Sensitivity Analysis of the Current, the Electrostatic Capacity, and the Far-Field of the Potential with Respect to Small Perturbations in the Surface of a Conductor

TL;DR

This paper rigorously derives asymptotic expansions for current, electrostatic capacity, and far-field of potential due to small surface perturbations of a conductor, and analyzes the sensitivity of the NP operator's eigenvector.

Contribution

It introduces a systematic method using layer potentials and field expansion to analyze how small surface changes affect electrostatic quantities and the NP operator's eigenstructure.

Findings

Asymptotic formulas for current, capacity, and far-field under surface perturbations.

Sensitivity analysis of the NP operator's eigenvector to shape changes.

Rigorous derivation using layer potential and field expansion techniques.

Abstract

We derive asymptotic expansions of the current, the electrostatic capacity, and the far-field of the electrostatic potential resulting from small perturbations in the shape of an isolated conductor with $\mathcal{C}^2$-surface. Our derivation is rigourous by using systematic way, based on layer potential techniques and the field expansion (FE) method. We then use these results to study the sensibility analysis of the first eigenvector of the $L^2$-adjoint of the Neumann-Poincar\'e (NP) operator with respect to small perturbations in the surface of its domain.