January  2015, 14(1): 83-106. doi: 10.3934/cpaa.2015.14.83

Mean value properties of fractional second order operators

1. 

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

Received  February 2014 Revised  April 2014 Published  September 2014

In this paper we introduce a method to define fractional operators using mean value operators. In particular we discuss a geometric approach in order to construct fractional operators. As a byproduct we define fractional linear operators in Carnot groups, moreover we adapt our technique to define some nonlinear fractional operators associated with the $p-$Laplace operators in Carnot groups.
Citation: Fausto Ferrari. Mean value properties of fractional second order operators. Communications on Pure & Applied Analysis, 2015, 14 (1) : 83-106. doi: 10.3934/cpaa.2015.14.83
References:
[1]

C. Bjorland, L. Caffarelli and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian,, \emph{Comm. Pure Appl. Math., 65 (2012), 337.  doi: 10.1002/cpa.21379.  Google Scholar

[2]

J. Bliedtner and W. Hansen, Potential Theory, An Analytic and Probabilistic Approach to Balayage,, Universitext, (1986).  doi: 10.1007/978-3-642-71131-2.  Google Scholar

[3]

K. Bogdan and T. .Zak, On Kelvin transformation,, \emph{Journal of Theoretical Probability}, 19 (2006), 89.  doi: 10.1007/s10959-006-0003-8.  Google Scholar

[4]

A. Bonfiglioli and E. Lanconelli, Subharmonic functions in sub-Riemannian settings,, \emph{J. Eur. Math. Soc.}, 15 (2013), 387.  doi: 10.4171/JEMS/364.  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]

L. Capogna, D. Danielli, S. Pauls and J. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem,, Birkh\, (2006).   Google Scholar

[7]

G. Citti, N. Garofalo and E. Lanconelli, Harnack's inequality for sum of squares of vector fields plus a potential,, \emph{Amer. J. Math.}, 115 (1993), 699.  doi: 10.2307/2375077.  Google Scholar

[8]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar

[9]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, \emph{Bull. Sci. Math.}, 136 (2012), 521.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[10]

E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations,, \emph{Comm. Partial Differential Equations}, 7 (1982), 77.  doi: 10.1080/03605308208820218.  Google Scholar

[11]

F. Ferrari and B. Franchi, Harnack inequality for fractional sub-Laplacians in Carnot groups,, \emph{preprint}, ().   Google Scholar

[12]

F. Ferrari, Q. Liu and J. J. Manfredi, On the characterization of p-harmonic functions on the Heisenberg group by mean value properties,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 2279.  doi: 10.3934/dcds.2014.34.2779.  Google Scholar

[13]

F. Ferrari and A. Pinamonti, Characterization by asymptotic mean formulas of $q-$harmonic functions in Carnot groups,, \emph{Potential Anal.}, (2014), 11118.  doi: 10.2478/agms-2013-0001.  Google Scholar

[14]

F. Ferrari and I. E. Verbitsky, Radial fractional Laplace operators and Hessian inequalities,, \emph{J. Differential Equations}, 253 (2012), 244.  doi: 10.1016/j.jde.2012.03.024.  Google Scholar

[15]

B. Franchi, R. Serapioni and F. Serra Cassano, On the structure of finite perimeter sets in step $2$ Carnot groups,, \emph{J. Geom. Anal.}, 13 (2003), 421.  doi: 10.1007/BF02922053.  Google Scholar

[16]

W. Fulks, An approximate Gauss mean value theorem,, \emph{Pacific J. Math.}, 14 (1964), 513.   Google Scholar

[17]

N. Garofalo and E. Lanconelli, Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type,, \emph{Trans. Amer. Math. Soc.}, 321 (1990), 775.  doi: 10.2307/2001585.  Google Scholar

[18]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition, (1998).   Google Scholar

[19]

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

[20]

D. Hartenstine and M. Rudd, Statistical functional equations and p-harmonious functions,, \emph{Adv. Nonlinear Stud.}, 13 (2013), 191.   Google Scholar

[21]

