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Systems involving mean value formulas on trees

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

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: [email protected] [email protected]
Carolina A. Mosquera
Affiliation:
Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina e-mail: [email protected] [email protected]
Julio D. Rossi*
Affiliation:
Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina e-mail: [email protected] [email protected]
*
e-mail: [email protected]
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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 , Volume 77 , Issue 2 , April 2025 , pp. 351 - 383
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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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).

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