On a three-dimensional free boundary problem modeling electrostatic mems

We consider the dynamics of an electrostatically actuated thin elastic plate being clamped at its boundary above a rigid plate. The model includes the harmonic electrostatic potential in the three-dimensional time-varying region between the plates along with a fourth-order semilinear parabolic equation for the elastic plate deflection which is coupled to the square of the gradient trace of the electrostatic potential on this plate. The strength of the coupling is tuned by a parameter $\lambda$ proportional to the square of the applied voltage. We prove that this free boundary problem is locally well-posed in time and that for small values of $\lambda$ solutions exist globally in time. We also derive the existence of a branch of asymptotically stable stationary solutions for small values of $\lambda$ and non-existence of stationary solutions for large values thereof, the latter being restricted to a disc-shaped plate.