D. Hartenstine and M. Rudd, Kelvin transform for $\alpha-$ harmonic functions in regular domains,, \emph{Demostratio Mathematica}, XLV (2012), 361.   Google Scholar

[22]

B. Kawohl, J. J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages,, \emph{J. Math. Pures Appl.}, 97 (2012), 173.  doi: 10.1016/j.matpur.2011.07.001.  Google Scholar

[23]

V. Julin and P. Juutinen, A new proof for the equivalence of a weak and viscosity solutions for the $p-$laplace equation,, \emph{Comm. Partial Differential Equations}, 37 (2012), 934.  doi: 10.1080/03605302.2011.615878.  Google Scholar

[24]

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

[25]

N. S. Landkof, Foundations of Modern Potential Theory,, Translated from the Russian by A. P. Doohovskoy. Die Grundlehren der mathematischen Wissenschaften, (1972).   Google Scholar

[26]

P. Lindqvist, Notes on the p-Laplace equation,, Report. University of Jyv\, (2006).   Google Scholar

[27]

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

[28]

J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of p-harmonious functions,, \emph{Proc. Amer. Math. Soc.}, 138 (2010), 881.  doi: 10.1090/S0002-9939-09-10183-1.  Google Scholar

[29]

J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for p-harmonic functions,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sci.}, 11 (2012), 215.   Google Scholar

[30]

R. Monti and F. Serra Cassano, Surface measures in Carnot-Carathodory spaces,, \emph{Calc. Var. Partial Differential Equations}, 13 (2001), 339.  doi: 10.1007/s005260000076.  Google Scholar

[31]

K. Michalik and M. Ryznar, Asymptotic statistical characterizations of p-harmonic functions of two variables,, \emph{Rocky Mountain J. Math.}, 41 (2011), 493.  doi: 10.1216/RMJ-2011-41-2-493.  Google Scholar

[32]

I. Netuka and J. Veselý, Mean value property and harmonic functions,, Classical and modern potential theory and applications (Chateau de Bonas, (1993), 359.   Google Scholar

[33]

C. Pucci and G. Talenti, Elliptic (second-order) partial differential equations with measurable coefficients and approximating integral equations,, \emph{Advances in Math., 19 (1976), 48.   Google Scholar

[34]

M. Riesz, Intégrales de Riemann-Liouville et potentiels,, \emph{Acta Szeged}, 9 (1938), 1.   Google Scholar

show all references

References:
[1]

C. Bjorland, L. Caffarelli and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian,, \emph{Comm. Pure Appl. Math., 65 (2012), 337.  doi: 10.1002/cpa.21379.  Google Scholar

[2]

J. Bliedtner and W. Hansen, Potential Theory, An Analytic and Probabilistic Approach to Balayage,, Universitext, (1986).  doi: 10.1007/978-3-642-71131-2.  Google Scholar

[3]

K. Bogdan and T. .Zak, On Kelvin transformation,, \emph{Journal of Theoretical Probability}, 19 (2006), 89.  doi: 10.1007/s10959-006-0003-8.  Google Scholar

[4]

A. Bonfiglioli and E. Lanconelli, Subharmonic functions in sub-Riemannian settings,, \emph{J. Eur. Math. Soc.}, 15 (2013), 387.  doi: 10.4171/JEMS/364.  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]

L. Capogna, D. Danielli, S. Pauls and J. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem,, Birkh\, (2006).   Google Scholar

[7]

G. Citti, N. Garofalo and E. Lanconelli, Harnack's inequality for sum of squares of vector fields plus a potential,, \emph{Amer. J. Math.}, 115 (1993), 699.  doi: 10.2307/2375077.  Google Scholar

[8]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar

[9]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, \emph{Bull. Sci. Math.}, 136 (2012), 521.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[10]

E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations,, \emph{Comm. Partial Differential Equations}, 7 (1982), 77.  doi: 10.1080/03605308208820218.  Google Scholar

[11]

F. Ferrari and B. Franchi, Harnack inequality for fractional sub-Laplacians in Carnot groups,, \emph{preprint}, ().   Google Scholar

