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

Pier Domenico Lamberti; Michele Zaccaron; Pier Domenico Lamberti; Michele Zaccaron

doi:10.3934/mine.2023018

Mathematics in Engineering

2023, Volume 5, Issue 1: 1-31. doi: 10.3934/mine.2023018

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

Pier Domenico Lamberti ^{1
,
,},
Michele Zaccaron ^{2
,
,}

1.
Dipartimento di Tecnica e Gestione dei Sistemi Industriali (DTG), University of Padova, Stradella S. Nicola 3, 36100 Vicenza, Italy
2.
Dipartimento di Matematica 'Tullio Levi-Civita', University of Padova, Via Trieste 63, 35121 Padova, Italy

Received: 22 September 2021 Revised: 25 January 2022 Accepted: 26 January 2022 Published: 04 March 2022

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.
- Maxwell's equations,
- spectral stability,
- cavities,
- shape sensitivity,
- boundary homogenization
Citation: Pier Domenico Lamberti, Michele Zaccaron. Spectral stability of the curlcurl operator via uniform Gaffney inequalities on perturbed electromagnetic cavities[J]. Mathematics in Engineering, 2023, 5(1): 1-31. doi: 10.3934/mine.2023018

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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.

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