Stationary states to a free boundary transmission problem for an electrostatically actuated plate

TL;DR

This paper analyzes a free boundary transmission problem modeling an electrostatically actuated plate, deriving formulas for energy derivatives, constructing minimizers, and establishing energy monotonicity.

Contribution

It introduces a representation formula for the electrostatic energy derivative and constructs energy minimizers for the problem.

Findings

Derived a formula for the derivative of electrostatic energy

Constructed energy minimizers for the free boundary problem

Proved monotonicity of electrostatic energy with respect to deflection

Abstract

A two-dimensional free boundary transmission problem arising in the modeling of an electrostatically actuated plate is considered and a representation formula for the derivative of the associated electrostatic energy with respect to the deflection of the plate is derived. The latter paves the way for the construction of energy minimizers and also provides the Euler-Lagrange equation satisfied by these minimizers. A by-product is the monotonicity of the electrostatic energy with respect to the deflection.