An obstacle problem for Tug-of-War games

TL;DR

This paper studies the obstacle problem for the infinity Laplace equation, establishing existence, uniqueness, and a game-theoretic interpretation through obstacle tug-of-war.

Contribution

It introduces a new obstacle tug-of-war game and proves the convergence of its value functions to the solution of the obstacle problem.

Findings

Existence and uniqueness of the super infinity-harmonic function above the obstacle.

The super infinity-harmonic function is the limit of obstacle tug-of-war game value functions.

The function is infinity harmonic where it lies above the obstacle.

Abstract

We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinity-harmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above the obstacle. Moreover, we show that this function is the limit of value functions of a game we call obstacle tug-of-war.