On an elliptic-parabolic MEMS model with two free boundaries

TL;DR

This paper analyzes a complex MEMS model with two free boundaries, demonstrating local and global well-posedness, stability of steady states, and convergence to a simplified model in the small aspect ratio limit.

Contribution

It introduces a novel elliptic-parabolic free boundary MEMS model with two membranes and establishes its well-posedness, stability, and asymptotic behavior.

Findings

Solutions exist globally for small voltages

Steady states are asymptotically stable

Solutions converge to the small aspect ratio model

Abstract

We discuss an evolution free boundary problem of mixed type with two free boundaries modeling an idealized electrostatically actuated MEMS device. While the electric potential is the solution of an elliptic equation, the dynamics of the membranes' displacement is modeled by two parabolic equations. It is shown that the model is locally well-posed in time and that solutions exist globally for small source voltages whereas non-existence holds for large voltage values. Moreover, our model possesses a steady state solution that is asymptotically stable. Finally, we show that in the vanishing aspect ratio limit, solutions of the model converge towards solutions of the associated small aspect ratio problem.