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].
Categories: math.AP
