Small volume expansion of the splitting of multiple Neumann Laplacian eigenvalues due to a grounded inclusion in two dimensions

TL;DR

This paper derives the first terms of the asymptotic expansion for how Neumann Laplacian eigenvalues split when a small grounded inclusion is introduced in two dimensions, providing explicit formulas and decay rates.

Contribution

It introduces explicit formulas for the eigenvalue splitting due to small inclusions, including decay rates and dependence on eigenfunctions, size, and position.

Findings

Eigenvalue splitting formulas derived for small inclusions.

One eigenvalue decays like O(1/log(ε)), the other like O(ε^2).

Explicit dependence on unperturbed eigenfunctions and inclusion parameters.

Abstract

The first terms of the small volume asymptotic expansion for the splitting of Neumann boundary condition Laplacian eigenvalues due to a grounded inclusion of size {\epsilon} are derived. An explicit formula to compute the first term from the eigenvalues and eigenfunctions of the unperturbed domain, the inclusion size and position is given. As a consequence, when an eigenvalue of double multiplicity splits in two distinct eigenvalues, one decays like O(1/log({\epsilon})), the other like O({\epsilon}^2).