On the comparison principle for a nonlocal infinity Laplacian

TL;DR

This work addresses the uniqueness of viscosity solutions to the nonlocal infinity Laplacian equation $\mathcal{L}_{\infty} u = f$ in a bounded domain $\Omega$ with a sign-definite right-hand side $f \le 0$. It develops a viscosity-solution framework for the nonlocal, non-divergence operator, decomposes $\mathcal{L}_{\infty} u$ into $\mathcal{L}_{\infty}^- u$ and $\mathcal{L}_{\infty}^+ u$, and proves a comparison principle for $0<\alpha<1$ using infimal convolution to regularize supersolutions. A key technical ingredient is that the infimum in $\mathcal{L}_{\infty}^- u(x)$ is attained outside $\Omega$, enabling a strict supersolution construction and a robust uniqueness result for a Dirichlet-type problem $\max\{\mathcal{L}_{\infty} u, \mathcal{L}_{\infty}^- u + \chi_D\}=0$ with exterior data; the paper also provides a corollary guaranteeing uniqueness for $\mathcal{L}_{\infty} u = f$ under exterior limits. The work advances nonlocal, nondivergence operator theory and supplies techniques (infimal convolution, exterior attainment) relevant to uniqueness in nonlocal fully nonlinear equations.

Abstract

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

Theorems & Definitions (11)

• Definition 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Definition 2.2
• Lemma 2.3
• proof
• Theorem 3.1
• proof : proof of Theorem \ref{['thm:comparison']}