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

TL;DR

This paper investigates the maximal and minimal solutions of a specific Aronsson equation, linking variational $L^{ abla}$ problems to game theory, and characterizes solutions that are neither minimizers nor game values.

Contribution

It establishes the maximal solution as the unique absolute minimizer and the minimal as the game-theoretic value, also characterizing intermediate solutions.

Findings

Maximal solution is the unique absolute minimizer.

Minimal solution corresponds to the game value from tug-of-war.

Characterization of solutions that are neither minimizers nor game values.

Abstract

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].