  We study Maxwell's equation as a theory for smooth $k$-forms on globally
hyperbolic spacetimes with timelike boundary as defined by Ak\'e, Flores and
Sanchez. In particular we start by investigating on these backgrounds the
D'Alembert - de Rham wave operator $\Box_k$ and we highlight the boundary
conditions which yield a Green's formula for $\Box_k$. Subsequently, we
characterize the space of solutions of the associated initial and boundary
value problem under the assumption that advanced and retarded Green operators
do exist. This hypothesis is proven to be verified by a large class of boundary
conditions using the method of boundary triples and under the additional
assumption that the underlying spacetime is ultrastatic. Subsequently we focus
on the Maxwell operator. First we construct the boundary conditions which
entail a Green's formula for such operator and then we highlight two
distinguished cases, dubbed $\delta\mathrm{d}$-tangential and
$\delta\mathrm{d}$-normal boundary conditions. Associated to these we introduce
two different notions of gauge equivalence and we prove that in both cases,
every equivalence class admits a representative abiding to the Lorenz gauge. We
use this property and the analysis of the operator $\Box_k$ to construct and to
classify the space of gauge equivalence classes of solutions of the Maxwell's
equations with the prescribed boundary conditions. As a last step and in the
spirit of future applications in the framework of algebraic quantum field
theory, we construct the associated unital $*$-algebras of observables proving
in particular that, as in the case of the Maxwell operator on globally
hyperbolic spacetimes with empty boundary, they possess a non-trivial center.
