Asymptotics of Some Nonlinear Eigenvalue Problems for a MEMS Capacitor: Part I: Fold Point Asymptotics

Several nonlinear eigenvalue problems modeling the steady-state deflection of an elastic membrane associated with a MEMS capacitor under a constant applied voltage are analyzed using formal asymptotic methods. The nonlinear eigenvalue problems under consideration represent various regular and singular perturbations of the basic membrane nonlinear eigenvalue problem ∆u = λ/(1 + u) in Ω with u = 0 on ∂Ω, where Ω is a bounded two-dimensional domain. The following three perturbations of this basic problem are considered; the addition of a bending energy term of the form −δ∆u; the effect of a fringing-field where λ is replaced by λ `