Bifurcation diagram of a Robin boundary value problem arising in MEMS

TL;DR

This paper analyzes the solution structure of a Robin boundary value problem modeling MEMS devices, revealing a critical voltage threshold that determines the number of steady-state solutions.

Contribution

It provides a rigorous analysis of the bifurcation diagram for a MEMS-related boundary value problem with Robin conditions, identifying the pull-in voltage and solution multiplicity.

Findings

Existence of a critical pull-in voltage for solution bifurcation

Two solutions below the critical voltage, one at the critical point, none above

Characterization of solution structure in MEMS model

Abstract

We consider a parabolic problem with Robin boundary condition which arises when the edge of a micro-electro-mechanical-system (MEMS) device is connected with a flexible nonideal support. Then via a rigorous analysis we investigate the structure of the solution set of the corresponding steady-state problem. We show that a critical value (the pull-in voltage) exists so that the system has exactly two stationary solutions when the applied voltage is lower than this critical value, one stationary solution for applying this critical voltage, and no stationary solution above the critical voltage.