A new family of first-order boundary conditions for the Maxwell system: derivation, well-posedness and long-time behavior

Abstract

This work deals with the time-dependent Maxwell system in the case of TE-polarized electromagnetic waves, when associated with a family of first-order local boundary conditions. The boundary conditions are derived by using a micro-diagonalization method, actuated by the standard one of M.E. Taylor and involving pseudodifferential technics. The conditions differ from an arbitrary function and any of them leads to a well-posed mixed problem that is described by a continuous semi-group. The arbitrary function can be seen as a parameter and an asymptotic analysis in time shows that it can be chosen so that the resulting boundary condition is absorbing: the system is related to an energy functional that converges towards zero as time tends to infinity. By involving an invariant space for the Maxwell system, the limit state can be explicitly written as a solution to a boundary-value problem depending on the initial data. The long time behavior of the solution is then completely analyzed.