On the comparison principle for a nonlocal infinity Laplacian

Technical Summary: On the Comparison Principle for a Nonlocal Infinity Laplacian

Problem Statement
This work addresses the uniqueness of viscosity solutions to the nonlocal Poisson equation involving the nonlocal infinity Laplacian operator, defined as:
$$L_\infty u = f \quad \text{in } \Omega,$$
where $\Omega \subset \mathbb{R}^n$ is a bounded domain, and $f \in C(\Omega)$ is a bounded function satisfying $f \leq 0$. The operator $L_\infty$ is defined for $\alpha \in (0, 1]$ as:
$$L_\infty u(x) := \sup_{y \in \mathbb{R}^n} \frac{u(y) - u(x)}{|x - y|^\alpha} + \inf_{y \in \mathbb{R}^n} \frac{u(y) - u(x)}{|x - y|^\alpha}.$$
The study focuses on the case where the right-hand side $f$ does not change sign (specifically $f \leq 0$). While existence and uniqueness for the homogeneous case ($f=0$) and the inhomogeneous case with strictly positive/negative $f$ in the local setting have been established in prior literature, the nonlocal case with a sign-preserving $f$ presents specific challenges regarding the comparison principle. The problem is posed in the viscosity sense due to the non-divergence form of the operator, which precludes a natural weak formulation.

Methodology
The primary methodology relies on establishing a comparison principle between viscosity supersolutions and subsolutions. The proof strategy involves several technical components:

1. Viscosity Solution Framework: The paper utilizes the standard definition of viscosity solutions adapted for nonlocal operators, employing test functions that touch the solution from above or below. The solutions are assumed to be bounded on $\mathbb{R}^n \setminus \Omega$ and to possess limits at infinity.
2. Strong Maximum Principle Variant: A key preliminary result (Lemma 2.1) establishes that if a viscosity supersolution attains its minimum within the domain $\Omega$ (under specific boundary conditions), the function must be constant. This serves as a nonlocal analogue to the strong maximum principle.
3. Localization of the Infimum: A critical technical contribution is Lemma 2.2, which demonstrates that for a bounded viscosity supersolution $u$ to $L_\infty u = f$ with $f \leq 0$, the infimum component of the operator, $L^-_\infty u(x)$, is attained strictly outside the domain $\Omega$ (i.e., in $\mathbb{R}^n \setminus \Omega$). This property is derived using a barrier function argument similar to Lemma 26 in Lindgren and Lindqvist [20].
4. Infimal Convolution: To handle the lack of smoothness in viscosity solutions, the author employs the infimal convolution technique (Definition 2.2). It is shown (Lemma 2.3) that the infimal convolution $u_\epsilon$ of a viscosity supersolution is a supersolution in the classical sense within a slightly smaller subdomain $\Omega_\epsilon$. This regularization allows for the application of classical calculus arguments.
5. Strict Supersolution Construction: The proof of the comparison principle (Theorem 3.1) proceeds by contradiction. Assuming the existence of a subsolution $v$ and supersolution $u$ such that $v > u$ somewhere, the author constructs a strict supersolution $\tilde{\phi}_\epsilon$ by modifying the infimal convolution of $u$. This modification utilizes a cut-off function and relies on the fact that the infimum of the operator is attained outside $\Omega$. The construction ensures that $L_\infty \tilde{\phi}_\epsilon < f$ at the point of maximum difference, leading to a contradiction with the definition of $v$ as a subsolution.

Key Contributions and Results

• Comparison Principle (Theorem 3.1): The main result proves that if $u$ is a viscosity supersolution and $v$ is a viscosity subsolution to $L_\infty u = f$ with $f \leq 0$, and if $u \geq v$ on $\mathbb{R}^n \setminus \Omega$ (with appropriate behavior at infinity), then $u \geq v$ throughout $\Omega$.
• Uniqueness (Corollary 3.2): As a direct consequence of the comparison principle, the paper establishes the uniqueness of viscosity solutions to the nonlocal infinity Laplacian equation when the right-hand side is non-positive and bounded.
• Technical Distinction: The paper distinguishes itself from previous works (such as Chambolle et al. [8]) by working in the full space $\mathbb{R}^n$ rather than a bounded domain with explicit boundary data, and by utilizing infimal convolution rather than mollification. Furthermore, it addresses the case where $f$ is not identically zero, implying that the supremum component of the operator is not necessarily attained outside $\Omega$, requiring a more nuanced analysis of the infimum component.

Significance and Scope
The paper claims to resolve the uniqueness question for the nonlocal infinity Laplacian with a sign-preserving source term, a case that remains open for the local inhomogeneous infinity Laplacian when $f \geq 0$. The results are derived strictly within the framework of viscosity solutions and do not address the existence of solutions. The author notes that the methods used for $f \leq 0$ can be adapted for $f \geq 0$. The work relies on the specific structure of the nonlocal operator and the properties of viscosity supersolutions, particularly the behavior of the infimum term relative to the domain boundary. The research is supported by VR Grant 2016-03639 and acknowledges contributions from Erik Lindgren and Mikko Parviainen regarding the problem formulation and potential generalizations.