Blow up analysis for a parabolic MEMS problem, I: Hölder estimate

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 Hölder 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 Hölder continuous solutions. These results are also used to prove a stratification theorem on the rupture set $\{u=0\}$.