Vanishing aspect ratio limit for a fourth-order MEMS model

Abstract

So far most studies on mathematical models for microelectromechanical systems (MEMS) are focused on the so-called small aspect ratio model which is a wave or beam equation with a singular source term. It is formally derived by setting the aspect ratio equal to zero in a more complex model involving a moving boundary. A rigorous justification of this derivation is provided here when bending is taken into account.Date: April 30, 2018. 1991 Mathematics Subject Classification. 35M30 -35R35 -35B25 -35Q74. Key words and phrases. MEMS -free boundary -small aspect ratio limit. Partially supported by the French-German PROCOPE project 30718Z.

Some singular equations modeling MEMS

“…Note that the regularity of ψ uε is no longer the same in the xand z-directions in the limit ε → 0 since (2.1) becomes degenerate elliptic. Hence, a cornerstone of the proof is to obtain estimates for the trace of ∂ z ψ uε on the upper boundary of Ω(u ε ) [92,93] is always well-defined but might reach the value −1 at some points. However, this cannot occur for d ∈ [1,7], and U λ stat 0 is then a classical solution to (4.8) with λ = λ stat 0 , see [47].…”

“…Theorem 5.6 (Vanishing Aspect Ratio Limit [30,92]). Let either β > 0, γ ≥ 0, τ ≥ 0 or β = γ = 0, τ > 0.…”

“…(iii) Estimate the difference between ψ uε and (x, z) → (1+z)/(1+u ε (x)) in a Sobolev space by positive powers of ε, which, in turn, provides an estimate of g ε (u ε ) − (1 + u ε ) −2 and shows that it converges to zero as ε → 0. Once these three properties are established, a compactness argument is used to obtain the following convergence result: Theorem 4.4 (Vanishing Aspect Ratio Limit [92,93]). There are a sequence ε k → 0, λ 0 > 0, and a (smooth) stationary solution u 0 to the stationary vanishing aspect ratio equation (4.5) with λ = λ 0 such that lim…”

“…Let us emphasize that Theorem 4.4 above is valid either when β > 0 [92] or when β = 0 and τ > 0 [93]. However, the starting point of the analysis and actually the derivation of properties (i)-(iii) listed above differ in both cases.…”

“…Note that the regularity of ψ uε is no longer the same in the xand z-directions in the limit ε → 0 since (2.1) becomes degenerate elliptic. Hence, a cornerstone of the proof is to obtain estimates for the trace of ∂ z ψ uε on the upper boundary of Ω(u ε ) [92,93].…”