Nonlocal eigenvalue problems and superposition operators
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.
Cite
@article{arxiv.2602.18035,
title = {Nonlocal eigenvalue problems and superposition operators},
author = {Serena Dipierro and Edoardo Proietti Lippi and Caterina Sportelli and Enrico Valdinoci},
journal= {arXiv preprint arXiv:2602.18035},
year = {2026}
}