Appl Math Optim 55:185–201 (2007)
2007 Springer Science+Business Media, Inc.
Continuous Observability for the Anisotropic Maxwell System
Matthias M. Eller
Department of Mathematics, Georgetown University,
Washington, DC 20057, USA
Abstract. A boundary observability inequality for the homogeneous Maxwell
system with variable, anisotropic coefﬁcients is proved. The result implies unique-
ness for an ill-posed Cauchy problem for Maxwell’s system. Both results are so far
known only in the special case of isotropic coefﬁcients, i.e., when Maxwell’s system
reduces to a vector wave equation. Here the analysis has been carried out for the
ﬁrst-order system directly without references to the wave equation.
Key Words. Maxwell’s system, Anisotropic media, Boundary control.
AMS Classiﬁcation. 35, 49.
1. Introduction and Main Result
Let ⊂ R
be an open, bounded set with a C
boundary ∂. Let e = e(t, x) and h =
h(t, x) be vector-valued functions denoting the electric ﬁeld intensity and the magnetic
ﬁeld intensity, respectively. The electric permeability and the magnetic permittivity are
denoted by ε and µ and are assumed to be real, positive deﬁnite, symmetric 3 × 3
matrices. Assume that is ﬁlled with a medium of zero conductivity and assume that
there are no electrical charges in . The evolution of the electromagnetic ﬁeld over the
time interval (0, T ) in is governed by the homogeneous Maxwell’s equations
(εe) −∇×h = 0,∂
(µh) +∇×e = 0
∇·(εe) = 0, ∇·(µh) = 0
= (0, T ) × , (1.1)
This research was supported by the National Science Foundation through Grant DMS-0308731.