Asymptotic expansions of Stekloff eigenvalues for perturbations of inhomogeneous media

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.