A free boundary problem modeling electrostatic MEMS: I. Linear bending effects

Philippe Laurencot (1) , Christoph Walker (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IFAM - Institut für Angewandte Mathematik

The dynamical and stationary behaviors of a fourth-order evolution equation with clamped boundary conditions and a singular nonlocal reaction term, which is coupled to an elliptic free boundary problem on a non-smooth domain, are investigated. The equation arises in the modeling of microelectromechanical systems (MEMS) and includes two positive parameters $\lambda$ and $\varepsilon$ related to the applied voltage and the aspect ratio of the device, respectively. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as a criterion for global existence excluding the occurrence of finite time singularities which are not physically relevant. Existence of a stable steady state is shown for sufficiently small~$\lambda$. Non-existence of steady states is also established when $\varepsilon$ is small enough and $\lambda$ is large enough (depending on $\varepsilon$).

Mots clés: MEMS; free boundary problem; fourth-order operator; well-posedness; bending; non-existence

Domaines: Equations aux dérivées partielles [math.AP]

Philippe Laurencot, Christoph Walker. A free boundary problem modeling electrostatic MEMS: I. Linear bending effects. Mathematische Annalen, 2014, 360, pp.307-349. ⟨10.1007/s00208-014-1032-8⟩. ⟨hal-00855094⟩