On the Partial Differential Equations of Electrostatic MEMS Devices with Effects of Casimir Force

We analyze pull-in instability of electrostatically actuated microelectromechanical systems, and we find that as the device size is reduced, the effect of the Casimir force becomes more important. In the miniaturization process there is a minimum size for the device below which the system spontaneously collapses with zero applied voltage. According to the mathematical analysis, we obtain a set U in the plane, such that elements of U correspond to minimal stable solutions of a two-parameter mathematical model. For points on the boundary of U, there exists weak solutions to this model, which are called extremal solutions. More refined properties of stable solutions-such as regularity, stability, uniqueness-are also established.