# On Fourth Order PDEs Modelling Electrostatic Micro-Electronical Systems

Micro-ElectroMechanical Systems (MEMS) and Nano-ElectroMechanical Systems (NEMS) are now a well established sector of contemporary technology. A key component of such systems is the simple idealized electrostatic device consisting of a thin and deformable plate that is held fixed along its boundary *TeX Embedding failed!*, where *TeX Embedding failed!* is a bounded domain in *TeX Embedding failed!* The plate, which lies below another parallel rigid grounded plate (say at level *TeX Embedding failed!*) has its upper surface coated with a negligibly thin metallic conducting film, in such a way that if a voltage l is applied to the conducting film, it deflects towards the top plate, and if the applied voltage is increased beyond a certain critical value *TeX Embedding failed!*, it then proceeds to touch the grounded plate. The steady-state is then lost, and we have a snap-through at a finite time creating the so-called pull-in instability. A proposed model for the deflection is given by the evolution equation

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Now unlike the model involving only the second order Laplacian (i.e., *TeX Embedding failed!*), very little is known about this equation. We shall explain how, besides the above practical considerations, the model is an extremely rich source of interesting mathematical phenomena.