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On qualitative properties of solutions to microelectromechanical systems with general permittivity

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Abstract

Qualitative properties of solutions to the evolution problem modelling microelectromechanical systems with general permittivity profile are investigated. The system couples a parabolic evolution problem for the displacement of a membrane with an elliptic free boundary value problem for the electric potential in the region between the membrane and a rigid ground plate. We briefly allude to results concerning local and global well-posedness and the small-apect ratio limit. However, the focus is here on proving non-positivity of the membrane displacement for the full moving boundary problem under certain boundary conditions on the potential, as well as the existence of finite-time singularities assuming to have a non-positive solution.

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Notes

  1. See Fig. 1 above for a sketch of the scaled system.

  2. Note that the nonlinearity in (1) is not induced by a local function but by a highly nonlocal operator.

  3. See Fig. 2 for numerical evidence.

  4. The subscript \(D\) in \(W^2_{q,D}(I)\) indicates homogeneous Dirichlet boundary conditions.

  5. In general, \(\tau > 0\) is obtained by a continued but finite application of Banach’s fixed point theorem.

  6. If \(f^{\prime }(x) = 0\) for some \(x \in I\), then the upper bound on \(\varepsilon \) in (14) is void.

  7. For the sake of better readability we suppress the variables in the calculations if no ambiguity is possible.

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Acknowledgments

I am grateful to Joachim Escher for proposing the research topic and for stimulating me with various exciting suggestions. In addition, I express my gratitude to the referee whose remarks improved the initial version of this contribution.

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  1. Institute of Applied Mathematics, Leibniz University of Hannover, Hannover, Germany

    Christina Lienstromberg

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  1. Christina Lienstromberg
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Correspondence to Christina Lienstromberg.

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Communicated by A. Constantin.

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Lienstromberg, C. On qualitative properties of solutions to microelectromechanical systems with general permittivity. Monatsh Math 179, 581–602 (2016). https://doi.org/10.1007/s00605-015-0744-5

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  • DOI: https://doi.org/10.1007/s00605-015-0744-5

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