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Dynamics of a free boundary problem with curvature modeling electrostatic MEMS


Authors: Joachim Escher, Philippe Laurençot and Christoph Walker
Journal: Trans. Amer. Math. Soc. 367 (2015), 5693-5719
MSC (2010): Primary 35R35, 35M33, 35Q74, 35B25, 74M05
DOI: https://doi.org/10.1090/S0002-9947-2014-06320-4
Published electronically: October 17, 2014
MathSciNet review: 3347187
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Abstract: The dynamics of a free boundary problem for electrostatically actuated microelectromechanical systems (MEMS) is investigated. The model couples the electric potential to the deformation of the membrane, the deflection of the membrane being caused by application of a voltage difference across the device. More precisely, the electrostatic potential is a harmonic function in the angular domain that is partly bounded by the deformable membrane. The gradient trace of the electric potential on this free boundary part acts as a source term in the governing equation for the membrane deformation. The main feature of the model considered herein is that, unlike most previous research, the small deformation assumption is dropped, and curvature for the deformation of the membrane is taken into account which leads to a quasilinear parabolic equation. The free boundary problem is shown to be well-posed, locally in time for arbitrary voltage values and globally in time for small voltage values. Furthermore, existence of asymptotically stable steady-state configurations is proved in case of small voltage values as well as non-existence of steady-state solutions if the applied voltage difference is large. Finally, convergence of solutions of the free boundary problem to the solutions of the well-established small aspect ratio model is shown.


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Additional Information

Joachim Escher
Affiliation: Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, D–30167 Hannover, Germany
Email: escher@ifam.uni-hannover.de

Philippe Laurençot
Affiliation: Institut de Mathématiques de Toulouse, UMR 5219, CNRS, Université de Toulouse, F–31062 Toulouse Cedex 9, France
Email: laurenco@math.univ-toulouse.fr

Christoph Walker
Affiliation: Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, D–30167 Hannover, Germany
Email: walker@ifam.uni-hannover.de

DOI: https://doi.org/10.1090/S0002-9947-2014-06320-4
Keywords: MEMS, free boundary problem, curvature, well-posedness, asymptotic stability, small aspect ratio limit
Received by editor(s): August 12, 2013
Published electronically: October 17, 2014
Additional Notes: The second author was partially supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02
Article copyright: © Copyright 2014 American Mathematical Society

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