# On Fourth Order PDEs Modelling Electrostatic Micro-Electronical Systems

Date: Wed, Jul 8, 2009

Location: University of New South Wales, Sydney, Australia

Conference: 1st PRIMA Congress

Subject: Mathematics, Partial Differential Equations

Class: Scientific

### Abstract:

Micro-ElectroMechanical Systems (MEMS) and Nano-ElectroMechanical Systems (NEMS) are now a well established sector of contemporary technology. A key component of such systems is the simple idealized electrostatic device consisting of a thin and deformable plate that is held fixed along its boundary $\partial \Omega$, where $\Omega$ is a bounded domain in $\mathbf{R}^2.$ The plate, which lies below another parallel rigid grounded plate (say at level $z=1$) has its upper surface coated with a negligibly thin metallic conducting film, in such a way that if a voltage l is applied to the conducting film, it deflects towards the top plate, and if the applied voltage is increased beyond a certain critical value $l^*$, it then proceeds to touch the grounded plate. The steady-state is then lost, and we have a snap-through at a finite time creating the so-called pull-in instability. A proposed model for the deflection is given by the evolution equation
$$\frac{\partial u}{\partial t} - \Delta u + d\Delta^2 u = \frac{\lambda f(x)}{(1-u^2)}\qquad\mbox{for}\qquad x\in\Omega, t\gt 0 $$
$$u(x,t) = d\frac{\partial u}{\partial t}(x,t) = 0 \qquad\mbox{for}\qquad x\in\partial\Omega, t\gt 0$$
$$u(x,0) = 0\qquad\mbox{for}\qquod x\in\Omega$$

Now unlike the model involving only the second order Laplacian (i.e., $d = 0$), very little is known about this equation. We shall explain how, besides the above practical considerations, the model is an extremely rich source of interesting mathematical phenomena.

Now unlike the model involving only the second order Laplacian (i.e., $d = 0$), very little is known about this equation. We shall explain how, besides the above practical considerations, the model is an extremely rich source of interesting mathematical phenomena.