Finite-time touchdown for a hinged electrostatic MEMS plate at large voltage
Let $D \subset \mathbb{R}^2$ be a bounded convex domain with $C^{4,\gamma}$ boundary for some $0 < \gamma < 1$, with exterior unit normal $\nu$ and curvature $\kappa \ge 0$, and fix $\varepsilon > 0$, $\beta > 0$, $\tau \ge 0$ and $-1 < \sigma < 1$. For a voltage parameter $\lambda > 0$, the deflection $u(t, x) > -1$ of a hinged elastic plate above a rigid ground plate at height $-1$ solves $$\partial_t u + \beta \Delta^2 u - \tau \Delta u = -\lambda \Bigl( \varepsilon^2 |\nabla_x \psi_u(t, x, u(t, x))|^2 + |\partial_z \psi_u(t, x, u(t, x))|^2 \Bigr) \quad \text{in } D,$$ $$u = \Delta u - (1 - \sigma) \kappa\, \partial_\nu u = 0 \quad \text{on } \partial D, \qquad u(0, \cdot) = 0,$$ where for each $t$ the electrostatic potential $\psi_u(t)$ solves $$\varepsilon^2 \Delta_x \psi_u + \partial_z^2 \psi_u = 0 \ \text{ in } \Omega(u(t)) = \{(x, z) : x \in D,\ -1 < z < u(t, x)\}, \qquad \psi_u(t, x, z) = \frac{1 + z}{1 + u(t, x)} \ \text{ on } \partial\Omega(u(t)).$$ Nik proved that this problem has a unique maximal solution on $[0, T_\lambda)$ with $u \in C([0, T_\lambda); H^3(D)) \cap C((0, T_\lambda); H^4(D)) \cap C^1((0, T_\lambda); L^2(D))$, $u(t)$ satisfying the hinged conditions, and $\psi_u(t) \in H^2(\Omega(u(t)))$; that $T_\lambda = \infty$ for $\lambda$ small enough; and that if $T_\lambda < \infty$, then $\liminf_{t \uparrow T_\lambda} \min_{\overline{D}} u(t) = -1$ (touchdown). For the stationary problem on the same domains, Nik proved that there is no solution with $u \in H^4(D)$, $\psi_u \in H^2(\Omega(u))$ and $u > -1$ once $\lambda$ is at least the principal eigenvalue of $\beta \Delta^2 - \tau \Delta$ with the hinged boundary conditions.
Problem. Is there a finite $\Lambda = \Lambda(D, \varepsilon, \beta, \tau, \sigma) > 0$ such that $T_\lambda < \infty$ for every $\lambda > \Lambda$?
Source: Nik (2021), in the introduction and at the end of Section 5, for general initial data. Stated by AIM — Open Applied Problems for the initial datum u = 0.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, finite-time singularities, free boundary problems, MEMS
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{nik2021parabolic,
author = {Nik, Katerina},
title = {On a free boundary model for three-dimensional {MEMS} with a hinged top plate {II}: parabolic case},
journal = {Commun. Pure Appl. Anal.},
volume = {20},
number = {10},
year = {2021},
pages = {3395--3417},
doi = {10.3934/cpaa.2021110},
eprint = {2103.06776},
archivePrefix = {arXiv},
}
@article{nik2021stationary,
author = {Nik, Katerina},
title = {On a free boundary model for three-dimensional {MEMS} with a hinged top plate: stationary case},
journal = {Port. Math.},
volume = {78},
number = {2},
year = {2021},
pages = {211--232},
doi = {10.4171/PM/2067},
eprint = {2103.06772},
archivePrefix = {arXiv},
}
@misc{aim-finite-time-touchdown-for-a-hinged-electrostatic-mems-plate-at-large-voltage,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Finite-time touchdown for a hinged electrostatic MEMS plate at large voltage}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/750-finite-time-touchdown-for-a-hinged-electrostatic-mems-plate-at-large-voltage.md},
note = {Entry 750 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01150, "Finite-time touchdown for a hinged electrostatic MEMS plate at large voltage", clopen.solutions.