Range of the first two eigenvalues of the laplacian

Source abstract

Abstract For each planar domain D of unit area, the first two Dirichlet eigenvalues of —∆ on D determine a point (λ1(D), λ2(D) in the (λ1, λ2) plane. As D varies over all such domains, this point varies over a set R which we determine. Its boundary consists of two semi-infinite straight lines and a curve connecting their endpoints. This curve is found numerially. We also show how to minimize the nth eigenvalue when the minimizing domain is diconnected. For n = 3 we show that the minimizing domain is connected and that λ3 is a local minimum for D a circular disc.