A game theoretical approximation for solutions to nonlinear systems with obstacle-type equations

Abstract

In this paper we find viscosity solutions to a coupled system composed by two equations, the first one is an obstacle type equation, min{−1 ∞u(x), (u − v)(x)} = 0, and the second one is −v(x) + v(x) − u(x) = h(x) in a smooth bounded domain with Dirichlet boundary conditions u(x) = f (x), v(x) = g(x) for x ∈ ∂. Here −1 ∞u is the ∞−Laplacian and −v is the standard Laplacian. This system is not variational and involves two different elliptic operators. Notice that in the first equation the obstacle is given by the second component of the system that also depends on the first component via the second equation (this system is fully coupled). 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 one of the players decides to play a round of a Tug-of-War game or to change boards and in the second board we play a random walk with the possibility of changing boards with a positive (but small) probability and a running payoff. We show that this game has two value functions (one for each board) that converge uniformly to the components of a viscosity solution to the PDE system.