On a three-dimensional free boundary problem modeling electrostatic mems

TL;DR

This paper models the dynamics of an electrostatically actuated elastic plate with a free boundary, analyzing existence, stability, and non-existence of solutions depending on the applied voltage parameter.

Contribution

It introduces a coupled 3D electrostatic and elastic plate model, proving local well-posedness, global existence for small voltages, and characterizing stationary solutions.

Findings

Existence of locally well-posed solutions in time.

Global solutions exist for small voltage parameters.

Stationary solutions are stable for small voltages and do not exist for large voltages.

Abstract

We consider the dynamics of an electrostatically actuated thin elastic plate being clamped at its boundary above a rigid plate. The model includes the harmonic electrostatic potential in the three-dimensional time-varying region between the plates along with a fourth-order semilinear parabolic equation for the elastic plate deflection which is coupled to the square of the gradient trace of the electrostatic potential on this plate. The strength of the coupling is tuned by a parameter $\lambda$ proportional to the square of the applied voltage. We prove that this free boundary problem is locally well-posed in time and that for small values of $\lambda$ solutions exist globally in time. We also derive the existence of a branch of asymptotically stable stationary solutions for small values of $\lambda$ and non-existence of stationary solutions for large values thereof, the latter being restricted to a disc-shaped plate.