M-dissipative boundary conditions and boundary tuples for Maxwell operators

For Maxwell operators (E, H) ->(i epsilon-1 backward difference x H, -i mu-1 backward difference x E) in Lipschitz domains, we describe all mdissipative boundary conditions and apply this result to generalized impedance and Leontovich boundary conditions including the cases of singular, degenerate, and randomized impedance coefficients. To this end we construct Riesz bases in the trace spaces associated with the curl-operator and introduce a modified version of boundary triple adapted to the specifics of Maxwell equations, namely, to the mixed-order duality of the related trace spaces. This provides a translation of the problem to operator-theoretic settings of abstract Maxwell operators. In particular, we show that Calkin reduction operators are naturally connected with Leontovich boundary conditions and provide an abstract version of impedance boundary condition applicable to other types of wave equations. Taking Friedrichs and Krein-von Neumann extensions of related boundary operators, it is possible to associate m-dissipative Maxwell operators to arbitrary non-negative measurable impedance coefficients. (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

This study focuses on Maxwell equations in Lipschitz domains, describing all dissipative boundary conditions and applying the results to generalized impedance and Leontovich boundary conditions, including cases of singular, degenerate, and randomized impedance coefficients. By constructing Riesz bases and modifying boundary triples, the problem is translated into the operator-theoretic settings of abstract Maxwell operators.