  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.
