A note on the Maxwell’s eigenvalues on thin sets

We analyse the Maxwell’s spectrum on thin tubular neighbourhoods of embedded surfaces of $\mathbb R^3$ . We show that the Maxwell’s eigenvalues converge to the Laplacian eigenvalues of the surface as the thin parameter tends to zero. To achieve this, we reformulate the problem in terms of the spectrum of the Hodge Laplacian with relative conditions acting on co-closed differential $1$ -forms. The result leads to new examples of domains where the Faber–Krahn inequality for Maxwell’s eigenvalues fails, examples of domains with any number of arbitrarily small eigenvalues, and underlines the failure of spectral stability under singular perturbations changing the topology of the domain. Additionally, we explicitly produce Maxwell’s eigenfunctions on product domains with the product metric, extending previous constructions valid in the Euclidean case.