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In this paper, we obtain the Faber-Krahn inequality for the first eigenvalue of the Robin Laplacian on bounded domain in Riemannian manifolds whose Ricci curvature satisfies Ric g ≥ (n − 1). The Faber-Krahn inequality also holds for the Robin Laplacian on bounded domain in hyperbolic space H n. [ABSTRACT FROM AUTHOR]

Abstract: Let Ω be a bounded domain in an n-dimensional Euclidean space . We study eigenvalues of an eigenvalue problem of a system of elliptic equations: Estimates for eigenvalues of the above eigenvalue problem are obtained. Furthermore, we obtain an upper bound on the th eigenvalue . We also obtain sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator on compact manifolds with boundary and positive Ricci curvature. [Copyright &y& Elsevier]

Abstract: For a bounded planar region in , we obtain the ratios of lower order eigenvalues of Laplace operator. Combining our results with the recursive formula in Cheng and Yang (2007) , we can obtain better upper bound of the -th () membrane eigenvalues. [Copyright &y& Elsevier]