Touchdown is the only finite time singularity in a three-dimensional MEMS model

TL;DR

This paper proves that in a 3D MEMS model, touchdown is the only finite-time singularity, using energy methods and semigroup smoothing effects related to the bi-Laplacian operator.

Contribution

It establishes the uniqueness of the finite-time singularity (touchdown) in a 3D MEMS model through novel analytical techniques.

Findings

Touchdown is the only finite-time singularity in the model.

The proof utilizes the energy structure and smoothing effects of the bi-Laplacian.

The approach confirms the singularity's uniqueness in the system.

Abstract

Touchdown is shown to be the only possible finite time singularity that may take place in a free boundary problem modeling a three-dimensional microelectromechanical system. The proof relies on the energy structure of the problem and uses smoothing effects of the semigroup generated in $L^1$ by the bi-Laplacian with clamped boundary conditions.