  Google Scholar

[8]

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

[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, (2007).   Google Scholar

[10]

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

[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.  Google Scholar

[12]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,, Reprint of the 1998 edition, (1998).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/S0036141000372179.  Google Scholar

[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.  doi: 10.1016/j.jfa.2013.02.009.  Google Scholar

[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.  doi: 10.1090/S0002-9939-09-10183-1.  Google Scholar

[17]

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

[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.  doi: 10.1016/0001-8708(76)90022-0.  Google Scholar

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