Generalized principal eigenvalues on $\mathbb{R}^d$ of second order elliptic operators with rough nonlocal kernels

TL;DR

This paper investigates the existence, properties, and implications of generalized principal eigenvalues for a broad class of nonlocal elliptic operators on , including singular kernels, with applications to maximum principles.

Contribution

It provides new conditions for the existence and uniqueness of principal eigenfunctions and characterizes the simplicity of the principal eigenvalue for nonlocal elliptic operators.

Findings

Existence of principal eigenfunctions under broad conditions

Necessary and sufficient conditions for maximum principles

Characterization of eigenvalue simplicity

Abstract

We study the generalized eigenvalue problem on the whole space for a class of integro-differential elliptic operators. The nonlocal operator is over a finite measure, but this has no particular structure. Some of our results even hold for singular kernels. The first part of the paper presents results concerning the existence of a principal eigenfunction. Then we present various necessary and/or sufficient conditions for the maximum principle to hold, and use these to characterize the simplicity of the principal eigenvalue.