Open Problems on Eigenvalues of the Laplacian

Abstract

In this paper a number of open problems for the low eigenvalues of the Laplacian on a Euclidean domain are discussed. These include problems with Neumann boundary conditions as well as Dirichlet. Some problems for the biharmonic operator (with "clamped" boundary conditions) are also included, particularly the analogs of the Faber-Krahn result for the vibrating clamped plate and buckling problems. In many of the problems presented here, it is expected that the conjectured inequality saturates at a disk in ]R2 (or a ball in higher dimensions). For the most part, the problems listed complement those found in a prior paper of R. Benguria and the author [10), which, for example, discusses the P6lya conjectures for the Dirichlet and Neumann eigenvalues of a Euclidean domain. References to related problem lists and discussions due to L.E. Payne [32), [33), [34) and S.-T. Yau [49), [50), [51) are also given. 1991 Mathematics Subject Classification: Primary 35P15j Secondary 49R05, 49R10. We shall consider the eigenvalues and eigenfunctions of a domain (= open connected set) n in Euclidean space }Rn. Throughout the discussion n will denote a bounded domain. We concentrate mainly on the fixed membrane problem,-D.u = AU u=o in n c }Rn, on an, (1) (2) though later we shall also bring in several other classical eigenvalue problems associated with the Laplacian. Recall that for a bounded domain, problem (1)-(2) has a real and purely discrete spectrum {Ad~l where (3)