Uniqueness of solutions for an elliptic equation modeling MEMS

TL;DR

This paper investigates the conditions under which solutions to a singular elliptic PDE modeling MEMS devices are unique, considering parameters like domain shape, dimension, and nonlinearity profile.

Contribution

It provides new insights into the uniqueness of solutions for a MEMS-related elliptic equation, accounting for various parameters and domain geometries.

Findings

Uniqueness depends on the parameter λ and domain geometry.

Existence of multiple solutions for certain parameter ranges.

Conditions for uniqueness are characterized in terms of domain and profile.

Abstract

We study the effect of the parameter $\lambda$, the dimension $N$, the profile $f$ and the geometry of the domain $\Omega \subset\mathbb{R}^N$, on the question of uniqueness of the solutions to the following elliptic boundary value problem with a singular nonlinearity: $$ 180pt {{array}{ll} -\Delta u= \frac{\lambda f(x)}{(1-u)^2} & \hbox{in}\Omega 0<u<1 &\hbox{in}\Omega u=0 &\hbox{on}\partial \Omega. {array}. 130pt (S)_{\lambda, f} $$ This equation has been proposed as a model for a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 below a rigid ground plate located at height z = 1.