On Fourth Order PDEs Modelling Electrostatic Micro-Electronical Systems

Nassif Ghoussoub
Wed, Jul 8, 2009
University of New South Wales, Sydney, Australia
1st PRIMA Congress

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.