On the Partial Differential Equations of Electrostatic MEMS Devices:   Stationary Case
  We analyze the nonlinear elliptic problem $\Delta u=\frac{\lambda f(x)}{(1+u)^2}$ on a bounded domain $\Omega$ 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 $\lambda$-- 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 lends to a nonlinear parabolic problem for the dynamic deflection of the elastic membrane which will be considered in forthcoming papers \cite{GG2} and \cite{GG3}. For now, we focus on the stationary equation where the challenge is to estimate $\lambda^*$ 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, the existence of $\lambda^*$ is established together with rigorous bounds. We show the existence of at least one steady-state when $\lambda < \lambda^*$ (and when $\lambda=\lambda^*$ in dimension $N< 8$) while none is possible for $\lambda>\lambda^*$. 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. 
Nassif Ghoussoub and Yujin Guo