Accurate Bounds for the Eigenvalues of the Laplacian and Applications to Rhombical Domains

摘要

The report concerns the eigenvalues and eigenfunctions of Laplace's differential operator on a bounded two-dimensional domain G with zero values on the boundary. The paper describes a new technique for determining the coefficients in the expansion of an eigenfunction in terms of particular eigenfunctions of the differential operator. The coefficients are chosen to make the sum of the expansion come close to satisfying the boundary conditions. As an example, the eigenvalues and eigenfunctions are determined for a rhombical membrane. (Author)