On an elliptic-parabolic MEMS model with two free boundaries

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 toward solutions of the associated small aspect ratio problem.

Keywords:

• MEMS
• free boundary problem
• local and global well-posedness
• non-existence
• asymptotic stability
• small aspect ratio limit

AMS Subject Classifications:

• 35R35
• 35M33
• 35B30
• 35Q74
• 74M05