Nonlocal Dynamic Problems with Singular Nonlinearities and Applications to MEMS

Abstract

We establish existence and regularity results for a time dependent fourth order integro-differential equation with a possibly singular nonlinearity which has applications in designing MicroElectroMechanicalSystems. The key ingredient in our approach, besides basic theory of hyperbolic equations in Hilbert spaces, exploits the Near Operators Theory introduced by Campanato

Electrostatic field in terms of geometric curvature in membrane MEMS devices

“…Analogously, for the other inequality of (62) (Fig. 2), referred to interval [20,140], with the same tolerance and d * = 0, and defining f 2 (H) = 10 2 λ 2 − H 6 − 1 we obtain f 2 (H) = 0 when H = 99. So, to guarantee the existence of the solution of the problem, it is necessary that sup|H| = 99 corresponding to 88.92 o in dimensionless conditions.…”

“…optimal shape synthesis, and then its solution is approximated by means of an algorithm of numerical minimization [17,18]. Some authors, in recent years, have gained expertise in the field of modeling of electrostatic actuators in MEMS in both steady cases and in the dynamical ones carrying out existence, uniqueness and regularity results by means of near operator theory even in presence of nonlinear singularities [19][20][21][22]. In these works, generalizing and deepening the research done in previous publications [23][24][25][26][27], it was considered a MEMS composed of two plates one of which is fixed and the other deformable, but clamped at boundary of a region Ω ∈ R N ; once a voltage drop is suitably applied, the ground plate deflects from the steady state u = 0 towards a fixed plate (ground plate) positioned at height u = 1.…”

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

“…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.…”

“…Note that the boundary conditions (5)- (6) are incorporated in the domain of the operator A(·). We now recall some important properties of A(·) from [9]: For ω > 0 and k ≥ 1 let H(W Let λ, µ > 0, q ∈ (2, ∞), ε ∈ (0, 1) and fix κ ∈ (0, 1/4).…”

Vanishing aspect ratio limit for a fourth-order MEMS model

“…(1. 5) In (1.3), the terms corresponding to mechanical forces are β∂ 4 x u, which accounts for plate bending, and (τ + a ∂ x u 2 2 ) ∂ 2 x u, which accounts for external stretching (τ > 0) and self-stretching due to moderately large oscillations (a > 0). The right-hand side of (1.3) describes the electrostatic forces exerted on the elastic plate which are proportional to the square of the trace of the (rescaled) gradient of the electrostatic potential on the elastic plate, the parameter λ depending on the square of the applied voltage difference before scaling.…”