Upper and lower bounds for the pull-in voltage and the pull-in distance for a generalized MEMS problem

Yan-Hsiou Cheng; Kuo-Chih Hung; Shin-Hwa Wang; Jhih-Jyun Zeng; Yan-Hsiou Cheng; Kuo-Chih Hung; Shin-Hwa Wang; Jhih-Jyun Zeng

doi:10.3934/mbe.2022321

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 7: 6814-6840. doi: 10.3934/mbe.2022321

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Research article

Upper and lower bounds for the pull-in voltage and the pull-in distance for a generalized MEMS problem

1.
Department of Mathematics and Information Education, National Taipei University of Education, Taipei 106, Taiwan
2.
Fundamental General Education Center, National Chin-Yi University of Technology, Taichung 411, Taiwan
3.
Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan

Received: 03 October 2021 Revised: 28 February 2022 Accepted: 27 April 2022 Published: 05 May 2022

We study upper and lower bounds for the pull-in voltage and the pull-in distance for the one-dimensional prescribed mean curvature problem arising in MEMS

$ \begin{equation*} \left \{\begin{array}{l} - \left( \frac{u^{ \prime } (x)}{\sqrt{1 +\left (u^{ \prime } (x)\right )^{2}}} \right)^{ \prime } = \frac{\lambda }{(1 -u)^{p}} , \ \ u <1 , \ \ -L <x <L, \\ u ( -L) = u (L) = 0, \end{array}\right . \end{equation*} $

where $ \lambda > 0 $ is a bifurcation parameter, and $ p, L > 0 $ are two evolution parameters. We further study monotonicity properties and asymptotic behaviors for the pull-in voltage and pull-in distance with respect to positive parameters $ p $ and $ L $.
- prescribed mean curvature problem,
- global bifurcation diagram,
- positive solution,
- pull-in voltage,
- pull-in distance,
- MEMS
Citation: Yan-Hsiou Cheng, Kuo-Chih Hung, Shin-Hwa Wang, Jhih-Jyun Zeng. Upper and lower bounds for the pull-in voltage and the pull-in distance for a generalized MEMS problem[J]. Mathematical Biosciences and Engineering, 2022, 19(7): 6814-6840. doi: 10.3934/mbe.2022321

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Abstract

We study upper and lower bounds for the pull-in voltage and the pull-in distance for the one-dimensional prescribed mean curvature problem arising in MEMS

$ \begin{equation*} \left \{\begin{array}{l} - \left( \frac{u^{ \prime } (x)}{\sqrt{1 +\left (u^{ \prime } (x)\right )^{2}}} \right)^{ \prime } = \frac{\lambda }{(1 -u)^{p}} , \ \ u <1 , \ \ -L <x <L, \\ u ( -L) = u (L) = 0, \end{array}\right . \end{equation*} $

where $ \lambda > 0 $ is a bifurcation parameter, and $ p, L > 0 $ are two evolution parameters. We further study monotonicity properties and asymptotic behaviors for the pull-in voltage and pull-in distance with respect to positive parameters $ p $ and $ L $.

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