C. Esteve, Ph Souplet. Quantitative touchdown localization for the MEMS problem with variable dielectric permittivity, NONLINEARITY, Vol. 31, No. 11 (2018). DOI: 10.1088/1361-6544
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 f, which had long remained open, was answered negatively in Guo and Souplet (2015 SIAM J. Math. Anal. No. 47, pp. 614–25) for the case of interior points, and we then showed in Esteve and Souplet (2017 arXiv:1706.04375) that touchdown can actually be ruled out in subregions of Ω where f 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 ρ between 
and its maximum, below which no touchdown occurs outside of the bumps. The ratio ρ 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 ρ 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.