A game theoretical approximation for a parabolic/elliptic system with different operators

Alfredo Miranda; Julio D. Rossi

doi:10.3934/dcds.2022034

Article Contents

2023, Volume 43, Issue 3&4: 1625-1656. Doi: 10.3934/dcds.2022034

This issue Previous Article Radon measure-valued solutions of unsteady filtration equations Next Article Hölder continuity of solutions for a class of drift-diffusion equations

A game theoretical approximation for a parabolic/elliptic system with different operators

Alfredo Miranda and
Julio D. Rossi^,

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

^*Corresponding author: Julio D. Rossi
^*Corresponding author: Julio D. Rossi

To Juan Luis Vazquez in his 75th anniversary with our best wishes.

Received: June 2021

Revised: February 2022

Early access: March 2022

Published: March 2023

partially supported by CONICET grant PIP GI No 11220150100036CO (Argentina), PICT-2018-03183 (Argentina) and UBACyT grant 20020160100155BA (Argentina).

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Abstract

In this paper we find viscosity solutions to a coupled system composed by two equations, the first one is parabolic and driven by the infinity Laplacian while the second one is elliptic and involves the usual Laplacian. We prove that there is a two-player zero-sum game played in two different boards with different rules in each board (in the first one we play a Tug-of-War game taking the number of plays into consideration and in the second board we move at random) whose value functions converge uniformly to a viscosity solution to the PDE system.

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Mathematics Subject Classification: Primary: 35D40, 35M13, 91A05.

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