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The Dirichlet problem     $$ \left\{ \begin{array}{l}\Delta _\infty u - |Du|^2 = 0 \quad {\rm on} \, \Omega \subset {{\mathbb R}^n} \\ u|\partial \Omega = g \\\end{array} \right. $$    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 $${H(p,z)={\frac{1}{2}}|p|^2-z}$$ from calculus of variations in L ∞ 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 (Commun Partial Differ Equ 26(11–12):2323–2337, 2001).

Maximal and minimal solutions of an Aronsson equation: L ∞ variational problems versus the game theory