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

Abstract

In this paper we prove that a function ${ u\in\mathcal{C}(\overline{\Omega})}$ is the continuous value of the Tug-of-War game described in Y. Peres et al. (J. Am. Math. Soc., 2008, to appear) if and only if it is the unique viscosity solution to the infinity Laplacian with mixed boundary conditions

$$\left\{ \begin{aligned}-\Delta_{\infty}u(x)=0 \quad & {\rm in} \, \Omega,\\ \frac{\partial u}{\partial n}(x)=0 \quad \quad & {\rm on} \, \Gamma_N,\\ u(x)=F(x) \quad & {\rm on}\, \Gamma_D. \end{aligned} \right.$$

By using the results in Y. Peres et al. (J. Am. Math. Soc., 2008, to appear), it follows that this viscous PDE problem has a unique solution, which is the unique absolutely minimizing Lipschitz extension to the whole ${\overline{\Omega}}$ (in the sense of Aronsson (Ark. Mat. 6:551–561, 1967) and Y. Peres et al. (J. Am. Math. Soc., 2008, to appear)) of the Lipschitz boundary data ${F:\Gamma_D \to \mathbb R }$.