Asymptotic and numerical analysis of linear and nonlinear eigenvalue problems

We study two separate problems. They are related conceptually but the subject and techniques differ sufficiently that a division is natural. The first deals with a linear eigenvalue problem in PDE and geometric optimization, while the second deals with a generalized non-linear eigenvalue problem from chemical reactors and the effects on criticality due to several classes of perturbations.;First, we consider the Dirichlet Laplacian on a bounded planar domain. We combine perturbation theory and a numerical geometric optimization procedure to find the domains which minimize the second eigenvalue, given a fixed total area and a fixed first eigenvalue. This allows us to characterize completely the range of values the first two eigenvalues can take as the domain is varied. We show the method to be applicable to higher eigenvalues as well and prove that optimization calculations need only be performed on simple domains.;Second, we consider a PDE from steady-state combustion theory. The response of a chemical reactor can vary discontinuously as parameters are changed. We study the change in critical values of the parameters, which mark the transition to a state where discontinuous behavior can occur, under three classes of perturbations. Regular perturbation theory suffices to analyze changes in the reactor geometry or in the external temperature field, but singular perturbation theory is needed to analyze changes in the boundary conditions. The theory of strong localized perturbations is applied to handle the inclusion of cooling or insulating rods in the interior of the reactor. We use numerical methods based on our analytic results to compute the changes in critical values in several cases.