New universal inequalities for eigenvalues of elliptic operators in weighted divergence form on smooth metric measure spaces
Abstract In this paper, we investigate the eigenvalue problem of elliptic operators in weighted divergence form on smooth metric measure spaces. Firstly, we give a general inequality for eigenvalues of the elliptic operators in weighted divergence form on a compact smooth metric measure space with b...
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SpringerOpen
2025-06-01
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| Series: | Journal of Inequalities and Applications |
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| Online Access: | https://doi.org/10.1186/s13660-025-03328-0 |
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| author | Yanli Li Feng Du |
| author_facet | Yanli Li Feng Du |
| author_sort | Yanli Li |
| collection | DOAJ |
| description | Abstract In this paper, we investigate the eigenvalue problem of elliptic operators in weighted divergence form on smooth metric measure spaces. Firstly, we give a general inequality for eigenvalues of the elliptic operators in weighted divergence form on a compact smooth metric measure space with boundary (possibly empty). Then, applying this general inequality, we get some new universal inequalities for the eigenvalues of fourth-order elliptic operators in weighted divergence form on smooth metric measure spaces. Also, using these general inequalities and three generalized Cheeger–Gromoll splitting theorems, we give some new universal inequalities for the eigenvalues of vibration problem for a clamped plate on the smooth metric measure spaces that satisfy some curvature conditions. Moreover, our result can reveal the relationship between the ( k + 1 ) $(k + 1)$ -th eigenvalue and the first k eigenvalues in a relatively quick way. |
| format | Article |
| id | doaj-art-9c14d74152d74e5bb89f844ab4c0979a |
| institution | Kabale University |
| issn | 1029-242X |
| language | English |
| publishDate | 2025-06-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj-art-9c14d74152d74e5bb89f844ab4c0979a2025-08-20T03:45:33ZengSpringerOpenJournal of Inequalities and Applications1029-242X2025-06-012025111910.1186/s13660-025-03328-0New universal inequalities for eigenvalues of elliptic operators in weighted divergence form on smooth metric measure spacesYanli Li0Feng Du1School of New Energy, Jingchu University of TechnologySchool of Mathematics and Physics, Jingchu University of TechnologyAbstract In this paper, we investigate the eigenvalue problem of elliptic operators in weighted divergence form on smooth metric measure spaces. Firstly, we give a general inequality for eigenvalues of the elliptic operators in weighted divergence form on a compact smooth metric measure space with boundary (possibly empty). Then, applying this general inequality, we get some new universal inequalities for the eigenvalues of fourth-order elliptic operators in weighted divergence form on smooth metric measure spaces. Also, using these general inequalities and three generalized Cheeger–Gromoll splitting theorems, we give some new universal inequalities for the eigenvalues of vibration problem for a clamped plate on the smooth metric measure spaces that satisfy some curvature conditions. Moreover, our result can reveal the relationship between the ( k + 1 ) $(k + 1)$ -th eigenvalue and the first k eigenvalues in a relatively quick way.https://doi.org/10.1186/s13660-025-03328-0EigenvalueUniversal inequalitiesDrifting LaplacianElliptic operators in weighted divergence formSmooth metric measure spaceGeneralized Cheeger |
| spellingShingle | Yanli Li Feng Du New universal inequalities for eigenvalues of elliptic operators in weighted divergence form on smooth metric measure spaces Journal of Inequalities and Applications Eigenvalue Universal inequalities Drifting Laplacian Elliptic operators in weighted divergence form Smooth metric measure space Generalized Cheeger |
| title | New universal inequalities for eigenvalues of elliptic operators in weighted divergence form on smooth metric measure spaces |
| title_full | New universal inequalities for eigenvalues of elliptic operators in weighted divergence form on smooth metric measure spaces |
| title_fullStr | New universal inequalities for eigenvalues of elliptic operators in weighted divergence form on smooth metric measure spaces |
| title_full_unstemmed | New universal inequalities for eigenvalues of elliptic operators in weighted divergence form on smooth metric measure spaces |
| title_short | New universal inequalities for eigenvalues of elliptic operators in weighted divergence form on smooth metric measure spaces |
| title_sort | new universal inequalities for eigenvalues of elliptic operators in weighted divergence form on smooth metric measure spaces |
| topic | Eigenvalue Universal inequalities Drifting Laplacian Elliptic operators in weighted divergence form Smooth metric measure space Generalized Cheeger |
| url | https://doi.org/10.1186/s13660-025-03328-0 |
| work_keys_str_mv | AT yanlili newuniversalinequalitiesforeigenvaluesofellipticoperatorsinweighteddivergenceformonsmoothmetricmeasurespaces AT fengdu newuniversalinequalitiesforeigenvaluesofellipticoperatorsinweighteddivergenceformonsmoothmetricmeasurespaces |