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

Abstract

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).