Precise bounds and asymptotics for the first Dirichlet eigenvalue of triangles and rhombi (Q2460025)

For people interested in properties of eigenvalues it is an astonished fact that the first Dirichlet eigenvalue of the Laplacian is known only for very few planar domains. Furthermore, it is still an open problem for such domains as isosceles triangles and rhombi, which can be described in terms of a single parameter in order to reduce the problem to an ordinary differential equation with respect to a function of this parameter. For that reason approximations of eigenvalues are important. The aim of this paper is to give precise approximations of the first Dirichlet eigenvalue close to singular limit cases for these two types of domains.