A parabolic free boundary problem modeling electrostatic MEMS

TL;DR

This paper analyzes a free boundary problem modeling electrostatic MEMS, establishing well-posedness, stability for small voltages, and the effects of high voltages and aspect ratio limits.

Contribution

It introduces a mathematical model for electrostatic MEMS involving a free boundary problem and proves local well-posedness, stability at small voltages, and the small aspect ratio limit.

Findings

Solutions exist globally for small voltages

High voltages lead to non-existence of global solutions

Stable steady-state solutions are found for small voltages

Abstract

The evolution problem for a membrane based model of an electrostatically actuated microelectromechanical system (MEMS) is studied. The model describes the dynamics of the membrane displacement and the electric potential. The latter is a harmonic function in an angular domain, the deformable membrane being a part of the boundary. The former solves a heat equation with a right hand side that depends on the square of the trace of the gradient of the electric potential on the membrane. The resulting free boundary problem is shown to be well-posed locally in time. Furthermore, solutions corresponding to small voltage values exist globally in time while global existence is shown not to hold for high voltage values. It is also proven that, for small voltage values, there is an asymptotically stable steady-state solution. Finally, the small aspect ratio limit is rigorously justified.