Homogenization and nonselfadjoint spectral optimization for dissipative Maxwell eigenproblems

Abstract

The homogenization of eigenvalues of non-Hermitian Maxwell operators is studied by the H-convergence method. It is assumed that the Maxwell systems are equipped with suitable m-dissipative boundary conditions, namely, with Leontovich or generalized impedance boundary conditions of the form \(n \times E = Z [(n \times H )\times n ] \). We show that, for a wide class of impedance operators \(Z\), the nonzero spectrum of the corresponding Maxwell operator is discrete. To this end, a new continuous embedding theorem for domains of Maxwell operators is obtained. We prove the convergence of eigenvalues to an eigenvalue of a homogenized Maxwell operator under the assumption of the H-convergence of the material tensor-fields. This result is applied then to the existence of optimizers for eigenvalue optimization problems and to the existence of an eigenvalue-free region around zero. Connections with unique (and nonunique) continuation results are discussed.