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

We prove spectral stability results for the curlcurl 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.

Automated summary

This paper proves the spectral stability of the curlcurl operator subject to electric boundary conditions on a cavity under boundary perturbations. The cavities are assumed to be sufficiently smooth with weak restrictions on the strength of the perturbations. The methods involve the construction of suitable Piola-type transformations and the proof of uniform Gaffney inequalities using uniform a priori H-2 estimates for the Poisson problem of the Dirichlet Laplacian. Connections to boundary homogenization problems are also mentioned.