Open AccessSpectral stability of the curlcurl operator via uniform Gaffney inequalities on perturbed electromagnetic cavities
Author(s) -
Pier Domenico Lamberti,
Michele Zaccaron
Publication year - 2022
Publication title -
mathematics in engineering
Language(s) - English
Resource type - Journals
ISSN - 2640-3501
DOI - 10.3934/mine.2023018
Subject(s) - mathematics , curl (programming language) , mathematical analysis , sobolev space , homogenization (climate) , dirichlet boundary condition , a priori estimate , boundary value problem , variational inequality , operator (biology) , dirichlet distribution , laplace operator , a priori and a posteriori , boundary (topology) , pure mathematics , biodiversity , ecology , biochemistry , chemistry , philosophy , epistemology , repressor , computer science , transcription factor , gene , biology , programming language
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.