[12]

F. Ferrari, Q. Liu and J. J. Manfredi, On the characterization of p-harmonic functions on the Heisenberg group by mean value properties,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 2279.  doi: 10.3934/dcds.2014.34.2779.  Google Scholar

[13]

F. Ferrari and A. Pinamonti, Characterization by asymptotic mean formulas of $q-$harmonic functions in Carnot groups,, \emph{Potential Anal.}, (2014), 11118.  doi: 10.2478/agms-2013-0001.  Google Scholar

[14]

F. Ferrari and I. E. Verbitsky, Radial fractional Laplace operators and Hessian inequalities,, \emph{J. Differential Equations}, 253 (2012), 244.  doi: 10.1016/j.jde.2012.03.024.  Google Scholar

[15]

B. Franchi, R. Serapioni and F. Serra Cassano, On the structure of finite perimeter sets in step $2$ Carnot groups,, \emph{J. Geom. Anal.}, 13 (2003), 421.  doi: 10.1007/BF02922053.  Google Scholar

[16]

W. Fulks, An approximate Gauss mean value theorem,, \emph{Pacific J. Math.}, 14 (1964), 513.   Google Scholar

[17]

N. Garofalo and E. Lanconelli, Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type,, \emph{Trans. Amer. Math. Soc.}, 321 (1990), 775.  doi: 10.2307/2001585.  Google Scholar

[18]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition, (1998).   Google Scholar

[19]

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

[20]

D. Hartenstine and M. Rudd, Statistical functional equations and p-harmonious functions,, \emph{Adv. Nonlinear Stud.}, 13 (2013), 191.   Google Scholar

[21]

D. Hartenstine and M. Rudd, Kelvin transform for $\alpha-$ harmonic functions in regular domains,, \emph{Demostratio Mathematica}, XLV (2012), 361.   Google Scholar

[22]

B. Kawohl, J. J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages,, \emph{J. Math. Pures Appl.}, 97 (2012), 173.  doi: 10.1016/j.matpur.2011.07.001.  Google Scholar

[23]

V. Julin and P. Juutinen, A new proof for the equivalence of a weak and viscosity solutions for the $p-$laplace equation,, \emph{Comm. Partial Differential Equations}, 37 (2012), 934.  doi: 10.1080/03605302.2011.615878.  Google Scholar

[24]

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

[25]

N. S. Landkof, Foundations of Modern Potential Theory,, Translated from the Russian by A. P. Doohovskoy. Die Grundlehren der mathematischen Wissenschaften, (1972).   Google Scholar

[26]

P. Lindqvist, Notes on the p-Laplace equation,, Report. University of Jyv\, (2006).   Google Scholar

[27]

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

[28]

J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of p-harmonious functions,, \emph{Proc. Amer. Math. Soc.}, 138 (2010), 881.  doi: 10.1090/S0002-9939-09-10183-1.  Google Scholar

[29]

J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for p-harmonic functions,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sci.}, 11 (2012), 215.   Google Scholar

[30]

R. Monti and F. Serra Cassano, Surface measures in Carnot-Carathodory spaces,, \emph{Calc. Var. Partial Differential Equations}, 13 (2001), 339.  doi: 10.1007/s005260000076.  Google Scholar

[31]

K. Michalik and M. Ryznar, Asymptotic statistical characterizations of p-harmonic functions of two variables,, \emph{Rocky Mountain J. Math.}, 41 (2011), 493.  doi: 10.1216/RMJ-2011-41-2-493.  Google Scholar

[32]

I. Netuka and J. Veselý, Mean value property and harmonic functions,, Classical and modern potential theory and applications (Chateau de Bonas, (1993), 359.   Google Scholar

[33]

C. Pucci and G. Talenti, Elliptic (second-order) partial differential equations with measurable coefficients and approximating integral equations,, \emph{Advances in Math., 19 (1976), 48.   Google Scholar

[34]

M. Riesz, Intégrales de Riemann-Liouville et potentiels,, \emph{Acta Szeged}, 9 (1938), 1.   Google Scholar

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