Spectral stability of the $curl curl$ operator via uniform Gaffney inequalities on perturbed electromagnetic cavities

TL;DR

This paper establishes spectral stability for the curl curl operator in electromagnetic cavities under boundary perturbations, using variational methods, Piola transformations, and uniform Gaffney inequalities, with implications for boundary homogenization.

Contribution

It introduces a novel approach combining Piola transformations and uniform Gaffney inequalities to analyze spectral stability under boundary perturbations in electromagnetic cavities.

Findings

Spectral stability results for the curl curl operator under boundary perturbations.

Development of uniform Gaffney inequalities with $H^2$-estimates.

Connections to boundary homogenization problems.

Abstract

We prove spectral stability results for the $curl curl$ operator subject to electric boundary conditions on a cavity upon boundary perturbations. The cavities are assumed to be sufficiently smooth but we impose weak restrictions on the strength of the perturbations. The methods are of variational type and are based on two main ingredients: the construction of suitable Piola-type transformations between domains and the proof of uniform Gaffney inequalities obtained by means of uniform a priori $H^2$-estimates for the Poisson problem of the Dirichlet Laplacian. The uniform a priori estimates are proved by using the results of V. Maz'ya and T. Shaposhnikova based on Sobolev multipliers. Connections to boundary homogenization problems are also indicated.