Singular measure as principal eigenfunction of some nonlocal operators

TL;DR

This paper investigates the spectral properties of nonlocal operators, proving the existence of solutions involving singular measures and principal eigenfunctions, expanding understanding of eigenvalue problems in nonlocal PDEs.

Contribution

It establishes the existence of solutions in the space of signed measures for the principal eigenvalue problem of certain nonlocal operators, including explicit examples of singular measures.

Findings

Existence of solutions in the space of signed measures for the spectral problem.

Identification of principal eigenfunctions as absolutely continuous measures in some cases.

Explicit examples of singular measures solving the spectral problem.

Abstract

In this paper, we are interested in the spectral properties of the generalised principal eigenvalue of some nonlocal operator. That is, we look for the existence of some particular solution $(\lambda,\phi)$ of a nonlocal operator. $$\int_{\O}K(x,y)\phi(y)\, dy +a(x)\phi(x) =-\lambda \phi(x),$$ where $\O\subset\R^n$ is an open bounded connected set, $K$ a nonnegative kernel and $a$ is continuous. We prove that for the generalised principal eigenvalue $\lambda_p:=\sup \{\lambda \in \R \, |\, \exists \, \phi \in C(\O), \phi > 0 \;\text{so that}\; \oplb{\phi}{\O}+ a(x)\phi + \lambda\phi\le 0\}$ there exists always a solution $(\mu, \lambda_p)$ of the problem in the space of signed measure. Moreover $\mu$ a positive measure. When $\mu$ is absolutely continuous with respect to the Lebesgue measure, $\mu =\phi_p(x)$ is called the principal eigenfunction associated to $\lambda_p$. In some simple cases, we exhibit some explicit singular measures that are solutions of the spectral problem.