Abstract
In this paper, we study eigenvalues of elliptic operators in divergence form on compact Riemannian manifolds with boundary (possibly empty) and obtain a general inequality for them. By using this inequality, we prove universal inequalities for eigenvalues of elliptic operators in divergence form on compact domains of complete submanifolds in a Euclidean space, and of complete manifolds admitting special functions which include the Hadamard manifolds with Ricci curvature bounded below, a class of warped product manifolds, the product of Euclidean spaces with any complete manifold and manifolds admitting eigenmaps to a sphere.
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M. P. do Carmo was partially supported by FAPERJ; Q. Wang was partially supported by CNPq; C. Xia was partially supported by CNPq and FAPDF.
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do Carmo, M.P., Wang, Q. & Xia, C. Inequalities for eigenvalues of elliptic operators in divergence form on Riemannian manifolds. Annali di Matematica 189, 643–660 (2010). https://doi.org/10.1007/s10231-010-0129-2
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DOI: https://doi.org/10.1007/s10231-010-0129-2
Keywords
- Universal bounds
- Eigenvalues
- Elliptic operator
- Payne-Pólya-Weinberger-Yang type inequalities
- Submanifolds
- Hypersurfaces in space forms
- Warped manifolds