# UBC Mathematics Colloquium

## 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.