Payne-Pólya-Weinberger conjecture, Sperner’s inequality, biharmonic operator, bi-Laplacian, clamped plate problem, Rayleigh’s conjecture, buckling problem, the Pólya-Szegő conjecture, universal inequalities for eigenvalues, Hile-Protter inequality, H. C. Yang’s inequality. Short title: Isoperimetric and Universal Inequalities This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and bi-Laplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians and of the vibrating clamped plate problem (i.e., the biharmonic operator with “Dirichlet ” boundary conditions). We also discuss the known universal inequalities for the eigenvalues of the Dirichlet Laplacian and the vibrating clamped plate and buckling problems and go on to

2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518

We solve the biharmonic eigenvalue problem \Delta 2 u = u and the buckling plate problem \Delta 2 u = \Gamma\Deltau on the unit square using a highly accurate spectral Legendre-Galerkin method. We study the nodal lines for the first eigenfunction near a corner for the two problems. Five sign changes are computed and the results show that the eigenfunction exhibits a self similar pattern as one approaches the corner. The amplitudes of the extremal values and the coordinates of their location as measured from the corner are reduced by constant factors. These results are compared with the known asymptotic expansion of the solution near a corner. This comparison shows that the asymptotic expansion is highly accurate already from the first sign change as we have complete agreement between the numerical and the analytical results. Thus, we have an accurate description of the eigenfunction in the entire domain. AMS Subject Classification: 65N25. Key words: Biharmonic operator, eigenvalu...

In this paper we present an efficient, accurate and parallelizable direct method for the solution of the (indefinite) linear algebraic systems that arise in the solution of fourth order partial differential equations (PDEs) using mixed finite element approximations. The method is intended particularly for use when multiple right-hand sides occur, and when high accuracy is required in these solutions. The algorithm is described in some detail and its performance is illustrated through the numerical solution of a biharmonic eigenvalue problem where the smallest eigenpair is approximated using inverse iteration after discretization via the Ciarlet-Raviart mixed finite element method.