Exponential decay of the linear Maxwell system due to conductivity near the boundary
Richard Nutt, Roland Schnaubelt
TL;DR
This work proves exponential stabilization of the linear anisotropic Maxwell system in a bounded domain with boundary-localized conductivity. It develops a Helmholtz-type decomposition to separate charge-free and charge-bearing components and proves an observability-type estimate for the charge-free second-order subsystem using Morawetz multipliers. Under a non-trapping condition on $\varepsilon$ and $\mu$ and a strictly positive conductivity on a collar near the boundary, the authors establish exponential decay to zero for data with charges restricted to the damping region, and they extend observability/controllability results to the charge-free case. The results advance the theory for anisotropic, matrix-valued coefficients with localized damping and provide a rigorous framework for exponential decay in Maxwell systems with boundary damping.
Abstract
We study the anisotropic linear Maxwell system on a bounded domain $Ω$ with perfectly conducting boundary conditions. It is damped via a conductivity $σ$ which is strictly positive on a collar at the boundary. We prove that solutions decay exponentially to 0, if the fields have no magnetic charges on $Ω$ and no electric charges off the support of $σ$. Our approach relies on a splitting of the solution via a Helmholtz decomposition and an observability-type estimate for a related second-order system without charges, shown using Morawetz multipliers. Corresponding exact observability and controllability results are also established.