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A Parabolic Free Boundary Problem Modeling Electrostatic MEMS

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Abstract

The evolution problem for a membrane based model of an electrostatically actuated microelectromechanical system is studied. The model describes the dynamics of the membrane displacement and the electric potential. The latter is a harmonic function in an angular domain, the deformable membrane being a part of the boundary. The former solves a heat equation with a right-hand side that depends on the square of the trace of the gradient of the electric potential on the membrane. The resulting free boundary problem is shown to be well-posed locally in time. Furthermore, solutions corresponding to small voltage values exist globally in time, while global existence is shown not to hold for high voltage values. It is also proven that, for small voltage values, there is an asymptotically stable steady-state solution. Finally, the small aspect ratio limit is rigorously justified.

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    Joachim Escher & Christoph Walker

  2. Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université de Toulouse, 31062, Toulouse Cedex 9, France

    Philippe Laurençot

Authors
  1. Joachim Escher
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  2. Philippe Laurençot
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  3. Christoph Walker
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Correspondence to Philippe Laurençot.

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Communicated by C. Le Bris

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Escher, J., Laurençot, P. & Walker, C. A Parabolic Free Boundary Problem Modeling Electrostatic MEMS. Arch Rational Mech Anal 211, 389–417 (2014). https://doi.org/10.1007/s00205-013-0656-2

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  • DOI: https://doi.org/10.1007/s00205-013-0656-2

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