On a free boundary model for three-dimensional MEMS with a hinged top plate II: Parabolic case

Katerina Nik

doi:10.3934/cpaa.2021110

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2021, Volume 20, Issue 10: 3395-3417. Doi: 10.3934/cpaa.2021110 open access

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On a free boundary model for three-dimensional MEMS with a hinged top plate II: Parabolic case

Katerina Nik

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A–1090 Vienna, Austria

Received: January 2021

Revised: May 2021

Early access: June 2021

Published: October 2021

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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.

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Mathematics Subject Classification: 35K91, 35R35, 35M33, 35Q74, 35B44.

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Figure 1. Cross section of the idealized MEMS device

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