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arXiv · 2602.18035

Nonlocal eigenvalue problems and superposition operators

Serena Dipierro·Edoardo Proietti Lippi·Caterina Sportelli·Enrico Valdinoci
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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.

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Serena Dipierro, Edoardo Proietti Lippi, Caterina Sportelli, Enrico Valdinoci. 2026-02-20. Nonlocal eigenvalue problems and superposition operators. https://arxiv.org/abs/2602.18035

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