On a Free Boundary Model for Three-Dimensional MEMS with a Hinged Top Plate I: Stationary Case

A stationary free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a nonlocal fourth-order equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the free domain between the two plates. The strength of the coupling is tuned by a parameter $\lambda$ which is proportional to the square of the applied voltage difference. Existence of a stable stationary solution is established for small values of $\lambda$. Nonexistence of stationary solutions is obtained when $\lambda$ is large enough.