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

Fengwen Han, Tao Wang

TL;DR

The paper studies the discrete infinity Laplacian $\Delta_{\infty}$ on infinite graphs and unbounded subgraphs, addressing the Dirichlet problem with boundary data on $\delta U$ and establishing existence and uniqueness of sublinear solutions for the homogeneous case $f=0$. It develops a graph-based framework, including subharmonic characterizations and an exhaustion method, to derive existence and comparison principles that extend to unbounded Euclidean domains for $-\Delta_{\infty}u=0$ with Lipschitz boundary data. For inhomogeneous equations on trees with bounded boundary, it provides a sublinear-uniqueness result for $f\ge 0$, a downward-path representation formula, and a necessary (but not sufficient) condition for existence; it further reduces the general tree case to bounded-width graphs using previous results. Overall, the work connects discrete tug-of-war perspectives and Perron-type methods to infinity Laplace equations on unbounded domains, broadening the link between graph-based models and continuous viscosity theory.

Abstract

We study the Dirichlet problem of the following discrete infinity Laplace equation on unbounded subgraphs \begin{equation*} Δ_{\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.

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

TL;DR

The paper studies the discrete infinity Laplacian on infinite graphs and unbounded subgraphs, addressing the Dirichlet problem with boundary data on and establishing existence and uniqueness of sublinear solutions for the homogeneous case . It develops a graph-based framework, including subharmonic characterizations and an exhaustion method, to derive existence and comparison principles that extend to unbounded Euclidean domains for with Lipschitz boundary data. For inhomogeneous equations on trees with bounded boundary, it provides a sublinear-uniqueness result for , a downward-path representation formula, and a necessary (but not sufficient) condition for existence; it further reduces the general tree case to bounded-width graphs using previous results. Overall, the work connects discrete tug-of-war perspectives and Perron-type methods to infinity Laplace equations on unbounded domains, broadening the link between graph-based models and continuous viscosity theory.

Abstract

We study the Dirichlet problem of the following discrete infinity Laplace equation on unbounded subgraphs \begin{equation*} Δ_{\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 (), 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 on trees.

Paper Structure

This paper contains 4 sections, 13 theorems, 100 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. For homogeneous equations
  4. Inhomogeneous equations on trees

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^N$ be bounded and $\partial \Omega$ be bounded. Let $u, v \in C(\overline{\Omega})$, and $\Delta_{\infty}u \geq 0$, $\Delta_{\infty}v \leq 0$ in $\Omega$. Assume also that and Then for $x \in \Omega$,

Figures (3)

  • Figure 1: Infinity subharmonic is not equivalent to the condition \ref{['equ:counterexamplecondition']}.
  • Figure 2: The definition of $O_{\varepsilon}$ cannot be changed to $\{x \in U: L(u, x) < \varepsilon\}$.
  • Figure 3: A function satisfying \ref{['equ:sublinearsolution']} is not necessarily sublinear.

Theorems & Definitions (29)

  • Theorem 1.1: crandall2007uniqueness
  • Theorem 1.2: han2025discrete
  • Theorem 1.3: han2025discrete
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Proposition 2.1
  • ...and 19 more