Hostname: page-component-7bb8b95d7b-5mhkq Total loading time: 0 Render date: 2024-09-18T05:09:37.911Z Has data issue: false hasContentIssue false
  • English
  • Français

Systems involving mean value formulas on trees

Part of: Elliptic equations and systems Low-dimensional dynamical systems Potential theory on metric spaces Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  03 January 2024

Alfredo Miranda ,
Carolina A. Mosquera  and
Julio D. Rossi Open the ORCID record for Julio D. Rossi [Opens in a new window]
Alfredo Miranda
Affiliation:
Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina e-mail: amiranda@dm.uba.ar mosquera@dm.uba.ar
Carolina A. Mosquera
Affiliation:
Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina e-mail: amiranda@dm.uba.ar mosquera@dm.uba.ar
Julio D. Rossi*
Affiliation:
Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina e-mail: amiranda@dm.uba.ar mosquera@dm.uba.ar
*
e-mail: jrossi@dm.uba.ar
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study the Dirichlet problem for systems of mean value equations on a regular tree. We deal both with the directed case (the equations verified by the components of the system at a node in the tree only involve values of the unknowns at the successors of the node in the tree) and the undirected case (now the equations also involve the predecessor in the tree). We find necessary and sufficient conditions on the coefficients in order to have existence and uniqueness of solutions for continuous boundary data. In a particular case, we also include an interpretation of such solutions as a limit of value functions of suitable two-players zero-sum games.

Keywords

Regular treessystemsmean value formula

MSC classification

Primary: 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
Secondary: 35R30: Inverse problems 31E05: Potential theory on metric spaces 37E25: Maps of trees and graphs
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 33
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

C. Mosquera is partially supported by grants UBACyT 20020170100430BA (Argentina), PICT 2018–03399 (Argentina), and PICT 2018–04027 (Argentina). A. Miranda and J. Rossi are partially supported by grants CONICET grant PIP GI No. 11220150100036CO (Argentina), PICT-2018-03183 (Argentina), and UBACyT grant 20020160100155BA (Argentina).

References

Alvarez, V., Rodríguez, J. M., and Yakubovich, D. V., Estimates for nonlinear harmonic “measures” on trees . Michigan Math. J. 49(2001), no. 1, 47–64.Google Scholar
Anandam, V., Harmonic functions and potentials on finite or infinite networks, Lecture Notes of the Unione Matematica Italiana, 12, Springer, Heidelberg; UMI, Bologna, 2011.Google Scholar
Bjorn, A., Bjorn, J., Gill, J. T., and Shanmugalingam, N., Geometric analysis on Cantor sets and trees . J. Reine Angew. Math. 725(2017), 63–114.CrossRefGoogle Scholar
Blanc, P. and Rossi, J. D., Game theory and partial differential equations, De Gruyter Series in Nonlinear Analysis and Applications, 31, Walter de Gruyter GmbH & Co KG, Berlin, 2019.Google Scholar
Del Pezzo, L. M., Mosquera, C. A., and Rossi, J. D., The unique continuation property for a nonlinear equation on trees . J. Lond. Math. Soc. (2) 89(2014), no. 2, 364–382.CrossRefGoogle Scholar
Del Pezzo, L. M., Mosquera, C. A., and Rossi, J. D., Estimates for nonlinear harmonic measures on trees . Bull. Braz. Math. Soc. (N.S.) 45(2014), no. 3, 405–432.Google Scholar
Del Pezzo, L. M., Mosquera, C. A., and Rossi, J.D., Existence, uniqueness and decay rates for evolution equations on trees . Port. Math. 71(2014), no. 1, 63–77.Google Scholar
Hartenstine, D. and Rudd, M., Asymptotic statistical characterizations of $p$ -harmonic functions of two variables . Rocky Mountain J. Math. 41(2011), no. 2, 493–504.Google Scholar
Hartenstine, D. and Rudd, M., Statistical functional equations and $p$ -harmonious functions . Adv. Nonlinear Stud. 13(2013), no. 1, 191–207.Google Scholar
Kaufman, R., Llorente, J. G., and Wu, J.-M., Nonlinear harmonic measures on trees . Ann. Acad. Sci. Fenn. Math. 28(2003), no. 2, 279–302.Google Scholar
Kaufman, R. and Wu, J.-M., Fatou theorem of $p$ -harmonic functions on trees . Ann. Probab. 28(2000), no. 3, 1138–1148.CrossRefGoogle Scholar
Kesten, H., Relations between solutions to a discrete and continuous Dirichlet problem. In: Durrett, R. and Kesten, H. (eds.), Random walks, Brownian motion, and interacting particle systems, Progress in Probability, 28, Birkhauser, Boston, MA, 1991, pp. 309–321.CrossRefGoogle Scholar
Lewicka, M., A course on tug-of-war games with random noise: Introduction and basic constructions, Universitext Book Series, Springer, Cham, 2020.Google Scholar
Manfredi, J. J., Oberman, A., and Sviridov, A.. Nonlinear elliptic PDEs on graphs . Differential Integral Equations 28 (2015), no. 1-2, 79–102.Google Scholar
Manfredi, J. J., Parviainen, M., and Rossi, J. D., An asymptotic mean value characterization for $p$ -harmonic functions . Proc. Amer. Math. Soc. 138(2010), 881–889.Google Scholar
Oberman, A., A convergent difference scheme for the infinity Laplacian: Construction of absolutely minimizing Lipschitz extensions . Math. Comp. 74(2005), no. 251, 1217–1230.Google Scholar
Oberman, A., Finite difference methods for the infinity Laplace and $p$ -Laplace equations. J. Comput. Appl. Math. 254 (2013), 65–80.Google Scholar
Peres, Y., Schramm, O., Sheffield, S., and Wilson, D., Tug-of-war and the infinity Laplacian . J. Amer. Math. Soc. 22(2009), 167–210.CrossRefGoogle Scholar
Peres, Y. and Sheffield, S., Tug-of-war with noise: A game theoretic view of the $p$ -Laplacian .Duke Math. J. 145(2008), no. 1, 91–120.Google Scholar
Sviridov, A. P., Elliptic equations in graphs via stochastic games. Ph. D. thesis, University of Pittsburgh. ProQuest LLC, Ann Arbor, MI, 2011, 53 pp.Google Scholar
Sviridov, A. P., $p$ -harmonious functions with drift on graphs via games . Electron. J. Differential Equations 2011(2011), no. 114, 11.Google Scholar