  In this work, we study several properties of the normal Lebesgue trace of
vector fields introduced by the second and third author in [18] in the context
of the energy conservation for the Euler equations in Onsager-critical classes.
Among several properties, we prove that the normal Lebesgue trace satisfies the
Gauss-Green identity and, by providing explicit counterexamples, that it is a
notion sitting strictly between the distributional one for measure-divergence
vector fields and the strong one for $BV$ functions. These results are then
applied to the study of the uniqueness of weak solutions for continuity
equations on bounded domains, allowing to remove the assumption in [15] of
global $BV$ regularity up to the boundary, at least around the portion of the
boundary where the characteristics exit the domain or are tangent. The proof
relies on an explicit renormalization formula completely characterized by the
boundary datum and the positive part of the normal Lebesgue trace. In the case
when the characteristics enter the domain, a counterexample shows that
achieving the normal trace in the Lebesgue sense is not enough to prevent
non-uniqueness, and thus a $BV$ assumption seems to be necessary for the
uniqueness of weak solutions.
