On the Partial Differential Equations of Electrostatic MEMS Devices: Stationary Case

Abstract

We analyze the nonlinear elliptic problem $\Delta u =\frac{\lambda f(x)}{(1+u)^2}$ on a bounded domain Ω of $R^N$ 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 $\lambda^*$ (pull‐in voltage). This creates a so‐called pull‐in instability, which greatly affects the design of many devices. The mathematical model leads to a nonlinear parabolic problem for the dynamic deflection of the elastic membrane, which will be considered in a forthcoming paper. Here, we focus on the stationary equation and on estimates for $\lambda^*$ 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, we establish upper and lower bounds for $\lambda^*$ in terms of the volume and shape of the domain, the dimension of the ambient space, as well as the permittivity profile. We analyze the first branch of stable steady states when $\lambda < \lambda^*$ and prove that a semistable (extremal) solution exists at $\lambda=\lambda^*$ in dimension $1\leq N\leq 7$, and that classical extremal solutions may not exist for dimension $N\geq 8$. More refined properties of stable steady states—such as regularity, stability, uniqueness, multiplicity, energy estimates, and comparison results—are also established. The analysis of branches of unstable solutions is more elaborate and is tackled in the companion paper [P. Esposito, N. Ghoussoub, and Y. Guo, Comm. Pure Appl. Math., (2006), to appear].