Maximal and minimal solutions of an Aronsson equation: $L^{\infty}$ variational problems versus the game theory

The Dirichlet problem $$\begin{cases} \Delta_{\infty}u-|Du|^2=0 \quad \text{on $\Omega\subset \Rset ^n$} u|_{\partial \Omega}=g \end{cases}$$ might have many solutions, where $\Delta_{\infty}u=\sum_{1\leq i,j\leq n}u_{x_i}u_{x_j}u_{x_ix_j}$. In this paper, we prove that the maximal solution is the unique absolute minimizer for $H(p,z)={1\over 2}|p|^2-z$ from calculus of variations in $L^{\infty}$ and the minimal solution is the continuum value function from the "tug-of-war" game. We will also characterize graphes of solutions which are neither an absolute minimizer nor a value function. A remaining interesting question is how to interpret those intermediate solutions. Most of our approaches are based on an idea of Barles-Busca [BB].