Finite Time Singularity in a MEMS Model Revisited

Abstract

A free boundary problem modeling a microelectromechanical system consisting of a fixed ground plate and a deformable top plate is considered, the plates being held at different electrostatic potentials. It couples a second order semilinear parabolic equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the device. The validity of the model is expected to break down in finite time when the applied voltage exceeds a certain value, a finite time singularity occurring then. This result, already known for non-positive initial configurations of the top plate, is here proved for arbitrary ones and thus now includes, in particular, snap-through instabilities.