On the Partial Differential Equations of Electrostatic MEMS Devices: Stationary Case
| dc.creator | Ghoussoub, Nassif | |
| dc.creator | Guo, Yujin | |
| dc.date | 2005-09-23 | |
| dc.date.accessioned | 2026-07-07T06:18:49Z | |
| dc.date.available | 2026-07-07T06:18:49Z | |
| dc.description | We analyze the nonlinear elliptic problem $Δu=\frac{λf(x)}{(1+u)^2}$ 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 $λ^*$ (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 \cite{GG2} and \cite{GG3}. 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. | |
| dc.description | 26 pages. Updated versions --if any-- of this author's papers can be downloaded at http://www.pims.math.ca/~nassif | |
| dc.identifier | https://arxiv.org/abs/math/0509534 | |
| dc.identifier | http://arxiv.org/abs/math/0509534 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/94864 | |
| dc.subject | Analysis of PDEs | |
| dc.title | On the Partial Differential Equations of Electrostatic MEMS Devices: Stationary Case | |
| dc.type | text |