A free boundary problem modeling electrostatic MEMS: I. Linear bending effects

TL;DR

This paper studies a complex mathematical model for electrostatic MEMS devices, analyzing their behavior through a fourth-order evolution equation with free boundary conditions, revealing conditions for stability and singularity formation.

Contribution

It introduces a new mathematical framework for MEMS modeling involving a coupled free boundary problem and provides well-posedness and stability criteria.

Findings

Existence of stable steady states for small voltage parameters.

Non-existence of steady states for large voltage and small aspect ratio.

Criteria for avoiding finite-time singularities.

Abstract

The dynamical and stationary behaviors of a fourth-order evolution equation with clamped boundary conditions and a singular nonlocal reaction term, which is coupled to an elliptic free boundary problem on a non-smooth domain, are investigated. The equation arises in the modeling of microelectromechanical systems (MEMS) and includes two positive parameters $\lambda$ and $\varepsilon$ related to the applied voltage and the aspect ratio of the device, respectively. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as a criterion for global existence excluding the occurrence of finite time singularities which are not physically relevant. Existence of a stable steady state is shown for sufficiently small $\lambda$. Non-existence of steady states is also established when $\varepsilon$ is small enough and $\lambda$ is large enough (depending on $\varepsilon$).