MAXWELL'S EQUATIONS AS A SCATTERING PASSIVE LINEAR SYSTEM?

摘要

We consider Maxwell's equations on a bounded domain Ω ? R~3 with Lipschitz boundary Γ, with boundary control and boundary observation. Relying on an abstract framework developed by us in an earlier paper, we define a scattering passive linear system that corresponds to Maxwell's equations and investigate its properties. The state of the system is [_D~B], where B and D are the magnetic and electric flux densities, and the state space of the system is X = E?E, where E = L~2(Ω;R~3). We assume that Γ0 and Γ1 are disjoint, relatively open subsets of Γ such that Γ_0∪Γ_1 = Γ. We consider Γ_0 to be a superconductor, which means that on Γ_0 the tangential component of the electric field is forced to be zero. The input and output space U consists of tangential vector fields of class L~2 on Γ_1. The input and output at any moment are suitable linear combinations of the tangential components of the electric and magnetic fields. The semigroup generator has the structure A =[L~ 0·G-γ~(-L)?Rγ ]P, where L = rot (with a suitable domain), γ is the tangential component trace operator restricted to Γ1, R is a strictly positive pointwise multiplication operator on U (that can be chosen arbitrarily), and P~(-1) = [ ~(μ 0)_(0 ε)] is another strictly positive pointwise multiplication operator (acting on X). The operator -G is pointwise multiplication with the conductivity g ≥ 0 of the material in Ω. The system is scattering conservative iff g = 0.