On semi-linear elliptic equation arising from Micro-Electromechanical Systems with contacting elastic membrane

TL;DR

This paper studies a nonlinear elliptic equation modeling contact phenomena in Micro-Electromechanical Systems, analyzing how boundary behavior influences solutions and the critical voltage parameter.

Contribution

It provides new insights into the boundary decay properties of solutions and the critical voltage in MEMS models with contact boundary conditions.

Findings

Boundary decay rate of solutions depends on parameters $eta$ and $ abla a$

Existence of a critical pull-in voltage $ar{ ext{voltage}}$ for solutions

Minimal solutions exhibit specific boundary behavior related to membrane contact

Abstract

This paper is concerned with the nonlinear elliptic problem $-\Delta u=\frac{\lambda }{(a-u)^2}$ on a bounded domain $\Omega$ of $\mathbb{R}^N$ with Dirichlet boundary conditions. This problem arises from Micro-Electromechanical Systems devices in the case that the elastic membrane contacts the ground plate on the boundary. We analyze the properties of minimal solutions to this equation when $\lambda>0$ and the function $a:\bar\Omega\to[0,1]$ satisfying $a(x)\ge \kappa{\rm dist}(x,\partial\Omega)^\gamma$ for some $\kappa>0$ and $\gamma\in(0,1)$. Our results show how the boundary decay of the membrane works on the solutions and pull-in voltage $\lambda$.