ar X iv : m at h / 05 09 53 4 v 1 [ m at h . A P ] 2 3 Se p 20 05 On the Partial Differential Equations of Electrostatic MEMS Devices : Stationary Case

We analyze the nonlinear elliptic problem ∆u = λf(x) (1+u)2 on a bounded domain Ω of R with Dirichlet boundary conditions. This equation models a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at −1. When a voltage –represented here by λ– is applied, the membrane deflects towards the ground plate and a snap-through may occur when it exceeds a certain critical value λ (pull-in voltage). This creates a so-called “pull-in instability” which greatly affects the design of many devices. The mathematical model lends to a nonlinear parabolic problem for the dynamic deflection of the elastic membrane which will be considered in forthcoming papers [11] and [12]. For now, we focus on the stationary equation where the challenge is to estimate λ in terms of material properties of the membrane, which can be fabricated with a spatially varying dielectric permittivity profile f . Applying analytical and numerical techniques, the existence of λ is established together with rigorous bounds. We show the existence of at least one steady-state when λ < λ (and when λ = λ in dimension N < 8) while none is possible for λ > λ. More refined properties of steady states –such as regularity, stability, uniqueness, multiplicity, energy estimates and comparison results– are shown to depend on the dimension of the ambient space and on the permittivity profile.