On the comparison principle for a nonlocal infinity Laplacian

TL;DR

This paper proves the uniqueness of viscosity solutions for a nonlocal infinity Laplace equation using a comparison principle, advancing the understanding of nonlocal PDEs in bounded domains.

Contribution

It establishes a comparison principle that guarantees the uniqueness of solutions for the nonlocal infinity Laplacian equation, a novel result in this area.

Findings

Uniqueness of viscosity solutions proven for the nonlocal infinity Laplacian.

Comparison principle established for the operator.

Results applicable to bounded domains with continuous functions.

Abstract

In this article, we prove the uniqueness of viscosity solutions to $\mathcal{L}_{\infty} u =f$ in $\Omega$, where $\mathcal{L}_{\infty}$ denotes the nonlocal infinity Laplace operator, $\Omega$ a bounded domain, and $f$ a continuous functions such that $f \leq 0$. Uniqueness is established through a comparison principle.