A parabolic free boundary problem modeling electrostatic MEMS

Joachim Escher (1) , Philippe Laurencot (2) , Christoph Walker (3)

1 IFAM - Institute for Applied Mathematics
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 IFAM - Institut für Angewandte Mathematik

The evolution problem for a membrane based model of an electrostatically actuated microelectromechanical system (MEMS) is studied. The model describes the dynamics of the membrane displacement and the electric potential. The latter is a harmonic function in an angular domain, the deformable membrane being a part of the boundary. The former solves a heat equation with a right hand side that depends on the square of the trace of the gradient of the electric potential on the membrane. The resulting free boundary problem is shown to be well-posed locally in time. Furthermore, solutions corresponding to small voltage values exist globally in time while global existence is shown not to hold for high voltage values. It is also proven that, for small voltage values, there is an asymptotically stable steady-state solution. Finally, the small aspect ratio limit is rigorously justified.

Mots clés: MEMS; free boundary problem; well-posedness; asymptotic stability; finite time singularity; small aspect ratio limit

Domaines: Equations aux dérivées partielles [math.AP]

Joachim Escher, Philippe Laurencot, Christoph Walker. A parabolic free boundary problem modeling electrostatic MEMS. Archive for Rational Mechanics and Analysis, 2014, 211, pp.389-417. ⟨10.1007/s00205-013-0656-2⟩. ⟨hal-00757167⟩