A new model for electrostatic MEMS with two free boundaries

TL;DR

This paper introduces a mathematical model for electrostatic MEMS devices with two free boundaries, analyzing existence of stationary solutions and their behavior as the aspect ratio diminishes.

Contribution

It develops a novel fixed point approach to prove stationary solutions and rigorously demonstrates convergence to a simplified model in the small aspect ratio limit.

Findings

Existence of stationary solutions for small voltages

Convergence of solutions to the small aspect ratio model

Mathematical validation of pull-in instability modeling

Abstract

A moving boundary problem with two free boundaries modeling a two-dimensional idealized MEMS device with pull-in instability is discussed. We use a fixed point argument to show that the model possesses stationary solutions for small source voltages. We also give a rigorous evidence that solutions of the model converge towards solutions of the associated small aspect ratio equation in the vanishing aspect ratio limit.