Quantitative touchdown localization for the MEMS problem with variable dielectric permittivity

TL;DR

This paper provides quantitative conditions for touchdown localization in MEMS models with variable dielectric permittivity, using finite-dimensional optimization and maximum principle techniques, with results applicable to practical MEMS design.

Contribution

It introduces a method to estimate the ratio of permittivity profiles that prevent touchdown outside specific regions, with explicit bounds derived from optimization problems.

Findings

Touchdown can be localized within permittivity bumps.

The ratio threshold for no touchdown outside bumps can be as high as 0.3.

Quantitative estimates are obtained via finite-dimensional optimization.

Abstract

We consider a well-known model for micro-electromechanical systems (MEMS) with variable dielectric permittivity, based on a parabolic equation with singular nonlinearity. We study the touchdown or quenching phenomenon. Recently, the question whether or not touchdown can occur at zero points of the permittivity profile, which had long remained open, was answered negatively by Guo and Souplet for the case of interior points, and we then showed that touchdown can actually be ruled out in subregions of the domain where the permittivity is positive but suitably small. The goal of this paper is to further investigate the touchdown localization problem and to show that, in one space dimension, one can obtain quite quantitative conditions. Namely, for large classes of typical, one-bump and two-bump permittivity profiles, we find good lower estimates of the ratio between f and its maximum, below which no touchdown occurs outside of the bumps. The ratio is rigorously obtained as the solution of a suitable finite-dimensional optimization problem (with either three or four parameters), which is then numerically estimated. Rather surprisingly, it turns out that the values of the ratio are not "small" but actually up to the order 0.3, which could hence be quite appropriate for robust use in practical MEMS design. The main tool for the reduction to the finite-dimensional optimization problem is a quantitative type I, temporal touchdown estimate. The latter is proved by maximum principle arguments, applied to a multi-parameter family of refined, nonlinear auxiliary functions with cut-off.