M-dissipative boundary conditions and boundary tuples for Maxwell operators

TL;DR

This paper characterizes all m-dissipative boundary conditions for Maxwell operators in Lipschitz domains, introducing new boundary triple methods and linking boundary conditions to operator extensions, including complex impedance cases.

Contribution

It develops a comprehensive framework for m-dissipative boundary conditions for Maxwell operators, including singular and randomized impedance, using novel boundary triple and Riesz basis techniques.

Findings

All m-dissipative boundary conditions for Maxwell operators are described.

Connection established between Calkin reduction operators and Leontovich boundary conditions.

Maxwell operators can be associated with arbitrary non-negative impedance coefficients.

Abstract

For Maxwell operators $(E,H) \to (i \epsilon^{-1} \nabla\times H, -i \mu^{-1} \nabla \times E)$ in Lipschitz domains, we describe all m-dissipative 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 for 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.