Lower order eigenvalues of Dirichlet Laplacian. (English) Zbl 1137.35050
Summary: 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.
MSC:
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 58C40 | Spectral theory; eigenvalue problems on manifolds |
Keywords:
eigenvalue problem; Riemannian manifold; projective space; Gram-Schmidt orthogonalization; trial functionsReferences:
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