Well-posedness of a quasilinear evolution problem modelling MEMS with general permittivity

Abstract

Subject of consideration is the analytical investigation of a coupled system of partial differential equations arising from the modelling of electrostatically actuated microelectromechanical systems with general permittivity profile. A quasilinear parabolic evolution problem for the displacement u of an elastic membrane is coupled with an elliptic free boundary value problem that determines the electrostatic potential \(\psi \) in the region between the elastic membrane and a rigid ground plate. The system is shown to be well-posed locally in time for all arbitrarily large values \(\lambda \) of the applied voltage, whereas small values of the applied voltage, which do not exceed a certain critical value \(\lambda _{*}\), do even allow globally in time existing solutions. In addition, conditions are specified which force solutions emerging from a non-positive initial deflection to stay non-positive as long as they exist.