RANGE OF THE FIRST 2 EIGENVALUES OF THE LAPLACIAN

For each planar domain D of unit area, the first two Dirichlet eigenva lues of -Delta on D determine a point (lambda(1)(D),lambda(2)(D) in th e (lambda(1),lambda(2)) plane. As D varies over all such domains, this point varies over a set R which we determine. Its boundary consists o f two semi-infinite straight lines and a curve connecting their endpoi nts. 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 lambda(3) is a, local minimum for D a circular disc.