Nonlocal eigenvalue problems and superposition operators

TL;DR

This paper investigates the spectral properties of mixed local and nonlocal operators, revealing how eigenvalues and eigenfunctions behave under domain disconnection, localization, and lower-order term variations, with implications for superposition operators.

Contribution

It provides new insights into eigenvalue problems involving mixed operators, including convergence results, sign-changing eigenfunctions on disconnected domains, and a comprehensive regularity theory.

Findings

Eigenfunctions associated with the first eigenvalue must change sign on disconnected domains.

The first eigenvalue of a union of disconnected domains is strictly smaller than that of individual components.

The regularity theory supports the analysis of eigenvalues and eigenfunctions.

Abstract

We study the spectral theory of mixed local and nonlocal operators with lower-order terms in the right-hand side of the equation. This kind of problems is motivated by the analysis of superposition operators of mixed order and with the "wrong sign" of the lower-order terms with respect to the classical elliptic theory. Our results include: -convergence to classical cases when the right-hand side of the eigenvalye equations "localizes", recovering the simplicity and sign-definiteness of eigenfunctions in the limit; -a detailed analysis of disconnected domains, showing that, unlike the classical case, any eigenfunction associated with the first eigenvalue must change sign, and that the first eigenvalue of a union of disconnected domains is strictly smaller than that of its individual components; -examples in which the first eigenvalue is either simple or non-simple in disconnected domains; -a regularity theory that underpins these results.