The eigenvalue problem for −Δu=λu with dirichlet boundary conditions for a certain class of two-dimensional regions

Abstract

By using the Weinstein method, eigenvalues and eigenfunctions of the equation −Δu=λu with Dirichlet boundary conditions are calculated for a certain class of regions. The regions are composed of unions of rectangles, and include L-shaped, single-notched and crossed rectangles. The method consists of determining the zeros of the Weinstein determinant W n (λ) . This function W n (λ) in turn is determined by the eigenvalues and eigenfunctions of −Δu=λu with mixed boundary conditions (Dirichlet and Neumann) for each component rectangle of the given region, which are easily calculated, and by trial functions p 1 , p 2 , …, p n which are easily chosen. Numerical results for two examples, an asymmetrical L-shaped region and a symmetrical single-notched region, are given, and are shown to be reasonably precise by comparison with results available in the literature. The method is applicable to many problems for a certain class of regions.