Homogenization and nonselfadjoint spectral optimization for dissipative Maxwell eigenproblems

Abstract The optimization of eigenvalues of non-Hermitian Maxwell operators is studied by the homogenization method. We obtain 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 used to prove the existence of optimizers for eigenvalue optimization problems and the existence of an eigenvalue-free region. As applications, a connection with the quantum optics problem of the design of high-Q resonators is discussed and a new way of the quantification of the unique (and nonunique) continuation property is suggested. It is assumed throughout the paper that Maxwell systems are equipped with suitable m-dissipative boundary conditions, namely, with Leontovich or generalized impedance boundary conditions $${\textbf{n}}\times {\textbf{E}}= Z [({\textbf{n}}\times {\textbf{H}})\times {\textbf{n}}] $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>E</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>[</mml:mo> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>H</mml:mi> <mml:mo>)</mml:mo> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> . Under accretivity and coercivity assumptions on the impedance operator Z , we obtain a new embedding theorem for the domain of the associated Maxwell operator M , which ensures, in particular, that the nonzero spectrum of the Maxwell operator M is discrete.