On the strong maximum principle for nonlocal operators

TL;DR

This paper establishes a strong maximum principle for weak supersolutions of nonlocal equations involving operators with nonnegative kernels, extending classical results to anisotropic and regional variants, and considering variable kernels.

Contribution

It introduces minimal positivity conditions on kernels for nonlocal operators, including anisotropic and regional variants, broadening the applicability of maximum principles.

Findings

Proves a strong maximum principle for a broad class of nonlocal operators.

Extends results to regional and variable kernel operators.

Includes highly anisotropic variants of the fractional Laplacian.

Abstract

In this paper we derive a strong maximum principle for weak supersolutions of nonlocal equations of the form $Iu=c(x) u$ in $\Omega$, where $\Omega\subset \mathbb{R}^N$ is a domain, $c\in L^{\infty}(\Omega)$ and $I$ is an operator of the form $Iu(x)=P.V.\int_{\mathbb{R}^N}(u(x)-u(y))j(x-y)\ dy$ with a nonnegative kernel function $j$. We formulate minimal positivity assumptions on $j$ corresponding to a class of operators which includes highly anisotropic variants of the fractional Laplacian. Somewhat surprisingly, this problem leads to the study of general lattices in $\mathbb{R}^N$. Our results extend to the regional variant of the operator $I$ and, under weak additional assumptions, also to the case of $x$-dependent kernel functions.