Blow up analysis for a parabolic MEMS problem. I: Hölder estimate (Q6588094)

The authors develop a blow-up analysis framework for a parabolic MEMS problem, providing a detailed characterization of how solutions \(u\) of this problem quench, i.e., vanish, in finite time, and elucidating the structure of the rupture set \(\{u=0\}.\) They first derive an optimal Hölder estimate for solutions to the parabolic problem, leveraging this estimate to construct a rigorous convergence theory for solution sequences of the underlying problem. These foundational results are subsequently applied to estimate the Hausdorff dimension of the rupture set \(\{u=0\}.\)