The onset of multi-valued solutions of a prescribed mean curvature equation with singular non-linearity

Abstract

The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation with singular non-linearity is analysed. The partial differential equation is a recently derived variant of a canonical model used in the modelling of micro-electromechanical systems. It is observed that the bifurcation curve of solutions terminates at single dead-end point, beyond which no classical solutions exist. A necessary condition for the existence of solutions is developed, revealing that this dead-end point corresponds to a blow-up in the solution's gradient at a point internal to the domain. By employing a novel asymptotic analysis in terms of two small parameters, an accurate characterization of this dead-end point is obtained. An arc length parameterization of the solution curve can be employed to continue solutions beyond the dead-end point; however, all extra solutions are found to be multi-valued. This analysis therefore suggests that the dead-end is a bifurcation point associated with the onset of multi-valued solutions for the system.

Shapes of large, static soap bubbles

“…It also limits to the point (V 0 , z(0)) = (−1, π/3), whose corresponding surface is a cone generated by rotating (r(t), z(t)) = (t, t − 1) around the vertical axis (see [27]). This limiting conical solution, which is a consequence of the singularity at z = −1 in the governing equation for the angle of inclination in (3.1a), facilitates the asymptotic expansion of the unstable branch for z(0) → −1 + using a boundary layer analysis [33]. One such parameterization is…”

“…It also limits to the point ( V 0 , z (0)) = ( − 1, π /3), whose corresponding surface is a cone generated by rotating ( r ( t ), z ( t )) = ( t , t − 1) around the vertical axis (see [27]). This limiting conical solution, which is a consequence of the singularity at z = −1 in the governing equation for the angle of inclination in (3.1a), facilitates the asymptotic expansion of the unstable branch for z (0) → −1 + using a boundary layer analysis [33]. One such parameterization is V0=π3−ε3/22πA3sin(Φ−tan−1(37)−72log⁡ε),1emzfalse(0false)=−1+ε for 0 < ε ≪ 1, where A and Φ are constants whose values are determined numerically via the far-field behaviour of the leading-order boundary layer problem, i.e.…”

Refinements to the study of electrostatic deflections: theory and experiment

“…When ε = 0, the model reduces the standard 'MEMS equation' and has studied by numerous authors [1,10,17,19,20,23]. The case where ε 0 has recently been studied too [3][4][5]. The key feature of model (1.1) is that it captures the pull-in instability via the following fact: for all ε > 0, there exists a λ * such that no solutions of boundary value problem (1.1) exist for any λ > λ * (see [18, §7.5.1] and [4]).…”

“…This ending point occurs when the derivative of solution becomes singular for some r in (0, 1), which is similar to the case for β = 0 and ε 0 and seemingly can be made rigorous with the same approach (cf. [3]). Note that this disappearance of solutions is similar to the behaviour of other mean curvature type equations (cf.…”

Regularized model of post-touchdown configurations in electrostatic MEMS: interface dynamics

“…In the canonical ε = 0 case, the second order formulation (1.1a) models the deformable surface as a membrane and has been featured more heavily in mathematical studies of MEMS [3,4,12,13,26,15,7,18,17], while the fourth order equation (1.1b), prevalent in engineering studies of MEMS as a beam description, gives much better quantitative agreement with experiments [36,2]. The fourth order system (1.1b) has recently attracted more mathematical attention as it exhibits many interesting oscillatory solution behaviors and presents significant analytical challenge due to the absence of a maximum principle [23,29,28,14,9,33].…”