Shape sensitivity analysis for electromagnetic cavities

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

“…and in this case 3 . See [35] for more details. Unfortunately, given two domains Ω and Ω, in general it is not possible to define explicitly a diffeomorphism between Ω and Ω (even if it is known a priori that the two domains are diffeomorphic).…”

“…, s . We will only show (35), since the computations to prove (36) are similar. We will first see that…”

“…Recall that τ is a fixed positive constant (one could normalise it by setting τ = 1 but we prefer to keep it as it is also with reference to other papers where it is important to have the possibility to use different values of τ , see for example [35,Remark 2.13]). In this section we prove that S ǫ compactly converges to S 0 as ǫ → 0, and this implies spectra stability.…”

“…It seems to us that very little is known in the literature. The case of domain perturbations of the form Φ(Ω) where Φ is a regular diffeomorphism from Ω to Φ(Ω) are considered in [31] and [35] where differentiability results and Hadamard type formulas for shape derivatives are proved. We also quote the pioneering work [30] where the Hadamard formula was found on the base of heuristic computations.…”

A few results on permittivity variations in electromagnetic cavities

“…For more recent papers we refer to, e.g., [3,5,11,29,38,25]. Incidentally, we note that in [24] Lamberti and the second named author have considered the eigenvalues of problem (3) with fixed and constant permittivity ε = I 3 on a variable domain and proved a real analytic dependence upon variation of the shape of the domain. To the best of the authors' knowledge, the dependence of the eigenvalues λ j [ε] upon perturbation of ε has not yet been investigated.…”