# UBC Mathematics Colloquium

## Nassif Ghoussoub

(UBC)

## Mathematical Analysis of Partial Differential Equations Modeling
Electrostatic MEMS

### Fri., Oct. 30, 2009, 3:00pm, MATX 1100

Abstract:

Micro-Electro-Mechanical Systems (MEMS) and Nano-Electro-Mechanical
Systems (NEMS), which combine electronics with miniature-size
mechanical devices, are basic ingredients of contemporary
technology. A key component of such systems is the simple
idealized electrostatic device consisting of a thin and deformable
plate, consisting of a dielectric material with a negligibly thin
conducting film on its lower surface, that is held fixed along its
boundary in the two dimensional plane. Above the
deformable plate lies a rigid grounded plate. As one applies a
positive voltage to the thin conducting film the deformable plate
deflects upwards towards the ground plate. If the voltage is
increased beyond a certain critical value then the deformable plate
touches the ground plate, in finite time, and we have the so-called
"pull in instability".

Unfortunately, models for electro-statically actuated micro-plates that
account for moderately large deflections are quite complicated and not
yet amenable to rigorous mathematical analysis. In the last 5
years, my students (Cowan, Esposito, Guo, Moradifam) and I, dealt with
much simplified models that still lead to interesting second and fourth
order nonlinear elliptic equations (in the stationary case) and to
nonlinear parabolic equations (in the dynamic case). The non-linearity
is of an inverse square type, which -- until recently – has not
received much attention as a mathematical problem. It was therefore
rewarding to see, besides the above practical considerations, that the
model is actually a very rich source of interesting mathematical
phenomena. Numerics and formal asymptotic analysis give lots of
information and point to many conjectures, but even in the most simple
idealized versions of electrostatic MEMS, one essentially needs the
full available arsenal of modern nonlinear analysis and PDE techniques
“to do" the required mathematics.