A mixed problem for the infinity laplacian via Tug-of-War games

Abstract

In this paper we prove that a function $u\in \mathcal{C}(\bar{\Omega })$ is the continuous value of the Tug-of-War game described in \cite{PSSW} if and only if it is the unique viscosity solution to the infinity laplacian with mixed boundary conditions {-Δ_{\infty}u(x)=0\quad & \text{in} Ω, \frac{\partial u}{\partial n}(x)=0\quad & \text{on} Γ_N, u(x)=F(x)\quad & \text{on} Γ_D. By using the results in \cite{PSSW}, it follows that this viscous PDE problem has a unique solution, which is the unique {\it absolutely minimizing Lipschitz extension} to the whole $\bar{\Omega }$ (in the sense of \cite{Aronsson} and \cite{PSSW}) of the boundary data $F:{\Gamma }_D\to \mathbb{R}$.

13 pages. Final version