Analysis of the singular solution branch of a prescribed mean curvature equation with singular nonlinearity

TL;DR

This paper investigates the solution structure of a singular nonlinear PDE related to MEMS, identifying a dead-end bifurcation point where solutions cease to exist and analyzing solution continuation beyond this point.

Contribution

It introduces a necessary condition for solution existence, characterizes the dead-end point as a blow-up, and employs asymptotic analysis to predict this critical point.

Findings

Bifurcation curve terminates at a dead-end point.

Solution derivative blows up at the dead-end point.

Asymptotic analysis accurately predicts the dead-end point.

Abstract

The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation (PDE) with singular non-linearity is analyzed. The PDE is a recently derived variant of a canonical model used in the modeling of Micro-Electro Mechanical Systems (MEMS). 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 which reveals that this dead-end point corresponds to a blow-up in the solution derivative at a point internal to the domain. Using asymptotic analysis, an accurate prediction 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.