Nonlocal eigenvalue problems and superposition operators
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910027945607168 |
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| author | Dipierro, Serena Lippi, Edoardo Proietti Sportelli, Caterina Valdinoci, Enrico |
| author_facet | Dipierro, Serena Lippi, Edoardo Proietti Sportelli, Caterina Valdinoci, Enrico |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18035 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nonlocal eigenvalue problems and superposition operators Dipierro, Serena Lippi, Edoardo Proietti Sportelli, Caterina Valdinoci, Enrico Analysis of PDEs 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. |
| title | Nonlocal eigenvalue problems and superposition operators |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.18035 |