Seminár z kvalitatívnej teórie diferenciálnych rovníc

Seminar on Qualitative Theory of Differential Equations

Thursday 28.11.2019 at 14:00 Lecture room M-223

Annalisa Iuorio (RICAM Linz)

Singular Perturbation Analysis of a Regularized MEMS Model

Abstract:

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.

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. In this talk we analyze

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.

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 is emphasized;

in particular, the focus is on the singular limit as both parameters

tend to zero. This is a joint work with Nikola Popovic and Peter Szmolyan.