Skip to main content
SpringerLink
Log in

Estimates for nonlinear harmonic measures on trees

  • Published:
Bulletin of the Brazilian Mathematical Society, New Series Aims and scope Submit manuscript

Abstract

In this paper we give some estimates for nonlinear harmonic measures on trees. In particular, we estimate in terms of the size of a set D the value at the origin of the solution to u(x) = F((x, 0),...,(x,m − 1)) for every x ∈ \(\mathbb{T}_m \), a directed tree with m branches with initial datum f + χD. Here F is an averaging operator on ℝm, x is a vertex of a directed tree \(\mathbb{T}_m \) with regular m-branching and (x, i) denotes a successor of that vertex for 0 ≤ i ≤ m − 1.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. Alvarez, J.M. Rodríguez and D.V. Yakubovich. Estimates for nonlinear harmonic “measures” on trees. Michigan Math. J., 49(1) (2001), 47–64.

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Aviles and J.J. Manfredi. On null sets of P-harmonic measures. Partial differential equations with minimal smoothness and applications (Chicago, IL, 1990), 33–36, IMA Vol. Math. Appl., 42, Springer, New York (1992).

    Article  MathSciNet  Google Scholar 

  3. A. Björn, J. Björn and N. Shanmugalingam. A problem of Baernstein on the equality of the p-harmonic measure of a set and its closure. Proc. Amer. Math. Soc., 134(2) (2006), 509–519 (electronic).

    Article  MathSciNet  MATH  Google Scholar 

  4. L.M. Del Pezzo, C.A. Mosquera and J.D. Rossi. The unique continuation property for a nonlinear equation on trees. Journal of the London Mathematical Society, 89(2) (2014), 364–382.

    Article  MathSciNet  MATH  Google Scholar 

  5. S. Granlund, P. Lindqvist and O. Martio. F-harmonic measure in space. Ann. Acad. Sci. Fenn. Ser. A I Math., 7(2) (1982), 233–247.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Heinonen, T. Kilpeläinen and O. Martio. Nonlinear potential theory of degenerate elliptic equations. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, (1993), vi+363 pp.

    Google Scholar 

  7. D. Hartenstine and M. Rudd. Asymptotic statistical characterizations of pharmonic functions of two variables. Rocky Mountain J. Math., 41(2) (2011), 493–504.

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Hartenstine and M. Rudd. Statistical functional equations and p-harmonious functions. Adv. Nonlinear Stud., 13(1) (2013), 191–207.

    MathSciNet  MATH  Google Scholar 

  9. R. Kaufman, J.G. Llorente and Jang-Mei Wu. Nonlinear harmonic measures on trees. Ann. Acad. Sci. Fenn. Math., 28(2) (2003), 279–302.

    MathSciNet  MATH  Google Scholar 

  10. R. Kaufman and Jang-Mei Wu. Fatou theorem of p-harmonic functions on trees. Ann. Probab., 28(3) (2000), 1138–1148.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Kurki. Invariant sets for [Math Processing Error]-harmonic measure (English summary). Ann. Acad. Sci. Fenn. Ser. A I Math., 20(2) (1995), 433–436.

    MathSciNet  MATH  Google Scholar 

  12. J.G. Llorente, J.J. Manfredi and J.M. Wu. p-harmonic measure is not additive on null sets (English summary). Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 4(2) (2005), 357–373.

    MathSciNet  MATH  Google Scholar 

  13. J.J. Manfredi, M. Parviainen and J.D. Rossi. On the definition and properties of p-harmonious functions. Annali della Scuola Normale Superiore di Pisa, Clase di Scienze. Vol. XI(2) (2012), 215–241.

    MathSciNet  Google Scholar 

  14. J.J. Manfredi, M. Parviainen and J.D. Rossi. An asymptotic mean value characterization for p-harmonic functions. Procc. American Mathematical Society. 138 (2010), 881–889.

    Article  MathSciNet  MATH  Google Scholar 

  15. J.J. Manfredi, A. Oberman and A. Sviridov. Nonlinear elliptic PDEs on graphs. Preprint.

  16. O. Martio. Potential theoretic aspects of nonlinear elliptic partial differential equations. Bericht [Report], 44. Universität Jyväskylä, Mathematisches Institut, Jyväskylä, (1989), 23 pp.

  17. Y. Peres, O. Schramm, S. Sheffield and D. Wilson. Tug-of-war and the infinity Laplacian. Selected works of Oded Schramm. Volume 1, 2, 595–638, Sel. Works Probab. Stat., Springer, New York (2011).

    Article  MathSciNet  Google Scholar 

  18. Y. Peres and S. Sheffield. Tug-of-war with noise: a game theoretic view of the p-Laplacian. Duke Math. J., 145(1) (2008), 91–120.

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Oberman. Finite Difference Methods for the infinity Laplace and p-Laplace equations. J. Comput. Appl. Math., 254 (2013), 65–80.

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Oberman. A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions. Math. Comp., 74 (2005) 251, 1217–1230.

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Rudd and H.A. Van Dyke. Median values, 1-harmonic functions, and functions of least gradient. Commun. Pure Appl. Anal., 12(2) (2013), 711–719.

    Article  MathSciNet  MATH  Google Scholar 

  22. A.P. Sviridov. Elliptic equations in graphs via stochastic games. Thesis (Ph.D.) University of Pittsburgh. ProQuest LLC, Ann Arbor, MI, (2011), 53 pp.

    Google Scholar 

  23. A.P. Sviridov. p-harmonious functions with drift on graphs via games. Electron. J. Differential Equations (2011), No. 114, 11 pp.

    MathSciNet  Google Scholar 

  24. T.H. Wolff. Gap series constructions for the p-Laplacian. Paper completed by John Garnett and Jang-Mei Wu. J. Anal. Math., 102 (2007), 371–394.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CONICET and Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Pabellon I, Ciudad Universitaria (1428), Buenos Aires, Argentina

    Leandro M. Del Pezzo & Carolina A. Mosquera

  2. Departamento de Análisis Matemático, Universidad de Alicante, Ap. correo 99, 03080, Alicante, Spain

    Julio D. Rossi

Authors
  1. Leandro M. Del Pezzo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carolina A. Mosquera
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Julio D. Rossi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leandro M. Del Pezzo.

Additional information

Leandro M. Del Pezzo was partially supported by UBACyT 20020110300067 and CONICET PIP 5478/1438 (Argentina).

Carolina A. Mosquera was partially supported by UBACyT 20020100100638, PICT 0436 and CONICET PIP 112 200201 00398 (Argentina).

Julio D. Rossi was partially supported by MTM2011-27998 (Spain).

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Del Pezzo, L.M., Mosquera, C.A. & Rossi, J.D. Estimates for nonlinear harmonic measures on trees. Bull Braz Math Soc, New Series 45, 405–432 (2014). https://doi.org/10.1007/s00574-014-0056-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00574-014-0056-8

Keywords

Mathematical subject classification

Search

Navigation