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Extrinsic estimates for eigenvalues of the Laplace operator. (English) Zbl 1147.35060

Summary: For a bounded domain in a complete Riemannian manifold \(M^n\) isometrically immersed in an Euclidean space, we derive extrinsic estimates for eigenvalues of the Dirichlet eigenvalue problem of the Laplace operator, which depend on the mean curvature of the immersion. Further, we also obtain an upper bound for the \((k+1)\)th eigenvalue, which is the best possible in the meaning of order on \(k\).

MSC:

35P15 Estimates of eigenvalues in context of PDEs
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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