Shape sensitivity analysis of Neumann-Poincar\'e eigenvalues
Analysis of PDEs
2025-04-02 v1 Optimization and Control
Spectral Theory
Abstract
This paper concerns the eigenvalues of the Neumann-Poincar\'e operator, a boundary integral operator associated with the harmonic double-layer potential. Specifically, we examine how the eigenvalues depend on the support of integration and prove that the map associating the support's shape to the eigenvalues is real-analytic. We then compute its first derivative and present applications of the resulting formula. The proposed method allows for handling infinite-dimensional perturbation parameters for multiple eigenvalues and perturbations that are not necessarily in the normal direction.
Cite
@article{arxiv.2504.00696,
title = {Shape sensitivity analysis of Neumann-Poincar\'e eigenvalues},
author = {Matteo Dalla Riva and Pier Domenico Lamberti and Paolo Luzzini and Paolo Musolino},
journal= {arXiv preprint arXiv:2504.00696},
year = {2025}
}