Abstract
In this paper, we investigate an eigenvalue problem for the Dirichlet Laplacian on a domain in an n-dimensional compact Riemannian manifold. First we give a general inequality for eigenvalues. As one of its applications, we study eigenvalues of the Laplacian on a domain in an n-dimensional complex projective space, on a compact complex submanifold in complex projective space and on the unit sphere. By making use of the orthogonalization of Gram–Schmidt (QR-factorization theorem), we construct trial functions. By means of these trial functions, estimates for lower order eigenvalues are obtained.
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Qing-Ming Cheng research was partially supported by a Grant-in-Aid for Scientific Research from JSPS.
Hejun Sun and Hongcang Yang research were partially supported by NSF of China.
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Sun, H., Cheng, QM. & Yang, H. Lower order eigenvalues of Dirichlet Laplacian. manuscripta math. 125, 139–156 (2008). https://doi.org/10.1007/s00229-007-0136-9
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DOI: https://doi.org/10.1007/s00229-007-0136-9