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

Abstract

The Dirichlet problem $$ \begin{cases} Δ_{\infty}u-|Du|^2=0 \quad \text{on $Ω\subset \Rset ^n$} u|_{\partial Ω}=g \end{cases} $$ might have many solutions, where $\hat{I}{”}_{\infty }u={\sum }_{1\le i,j\le n}{u}_{x_i}{u}_{x_j}{u}_{x_i{x}_j}$. 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 $L^{\infty }$ 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].

To appear in "Cal. Var. Partial Differential Equations."