The eigenvalue problem for −Δu=λu with Dirichlet boundary conditions for two-dimensional regions. II. T-shaped, cross-shaped, and H-shaped regions

Abstract

This article deals with the calculation of the eigenvalues and eigenfunctions of the two-dimensional Laplacian with Dirichlet boundary condition for T-shaped, cross-shaped, and H-shaped regions. The method used is the Weinstein method (the intermediate method). The suitable base problem can be obtained through dividing the given region into rectangles and replacing Dirichlet boundary conditions by Neumann boundary conditions on a part of the boundaries. The intermediate problems can be determined by appropriate trial functions, and by the eigenvalues and eigenfunctions for each decomposed rectangle. The numerical results are considered to be reasonably precise. Special attention is paid to the dependence of the eigenvalues and eigenfunctions on parameters of the region. Some consideration is also given to the curve-veering phenomena observed frequently in the numerical results.