On geometric estimates for some problems arising from modeling pull-in voltage in MEMS

TL;DR

This paper establishes that for p-MEMS problems, the pull-in voltage is minimized by domain and permittivity symmetrization, using Talenti's comparison principle, applicable to various multidimensional and boundary value problems.

Contribution

It introduces a novel symmetrization approach to estimate pull-in voltage in p-MEMS problems, extending to multidimensional and elliptic boundary value problems.

Findings

Pull-in voltage minimized by domain symmetrization.

Method applies to multidimensional MEMS models.

Results extend to elliptic boundary value problems.

Abstract

In this paper for all $p>1$ we prove that the pull-in voltage of the $p$-MEMS (micro-electro mechanical systems) problems on a smooth bounded domain of $\mathbb R^{d}, d\geq1,$ is minimized by symmetrizing the domain and the permittivity profile. The proofs rely on some suitable version of Talenti's comparison principle. We also demonstrate our method to the multidimensional MEMS type problems on the whole space $\mathbb R^{d}, d\geq3,$ and the Dirichlet boundary value problems of second order uniformly elliptic differential operators.