A Compact Riemannian Manifold Of The Eigenvalue Problem,

In the first part of this paper, we give a upper bound of the (k+l)-th eigenvalue in terms of the first k eigenvalues of the eigenvalue problem for the Dirichlet biharmonic operator on a connected bounded domain in a complete Riemannian manifold, which generalizes the result of paper [4] and [14]. LetΩbe a connected bounded domain in M which is an n-dimensional complete Riemannian manifold, then the eigenvalue problem for the Dirichlet biharmonic operator, or the'Clamped Plate problem', onΩis: where△is the Laplacian on M and△2 is the biharmonic operator on M. Assuming that are eigenvalues of the above problem, we prove that there exists a constant H02 in terms of the upper bound of the mean curvature, which only depends on M andΩ,such that the following inequality holds: which generalizes the main results of paper [4] and [14]. In order to prove our result, we apply a similar technique as the one proposed by [4], which eliminates the unwanted terms in the deduction perfectly. By making use of Nash's theorem and introducing k free constants, we derive a universal bound for the (k+l)-th eigenvalue. In fact, the above inequality is one of the sharpest estimates currently for the eigenvalue problems for the Dirichlet biharmonic operators in virtue of the method of [4] which we follow by substituting coordinate components of the position vector of a complete Riemannian manifold for the coordinate functions of a Euclidean space. We also derive the corresponding estimates for the eigenvalues under some specific conditions, namely, M being a minimal submanifold of a convex hypersurface in a Euclidean space, a minimal submanifold of an ellipsoid or a minimal submanifold of a cylinder.In the second part of this paper, we give a new estimate on the lower bound of the first eigenvalue of the Laplacian on a 2-dimensional or 3-dimensional closed Riemannian manifold with positive Ricci curvature in terms of the diameter of the manifold and the lower bound of Ricci curvature, which sharpens one of the results of paper [10]. For an n-dimensional closed Riemannian manifold whose Ricci curvature has a positive lower bound (n-1)K for some constant K> 0, [10] gave both a lower bound of the the first eigenvalueλfor arbitrary n,λ≥π2/d2+0.31(n-1)K, and a better one,λ≥π2/d2+0.375(n-1)K, when n= 2, both of which are the best estimates currently for this eigenvalue problem on compact Riemannian manifold. In our paper, under the low-dimensional condition that n=2,3, we give a better estimate,λ≥π2/d2+0.425(n-1)K, mainly by a similar'Testing Function' method based on Yau's 'Gradient Estimation'and a classification discussion as that used in [10], noticing that the classification in [10] can be adjusted for a better estimate.