Some existence and uniqueness results for infinity Laplace equations on infinite graphs

TL;DR

This paper investigates the existence and uniqueness of solutions to the infinity Laplace equation on infinite graphs and unbounded domains, establishing key results for both homogeneous and nonhomogeneous cases.

Contribution

It provides new existence and uniqueness results for the infinity Laplace equation on infinite graphs and unbounded Euclidean domains, including sublinear solutions.

Findings

Existence and uniqueness of sublinear solutions for homogeneous infinity Laplace equations.

Extension of results to unbounded Euclidean domains.

Uniqueness for cases where f ≥ 0 on trees.

Abstract

We study the Dirichlet problem of the following discrete infinity Laplace equation on unbounded subgraphs \begin{equation*} \Delta_{\infty}u(x):=\inf_{y\sim x}u(y)+\sup_{y\sim x}u(y)-2u(x)=f(x). \end{equation*} For the homogeneous case ($f=0$), the existence and uniqueness of sublinear solutions are established. This result is applied to prove the existence and uniqueness of sublinear solutions for the homogeneous (normalized) infinity Laplace equations on unbounded Euclidean domains. Uniqueness is also shown for the case $f \geq 0$ on trees.