  Nonnegative measures that are solutions to a transport equation with
continuous coefficients have been widely studied. Because of the low regularity
of the associated vector field, there is no natural flow since nonuniqueness of
integral curves is the general rule. It has been known since the works by L.
Ambrosio [2] and L. Ambrosio and G. Crippa [3, 4] that such measures can be
described as a superposition of $\delta$-measures supported on integral curves.
In this article, motivated by some observability questions for the wave
equation, we are interested in such transport equations in the case of domains
with boundary. Associated with a wave equation with C^1-coefficients are
bicharacteristics that are integral curves of a continuous Hamiltonian vector
field. We first study in details their behaviour in the presence of a boundary
and define their natural generalisation that follows the laws of geometric
optics. Then, we introduce a natural class of transport equations with a source
term on the boundary, and we prove that any nonnegative measure satisfying such
an equation has a union of maximal generalized bicharacteristics for support.
This result is a weak form of the superposition principle in the presence of a
boundary. With its companion article [7], this study completes the proof of
wave observability generalizing the celebrated result of Bardos, Lebeau, and
Rauch [5] in a low regularity framework where coefficients of the wave equation
(and associated metric) are C^1 and the boundary and the manifold are C^2.
