Discontinuous gradient constraints and the infinity Laplacian

TL;DR

This paper investigates a gradient constraint problem involving the infinity Laplacian, establishing existence and uniqueness under certain conditions, and exploring discrepancies between solutions from game theory and $L^p$-approximations.

Contribution

It introduces a new gradient constraint problem with the infinity Laplacian and analyzes solution existence, uniqueness, and differences arising from various approximation methods.

Findings

Solutions exist under a regularity condition.

Uniqueness is guaranteed when the regularity condition holds.

Discrepancies occur between game-theoretic and $L^p$-approximate solutions when the condition fails.

Abstract

Motivated by tug-of-war games and asymptotic analysis of certain variational problems, we consider a gradient constraint problem involving the infinity Laplace operator. We prove that this problem always has a solution that is unique if a certain regularity condition on the constraint is satisfied. If this regularity condition fails, then solutions obtained from game theory and $L^p$-approximation need not coincide.