A Parabolic Free Boundary Problem Modeling Electrostatic MEMS

Abstract

Abstract. 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.

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

“…This is in sharp contrast with the case β = 0 studied in [6], where the finiteness of T m could also be due to a blowup of the W 2 q (I)-norm of u(t) as t → T m . The additional regularity of u provided here by the fourth-order term β∂ 4 x u allows us to rule out the occurrence of this latter singularity.…”

“…Before describing more precisely the results of the analysis performed in this paper, let us first single out the main findings: the starting point is to establish the local well-posedness of (1.1)-(1.5) along with an extension criterion guaranteeing global existence. As already observed in [6,7], the right hand side of (1.1) is a nonlinear operator of, roughly speaking, order 3/2 (in the sense that it maps W 2 q (I) in W θ q (I) for all θ ∈ [0, 1/2) and q ∈ (2, ∞), see Proposition 2.1 below). Since it also becomes singular when u approaches −1, the extension criterion resulting from the fixed point argument leading to local well-posedness involves not only a lower bound on u, but also an upper bound on the norm of (u, γ 2 ∂ t u) in a suitable Sobolev space.…”

“…In this direction, let us recall that a classical technique to investigate the possible occurrence of finite time singularities is the so-called eigenfunction technique. Owing to the nonlocal character of the right hand side of (1.1), this technique does not seem to be appropriate here, but a nonlinear variant thereof introduced in [6] has proven to be successful and allowed us to show the finiteness of T m for sufficiently large λ in the second order parabolic case, that is, when γ = β = 0. We have yet been unable to develop it further to achieve a similar result when (γ, β) = (0, 0), in particular when β > 0, the fourth-order problem under investigation herein.…”

“…We thus shall take a different route in the spirit of the approach developed in [6,7] for the second-order parabolic version of (1.1)-(1.5) corresponding to the choice β = γ = 0 of the parameters. Let us also mention that a quasilinear variant of the parabolic case γ = 0 of (1.1)-(1.5) with β > 0 and curvature terms is investigated in the companion paper [19], where the small deformation assumption is discarded.…”

A fourth-order model for MEMS with clamped boundary conditions

“…Mathematical models have been derived to describe MEMS devices which lead to free boundary problems due to the deformable membrane [21]. Since these models are difficult to analyze mathematically (though recent contributions can be found in [7,9,15]), one often takes advantage of the small aspect ratio of the devices to reduce the free boundary problem to a single equation for the displacement, see [21]. More precisely, the small aspect ratio model describing the dynamics of the displacement u = u(t, x) of the membrane Ω ⊂ R d reads Here, γ 2 ∂ 2 t u and ∂ t u account, respectively, for inertia and damping effects, B∆ 2 u and −T ∆u are due to bending and stretching of the membrane, while −λ(1 + u) −2 reflects the action of the electrostatic forces in the small aspect ratio limit.…”