For each planar domain D of unit area, the first two Dirichlet eigenvalues of —∆ on D determine a point (λ 1 ( D ), λ 2 ( D ) in the (λ 1 , λ 2 ) 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 n th eigenvalue when the minimizing domain is diconnected. For n = 3 we show that the minimizing domain is connected and that λ 3 is a local minimum for D a circular disc.

ABSTRACT. We consider fold point behavior for nonlinear eigenvalue problems of the form Aito + XQF(UO,P) = 0, x e D; d u uo + buo = 0, x e dD.Here b and 0 are positive parameters. In circular cylindrical or spherical geometries, we assume that F (u, (3) is such that this problem has multiple radially symmetric solutions for some range of AQ only when 0 < /3 < (3Q. As (3 -»• (3Q from below, two simple fold points are assumed to coalesce producing a cubic fold point at AQ = A c o when (3 = j3o. We calculate the change in the cubic fold point location resulting from various classes of perturbations of this problem. The perturbations we consider are a small change in the boundary condition maintained on the boundary of a circular cylindrical or a spherical domain, a small distortion of the boundary of a circular cylindrical domain, and the removal of a small hole from a domain. In each case, we derive asymptotic expansions for the location of the perturbed cubic fold point in terms of a small parameter e measuring the size of the perturbation. A numerical scheme is then formulated to evaluate the coefficients in these expansions and the method is illustrated for the combustion nonlinearity F(u,(3) = exp[w/(l + /3u)]. In certain cases, we compare our expansions for the perturbed cubic fold point location with some previous results and with results obtained from full numerical solutions to the perturbed problems. The significance of these results for the perturbed cubic fold point location are that they can provide the first step in characterizing, at least locally, the alteration of the boundary in parameter space that separates regions where the nonlinear eigenvalue problem has multiple solutions from regions where the problem has only unique solutions.