Estimates on Pull-in Distances in MEMS Models and other Nonlinear Eigenvalue Problems

TL;DR

This paper provides mathematical bounds on the pull-in distance in MEMS models, analyzing how it depends on domain geometry, dimension, and permittivity profile, supported by rigorous proofs and numerical estimates.

Contribution

It offers the first rigorous mathematical estimates for the pull-in distance in nonlinear eigenvalue problems related to MEMS, connecting theory with observed phenomena and numerical results.

Findings

Upper and lower bounds for the extremal solution's L-infinity norm.

Dependence of pull-in distance on domain geometry and dimension.

Validation of numerical estimates through rigorous proofs.

Abstract

Motivated by certain mathematical models for Micro-Electro-Mechanical Systems (MEMS), we give upper and lower $L^\infty$ estimates for the minimal solutions of nonlinear eigenvalue problems of the form $-\Delta u = \lambda f(x) F(u)$ on a smooth bounded domain $ \Omega$ in $\IR^N$. We are mainly interested in the {\it pull-in distance}, that is the $L^\infty-$norm of the extremal solution $u^*$ and how it depends on the geometry of the domain, the dimension of the space, and the so-called {\it permittivity profile} $f$. In particular, our results provide mathematical proofs for various observed phenomena, as well as rigorous derivations for several estimates obtained numerically by Pelesko \cite{P}, Guo-Pan-Ward \cite{GPW} and others in the case of the MEMS non-linearity $F(u)=\frac{1}{(1-u)^2}$ and for power-law permittivity profiles $f(x)=|x|^\alpha$.