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A. Iuorio, N. Popovic, P. Szmolyan:
"Singular perturbation analysis of a regularized MEMS mode";
SIAM Journal on Applied Dynamical Systems, 18 (2019), 2; 661 - 708.

Micro-Electro Mechanical Systems (MEMS) are defined as very small structures that combine electrical and mechanical components on a common substrate. Here, the electrostatic-elastic case is considered, where an elastic membrane is allowed to deflect above a ground plate under the action of an electric potential, whose strength is proportional to a parameter $\lambda$. Such devices are commonly described by a parabolic partial differential equation that contains a singular nonlinear source term. The singularity in that term corresponds to the so-called touchdown phenomenon, where the membrane establishes contact with the ground plate. "Touchdown" is known to imply the non-existence of steady-state solutions and blow-up of solutions in finite time. We study a recently proposed extension of that canonical model, where such singularities are avoided due to the introduction of a regularizing term involving a small "regularization" parameter $\varepsilon$. Methods from dynamical systems and geometric singular perturbation theory, in particular the desingularization technique known as "blow-up", allow for a precise description of steady-state solutions of the regularized model, as well as for a detailed resolution of the resulting bifurcation diagram. The interplay between the two main model parameters $\varepsilon$ and $\lambda$ is emphasized; in particular, the focus is on the singular limit as both parameters tend to zero

Micro-Electro Mechanical Systems, touchdown, boundary value problem, blow-up method, bifurcation diagram, geometric singular perturbation theory

http://dx.doi.org/10.1137/18M1197552

https://www.researchgate.net/publication/332366441_Singular_Perturbation_Analysis_of_a_Regularized_MEMS_Model