## 1 Introduction  
Microelectromechanical sensors are currently used in automotive electronics, medical equipment, smart portable electronics, hard disk drives, and computer peripherals [1]. Electrostatically actuated MicroElectroMechanical Systems (MEMS) are a special class of MEMS that use Coulomb's force as the driving mechanism in sensing and actuation. Electrostatically actuated MEMS are used as accelerometers [2], pressure sensors [3], and mass flow and chemical sensors [4]. Electrostatically actuated MEMS also find applications in the characterization of the material properties of thin films, such as elastic constants and residual stresses, see for example [5]. In addition to sensing applications, electrostatically actuated MEMS are employed in high-frequency signal filtering [6] and relay switches [7]. A comprehensive review of modeling of electrostatically actuated MEMS can be found in [8]. An electrostatically actuated MEMS is generally comprised of a conductive deformable electrode suspended above a rigid grounded electrode [9]. A direct current (DC) voltage applied across the two bodies results in the deflection of the deformable electrode and the consequent change in the system capacitance. When an alternating current (AC) is superimposed on the DC voltage, harmonic motions of the system are obtained that can be used to realize resonant devices. The applied DC voltage has an upper limit beyond which the electrostatic force is not balanced by the elastic restoring force in the deformable conductor. Beyond this critical voltage, the deformable conductor snaps and touches the lower rigid plate. This phenomenon is known as the pull-in instability, and it was first experimentally observed in the late sixties [10; 11]. The critical displacement and the critical voltage associated with this instability are called the pull-in displacement and the pull-in voltage, respectively. Their accurate evaluation is crucial in the design of electrostatically actuated MEMS. In micro-mirrors [12] and micro-resonators [13], the designer avoids this instability in order to achieve stable motions; while in switching applications [7], the designer exploits this effect to optimize the performance of the device. For a wide class of electrostatic MEMS, the deformable electrode is a flat body whose thickness \(h\) is much smaller than its characteristic in-plane dimension \(\ell\)[14]. Such electrodes can be regarded as two-dimensional (2-D) plate-like bodies. Since \(h/\ell\ll 1\), an approximate distributed model can be employed, where the system kinematics is described only through the displacement of points on the movable electrode mid-surface, see for example [15]. Linear and nonlinear problems for microplates have been studied in [16; 17; 18; 19; 20; 21; 22; 23]. When the bending stiffness of the deformable electrode is negligible as compared to its in-plane stretching and \(g_{0}/\ell\ll 1\), where \(g_{0}\) is the initial gap, the electrode can be regarded as a linear elastic membrane. The membrane approximation is valid for \(\ell/h\geq 400\), see for example [24]. Linear micromembranes have been studied in [25; 26; 27; 28; 29]. As discussed in [25], the membrane approximation yields accurate results for many MEMS devices such as micro-pumps made of thin glassy polymers and grating light valves comprised of stretched thin ribbons. With the decrease in electrostatic MEMS dimensions from the micro to the nanoscale additional nanoscale surface forces, such as the Casimir force and the van der Waals force [30; 31; 32; 33], should be considered, see for example [34]. At small scales, the nanoscale surface forces may overcome elastic restoring actions in the device and lead to the plates' sticking during the fabrication process. van der Waals force and Casimir force can both be connected with the existence of zero-point vacuum oscillations of the electromagnetic field [31; 32; 33].