A variational approach to a stationary free boundary problem modeling MEMS

TL;DR

This paper introduces a variational method to find stationary solutions in a free boundary problem modeling electrostatically actuated MEMS devices, revealing multiple solutions and addressing singularities related to plate touchdown.

Contribution

It develops a novel variational framework for a complex MEMS model with a non-coercive energy, demonstrating existence of multiple stationary solutions.

Findings

Existence of at least two stationary solutions for certain voltages.

Construction of solutions via constrained minimization.

Analysis of singularities related to plate touchdown.

Abstract

A variational approach is employed to find stationary solutions to a free boundary problem modeling an idealized electrostatically actuated MEMS device made of an elastic plate coated with a thin dielectric film and suspended above a rigid ground plate. The model couples a non-local fourth-order equation for the elastic plate deflection to the harmonic electrostatic potential in the free domain between the elastic and the ground plate. The corresponding energy is non-coercive reflecting an inherent singularity related to a possible touchdown of the elastic plate. Stationary solutions are constructed using a constrained minimization problem. A by-product is the existence of at least two stationary solutions for some values of the applied voltage.