| In this paper, we study three problems of eigenvalue estimates: The first one is theeigenvalues of a fixed membrane problem; The second one is the eigenvalues of quadraticpolynomial of the Laplacian; The eigenvalues of the buckling problem are discussed in thethird problem.In the first chapter, we introduce the recent research on the eigenvalue estimates ofthe fixed membrane problem.In the second chapter, we introduce the eigenvalues of the clamped plate problem.Moreover, we consider lower order eigenvalues of quadratic polynomial of the Laplacianon a bounded domain in a complete Riemannian manifold and obtain universal bounds onthe (k+1)th eigenvalue in terms of the first kth eigenvalues which genelizes the results ofCheng-Huang-Wei in [20]. Finally, we also consider eigenvalues of quadratic polynomialoperator of the Laplacian on a bounded domain in a hyperbolic space which extends thosein [16,30].In the third chapter, the recent research on the buckling eigenvalue problem is intro-duced.In the last chapter, eigenvalues of operators in divergence form with a weight onRiemannian manifolds are studied. By the Rayleigh-Ritz inequality, we obtain universalinequalities for lower order eigenvalues of such operators. In particular, for lower ordereigenvalues of Laplacian, our results are sharp. |
