Exponential decay of the linear Maxwell system due to conductivity near the boundary

TL;DR

This paper proves exponential decay of solutions to the anisotropic linear Maxwell system with boundary conductivity, under specific charge conditions, using Helmholtz decomposition and Morawetz multipliers.

Contribution

It introduces a novel approach combining Helmholtz decomposition and observability estimates to analyze decay and controllability of Maxwell's equations with boundary conductivity.

Findings

Solutions decay exponentially under certain charge conditions.

Established exact observability and controllability results.

Used Morawetz multipliers for key estimates.

Abstract

We study the anisotropic linear Maxwell system on a bounded domain $\Omega$ with perfectly conducting boundary conditions. It is damped via a conductivity $\sigma$ 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 $\Omega$ and no electric charges off the support of $\sigma$. 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.