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

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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 in a non-smooth domain, are investigated. The equation arises in the modeling of microelectromechanical systems 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 \(\varepsilon \)).

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Notes

  1. There is a sign misprint in the proof of [17, Lemma 11].

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Acknowledgments

The work of Ph.L. was partially supported by the CIMI (Centre International de Mathématiques et d’Informatique) Excellence program and by the Deutscher Akademischer Austausch Dienst (DAAD) while enjoying the hospitality of the Institut für Angewandte Mathematik, Leibniz Universität Hannover.

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Authors and Affiliations

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

    Philippe Laurençot

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

    Christoph Walker

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  1. Philippe Laurençot
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Correspondence to Christoph Walker.

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Partially supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02.

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Laurençot, P., Walker, C. A free boundary problem modeling electrostatic MEMS: I. Linear bending effects. Math. Ann. 360, 307–349 (2014). https://doi.org/10.1007/s00208-014-1032-8

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