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

Abstract

Abstract. 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 λ and ε 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 λ. Non-existence of steady states is also established when ε is small enough and λ is large enough (depending on ε).

The Abstract Quasilinear Cauchy Problem for a MEMS Model with Two Free Boundaries

“…By (16), the displacements might also blow up in W 2 q (I) × W 2 q (I) contradicting the physical expectation that there is collision of the membranes in the interior of the device for finite maximal existence times. A similar ambiguity has been observed in [27]. In Section 3 we present proofs of the following theorems.…”

“…For v ≡ −1, the problem (1)-(8) models the evolution of a free membrane suspended above a fixed ground plate. Various analytical results on this type of a MEMS have been obtained in recent years: [8,14,16,19,20,22,27,28] refer to the parabolic problem, [6,13,21,23,27] discuss the problem with a hyperbolic evolution equation and in [7,26,[31][32][33] the stationary model is presented. The corresponding model with an additional curvature term is discussed in [9] and our derivation of (1)-(8) refines Laurençot's line of arguments therein.…”

Touchdown is the Only Finite Time Singularity in a Three-Dimensional MEMS Model

“…For the investigation of the dynamics of MEMS devices it is thus of great importance to rule out mathematically the norm blowup in finite time. In [11] this was done if D = (−1, 1) is one-dimensional, that is, in case the elastic part is a beam or a rectangular plate that is homogeneous in one direction. The situation considered herein, where D is an arbitrary two-dimensional (convex) domain, is more delicate.…”

A free boundary value problem modelling microelectromechanical systems with general permittivity

“…Whereas a multitude of them treats the case of a vanishing aspect ratio (see for instance [1][2][3][4][5][6][7]), the recent works of Escher, Laurençot and Walker address the coupled system. Moreover, the reader shall be referred to the works [10][11][12][13], each of them again assuming the permittivity to be constant but taking other different physical aspects into account. Moreover, the reader shall be referred to the works [10][11][12][13], each of them again assuming the permittivity to be constant but taking other different physical aspects into account.…”