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

Abstract

<abstract><p>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.</p></abstract>

Asymptotic expansion for solution of Maxwell equation in domain with highly oscillating boundary

“…The obtained formula may in addition extend the already derived for small volume inhomogeneities [14][15][16][17][18][19]. It should be noted that the interface problems with non oscillating boundary are studied in several works, see for examples [9,[11][12][13][20][21][22][23][24][25][26]. To the best of our knowledge, this is the first work which deals with the Maxwell oscillating interface problem.…”

On a Steklov spectrum in Electromagnetics

“…In this section we briefly present some of the results in [32] for problem (16). In the sequel Ω will denote a bounded domain in R 3 with sufficiently smooth boundary, say of class C 1,1 (see e.g., [33,Definition 1]). As done in [16] for analogous problems, we introduce a penalty term θ grad div u in the equation, where θ can be any positive number, in order to guarantee the coercivity of the quadratic form associated with the corresponding differential operator.…”

“…We note en passant that the standard eigenvalues of Maxwell's equations in a cavity are given by two families of positive numbers, one of which is the family of the squares of the zeros of the equation j l (k) = 0, see [17] , or [33,Appendix], for more details.…”

A few results on permittivity variations in electromagnetic cavities

“…Here we mention, without the sake of completeness, the monographs [8,16,20,34,36,41] and the classical papers [12,13,14] for a complete introduction to this field and a detailed discussion of both theoretic and applied problems in the mathematical theory of electromagnetism. 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.…”