  This is the first in a series of papers devoted to the blow up analysis for
the quenching phenomena in a parabolic MEMS equation. In this paper, we first
give an optimal H\"{o}lder estimate for solutions to this equation by using the
blow up method and some Liouville theorems on stationary two-valued caloric
functions, and then establish a convergence theory for sequences of uniformly
H\"{o}lder continuous solutions. These results are also used to prove a
stratification theorem on the rupture set $\{u=0\}$.
