Asymptotic expansions of Stekloff eigenvalues for perturbations of inhomogeneous media

TL;DR

This paper derives precise formulas for how Stekloff eigenvalues change under perturbations in inhomogeneous media, aiding nondestructive testing by linking eigenvalue shifts to material property changes.

Contribution

It introduces a modified Stekloff eigenvalue problem with smoothing, providing explicit perturbation formulas that account for medium heterogeneity and anisotropy.

Findings

Formulas accurately predict eigenvalue perturbations in inhomogeneous media

Sensitivity of eigenvalues is strongly affected by medium anisotropy

Numerical examples confirm theoretical estimates

Abstract

Eigenvalues arising in scattering theory have been envisioned as a potential source of target signatures in nondestructive testing of materials, whereby perturbations of the eigenvalues computed for a penetrable medium would be used to infer changes in its constitutive parameters relative to some reference values. We consider a recently introduced modification of the class of Stekloff eigenvalues, in which the inclusion of a smoothing operator guarantees that infinitely many eigenvalues exist under minimal assumptions on the medium, and we derive precise formulas that quantify the perturbation of a simple eigenvalue in terms of the coefficients of a perturbed inhomogeneous medium. These formulas rely on the theory of nonlinear eigenvalue approximation and regularity results for elliptic boundary-value problems with heterogeneous coefficients, the latter of which is shown to have a strong influence on the sensitivity of the eigenvalues corresponding to an anisotropic medium. A simple numerical example in two dimensions is used to verify the estimates and suggest future directions of study.