On a Free Boundary Model for Three-Dimensional MEMS with a Hinged Top Plate II: Parabolic Case

TL;DR

This paper models a 3D MEMS device with a hinged top plate using a coupled parabolic PDE system, proving local and global well-posedness and analyzing singularity formation.

Contribution

It introduces a new free boundary model for 3D MEMS with a hinged top plate and establishes well-posedness and singularity results for the system.

Findings

Proved local well-posedness of the model.

Established global existence for small voltages.

Identified touchdown as the only finite-time singularity.

Abstract

A parabolic free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a fourth-order semilinear parabolic equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the device. The strength of the coupling is tuned by a parameter $\lambda$ which is proportional to the square of the applied voltage difference. It is proven that the model is locally well-posed in time and that, for $\lambda$ sufficiently small, solutions exist globally in time. In addition, touchdown of the top plate on the ground plate is shown to be the only possible finite time singularity.